Fact-checked by Grok 2 weeks ago

Thermodynamic process

A thermodynamic process is the manner in which a thermodynamic system changes from an initial state to a final state due to interactions with its surroundings, involving the transfer of heat, work, or both, while connecting equilibrium states.^[1] These processes are central to the field of thermodynamics, which studies the relationships between heat, work, temperature, and energy in physical systems, governing phenomena from engine efficiency to phase changes in matter.^[2] Thermodynamic processes are classified based on constraints on system variables such as temperature, pressure, volume, or entropy, and they can be reversible (occurring infinitely slowly through equilibrium states, allowing reversal without net change) or irreversible (involving finite rates and dissipative effects like friction).^[1] A key subclass is the quasistatic process, which proceeds slowly enough for the system to remain in near-equilibrium at every stage, enabling ideal analysis with differential equations.^[2] Common types include: In practice, thermodynamic processes form the basis of cycles—closed paths returning the system to its initial state—such as the Carnot cycle for maximum efficiency or the Rankine cycle in steam power plants, enabling the conversion of thermal energy into mechanical work while adhering to the laws of thermodynamics.^[3] These concepts underpin technologies ranging from internal combustion engines to refrigeration systems, with process efficiency limited by the second law, which introduces entropy as a measure of irreversibility.^[1]

Fundamentals

Definition and Characteristics

A thermodynamic process is defined as the change in the macroscopic state of a thermodynamic system that occurs from one equilibrium state to another, typically involving transfers of energy in the form of heat or work.^[4] This evolution represents a sequence of events where the system's properties, such as pressure, volume, and temperature, vary between well-defined initial and final equilibrium conditions.^[5] Such processes are fundamental to thermodynamics, which studies energy transformations in physical systems, assuming the system can be isolated or interact with its surroundings in controlled ways.^[6] Key characteristics of thermodynamic processes include their representation as paths in the state space, where the state space is a multidimensional space defined by the system's thermodynamic variables like internal energy, entropy, and volume.^[5] These paths trace the succession of equilibrium states through which the system passes during the change, distinguishing processes from static conditions like steady states, where no net change occurs over time.^[1] Processes can be idealized, such as those occurring infinitely slowly to maintain equilibrium at every step, or real, where finite rates lead to non-equilibrium transients; however, thermodynamic analysis often focuses on the initial and final states rather than microscopic fluctuations.^[7] This path-dependent nature underscores that the total energy exchange depends on the specific route taken in state space.^[8] The concept of thermodynamic processes originated in the 19th century, primarily through the foundational work of Rudolf Clausius and William Thomson (Lord Kelvin), who developed the principles of energy conservation and transformation amid the industrial era's focus on heat engines.^[9] Clausius, in his 1850 formulation, introduced key ideas around heat and work in cyclic processes, while Kelvin's 1851 contributions emphasized the impossibility of perpetual motion, laying the groundwork for modern thermodynamics.^[10] These developments built on earlier caloric theories but shifted emphasis to mechanical equivalents of heat. Understanding thermodynamic processes presupposes familiarity with basic thermodynamic systems—such as closed systems that exchange energy but not matter, or open systems that do both—and the notion of thermodynamic equilibrium, where macroscopic properties are uniform and time-independent.^[5]

State Functions versus Path Functions

In thermodynamics, state functions are thermodynamic properties that depend solely on the current state of the system, independent of the path or history by which that state was reached. Examples include internal energy U, temperature T, pressure P, volume V, and entropy S. The change in a state function, such as \Delta U = U_\text{final} - U_\text{initial}, is thus determined only by the initial and final states, making it path-independent./Thermodynamics/Fundamentals_of_Thermodynamics/State_vs._Path_Functions)^[11]^[12] In contrast, path functions are quantities whose values depend on the specific process or path taken between states. Heat Q and work W are classic examples, as their magnitudes are given by integrals along the process path: Q = \int \delta Q and W = \int \delta W, where the inexact differentials \delta Q and \delta W reflect their path dependence. For instance, the heat transferred or work done to go from one state to another can vary significantly depending on whether the process is direct or involves intermediate steps, even if the endpoints are identical.^[13]/Thermodynamics/Fundamentals_of_Thermodynamics/State_vs._Path_Functions) The distinction between state and path functions is central to the first law of thermodynamics for closed systems, which states that the change in internal energy equals the heat added minus the work done by the system: \Delta U = Q - W. Here, \Delta U is path-independent as a state function, while Q and W adjust accordingly to satisfy the equation for any path between the same states. In infinitesimal form, this becomes dU = \delta Q - \delta W, where dU is an exact differential, highlighting how state changes are balanced by path-dependent transfers of energy. For example, compressing a gas slowly versus rapidly between the same initial and final volumes yields the same \Delta U but different values of Q and W.^[13]^[11]

Primary Classifications

Reversible and Irreversible Processes

A reversible thermodynamic process is an idealized process in which both the system and its surroundings can be restored to their initial states without any net change in the universe, occurring infinitely slowly through a series of equilibrium states with no dissipative effects such as friction or unrestrained heat transfer.^[14] In such processes, the system remains in thermodynamic equilibrium at every infinitesimal stage, allowing the direction of change to be reversed by an infinitesimal modification of the driving forces.^[15] This idealization serves as a limiting case for analyzing real processes, enabling the calculation of maximum work or heat transfer potentials.^[16] In contrast, an irreversible thermodynamic process is a real-world process where the system and surroundings cannot be returned to their initial states without external intervention or net changes, typically due to finite gradients in temperature, pressure, or other potentials, leading to dissipative phenomena like friction, viscous flow, or spontaneous mixing.^[14] These processes generate entropy in the universe, making reversal impossible without additional work input that further increases total entropy.^[17] All actual physical processes are irreversible to some degree, as perfect equilibrium maintenance is unattainable in finite time.^[18] The criterion for reversibility is that the process must maintain thermodynamic equilibrium throughout, which requires it to be quasi-static—proceeding through successive equilibrium states—but quasi-static conditions alone are insufficient without the absence of irreversibilities. Reversibility is tied to the second law of thermodynamics, where the total entropy change of the universe is zero for reversible processes and positive for irreversible ones.^[19] A classic example is the expansion of an ideal gas: in a reversible isothermal expansion, the gas expands slowly against a gradually decreasing external pressure, maintaining equilibrium and allowing the piston to be pushed back to compress the gas to its original state without net entropy change.^[14] Conversely, free expansion of the same gas into a vacuum is irreversible, as the gas rushes out spontaneously without doing work, generating entropy through unrestrained molecular motion that cannot be undone without external effort.^[20] For reversible processes, the infinitesimal heat transfer is related to entropy change by the equation

\delta Q_\text{rev} = T \, dS

where \delta Q_\text{rev} is the reversible heat transfer, T is the absolute temperature, and dS is the infinitesimal entropy change of the system.^[21] The second law quantifies irreversibility through the total entropy change of the universe:

\Delta S_\text{universe} = \Delta S_\text{system} + \Delta S_\text{surroundings} \geq 0

with equality holding only for reversible processes.^[17]

Quasi-static Processes

A quasi-static process in thermodynamics is defined as one that proceeds sufficiently slowly such that the system remains in internal thermodynamic equilibrium at every instant, passing through a continuous sequence of equilibrium states.^[22] This idealized process is approximated by an infinite number of infinitesimal steps, ensuring that deviations from equilibrium are negligible.^[1] Quasi-static processes exhibit well-defined paths in the thermodynamic state space because the system is always in a state describable by its equilibrium variables, such as pressure, volume, and temperature.^[23] While many quasi-static processes are reversible in the absence of dissipative effects like friction, others may be irreversible; for instance, a quasi-static compression involving sliding friction within the system generates entropy and cannot be perfectly reversed.^[24] In calculations, quasi-static processes are often assumed to be reversible for simplicity when dissipation is minimal.^[25] These processes are typically represented as smooth, continuous curves on thermodynamic diagrams, such as pressure-volume (PV) or temperature-entropy (TS) plots, which illustrate the trajectory through state space.^[22] In practice, quasi-static approximations are valuable for analyzing real-world slow processes, like the gradual compression of a gas in a piston-cylinder assembly, where the system's response time is much shorter than the process duration.^[26] During a quasi-static process, state variables evolve continuously, allowing the work done due to volume changes to be calculated precisely. For such volume work, the expression is:

W = \int P \, dV

where P is the system's pressure and dV is the infinitesimal volume change./University_Physics_II_-Thermodynamics_Electricity_and_Magnetism(OpenStax)/03%3A_The_First_Law_of_Thermodynamics/3.05%3A_Thermodynamic_Processes)

Common Specific Processes

Isothermal Processes

An isothermal process is a thermodynamic process in which the temperature of the system remains constant throughout the change in state.^[1] This constancy is typically maintained by placing the system in thermal contact with a large heat reservoir, or heat bath, at the same temperature, allowing heat to flow as needed to counteract any temperature variations during expansion or compression.^[27] Such processes are fundamental in understanding heat engines and refrigeration cycles, where temperature control is essential for efficiency. For an ideal gas undergoing an isothermal process, the internal energy change is zero because the internal energy depends solely on temperature: \Delta U = 0.^[28] From the first law of thermodynamics, \Delta U = Q - W, where W is the work done by the system, it follows that the heat absorbed by the system equals the work done by it: Q = W. For a reversible isothermal expansion of an ideal gas, the work done by the system is given by

W = nRT \ln\left(\frac{V_f}{V_i}\right),

where n is the number of moles, R is the gas constant, T is the constant temperature, and V_f and V_i are the final and initial volumes, respectively.^[28] This expression arises from integrating the work differential dW = P dV using the ideal gas law P = nRT / V. In the reversible case, the infinitesimal heat transfer is related to the entropy change by \delta Q = T dS.^[29] Isothermal processes find applications in phase changes, such as boiling or condensation, where the temperature remains fixed at the transition point (e.g., 100°C at standard pressure for water) while heat is added or removed as latent heat.^[30] They also occur in slow expansions or compressions of gases in contact with thermal reservoirs, as in certain stages of heat engines or chemical reactors where precise temperature control prevents unwanted side reactions.^[1] On a pressure-volume (PV) diagram, an isothermal process for an ideal gas traces a hyperbola following PV = nRT, reflecting the inverse relationship between pressure and volume at constant temperature.^[31] These processes can be quasi-static if performed slowly enough to maintain equilibrium at every stage.^[32]

Adiabatic Processes

An adiabatic process is a thermodynamic process in which no heat is transferred between the system and its surroundings, denoted as Q = 0.^[33] According to the first law of thermodynamics, the change in internal energy \Delta U equals the negative of the work done by the system, \Delta U = -W, where work arises solely from changes in volume or other forms of energy transfer.^[34] For an ideal gas undergoing such a process, compression increases the temperature as internal energy rises due to work done on the system, while expansion decreases the temperature as the gas performs work.^[34] In a reversible adiabatic process, which is typically modeled as quasi-static to maintain equilibrium, the process is isentropic, meaning the entropy change \Delta S = 0.^[35] For an ideal gas, this leads to the relations TV^{\gamma-1} = \text{constant} and PV^\gamma = \text{constant}, where \gamma = C_p / C_v is the ratio of specific heats at constant pressure and volume, respectively.^[34] These equations derive from integrating the first law under the assumption of reversibility, combining dU = C_v dT with the ideal gas law.^[35] Irreversible adiabatic processes, in contrast, involve non-equilibrium conditions and generate entropy. A classic example is the free expansion of an ideal gas into a vacuum, where no work is done (W = 0) and no heat is exchanged, resulting in \Delta U = 0 and thus a constant temperature.^[35] Despite the volume increase, the lack of work or heat transfer preserves the internal energy, which for an ideal gas depends only on temperature.^[36] Adiabatic processes find applications in rapid compressions within diesel engines, where fuel ignition occurs without significant heat loss during the compression stroke, enhancing efficiency.^[37] They also describe the propagation of sound waves in air, which occurs nearly adiabatically due to the high speed of sound relative to heat conduction.^[38] On a pressure-volume (PV) diagram, the path for a reversible adiabatic process appears as a steeper curve than that of a corresponding isothermal process, reflecting the greater pressure drop for a given volume change due to cooling during expansion.^[34] This steepness arises because the adiabatic condition prohibits heat addition to offset the temperature decrease.^[39]

Isobaric and Isochoric Processes

In thermodynamics, an isobaric process maintains constant pressure throughout, allowing volume to change as heat is added or removed, whereas an isochoric process keeps volume fixed, resulting in pressure variations with temperature changes.^[1] These processes are fundamental in analyzing heat engines and chemical reactions, distinct from others by their constraints on pressure or volume.^[40] On a pressure-volume (PV) diagram, an isobaric process appears as a horizontal line, reflecting unchanged pressure as volume expands or contracts, while an isochoric process is a vertical line, indicating constant volume with pressure shifts along the ordinate.^[41] For an isobaric process, the work done by the system is given by

W = P \Delta V,

where P is the constant pressure and \Delta V is the volume change; this equals the area under the horizontal line on the PV diagram.^[28] From the first law of thermodynamics, \Delta U = Q - W, the heat transfer Q at constant pressure equals the enthalpy change \Delta H = \Delta U + P \Delta V.^[42] For an ideal gas, this simplifies to Q = n [C_p](/page/Molar_heat_capacity) \Delta T, where n is the number of moles, C_p is the molar heat capacity at constant pressure, and \Delta T is the temperature change.^[43] Such processes commonly occur in open systems, such as heating a liquid in an uncovered container at atmospheric pressure, where expansion happens freely against constant external pressure.^[44] In an isochoric process, no work is performed since \Delta V = 0, so W = 0.^[45] Consequently, the first law yields Q = \Delta U, the change in internal energy. For an ideal gas, \Delta U = n C_v \Delta T, where C_v is the molar heat capacity at constant volume, thus Q = n C_v \Delta T.^[46] This process is typical in rigid containers, like heating a gas confined in a fixed-volume tank, where all added heat increases internal energy without expansion work.^[47] For ideal gases, the heat capacities relate via C_p = C_v + R, where R is the gas constant, arising from the difference in work contributions between constant-pressure and constant-volume conditions.^[48] Isobaric and isochoric processes often combine in thermodynamic cycles to model efficient energy conversion.^[49]

Advanced Process Types

Cyclic Processes

A cyclic process in thermodynamics consists of a sequence of thermodynamic processes that form a closed loop in the state space, returning the system to its initial thermodynamic state after completion. This closure ensures that all state functions, such as internal energy U, pressure P, volume V, temperature T, and entropy S, revert to their starting values.^[3]^[50] For a cyclic process in a closed system, the net change in internal energy is zero (\Delta U = 0), so the first law of thermodynamics implies that the net heat absorbed by the system equals the net work done by the system (Q_\text{net} = W_\text{net}). On a pressure-volume (PV) diagram, the net work output for a cycle is given by the area enclosed by the path; cycles traversed clockwise typically produce positive net work, as in heat engines. These processes are fundamental to devices like heat engines, which convert thermal energy to mechanical work, and refrigerators, which transfer heat against a temperature gradient using external work.^[51]^[52] Prominent examples include the Carnot cycle, introduced by Sadi Carnot in 1824, which comprises two reversible isothermal processes and two reversible adiabatic processes operating between a hot reservoir at temperature T_h and a cold reservoir at T_c. Another is the Otto cycle, which models spark-ignition internal combustion engines and features two isochoric processes (one for heat addition via combustion) along with adiabatic compression and expansion. For heat engines, the thermal efficiency is defined as \eta = W_\text{net} / Q_\text{in}, where Q_\text{in} is the heat input from the hot source; the Carnot cycle achieves the maximum possible efficiency of \eta = 1 - T_c / T_h, setting an upper bound for all reversible engines operating between the same temperatures.^[53]^[54]^[55] Quasi-static cyclic processes are idealized cycles where each constituent process occurs infinitely slowly, maintaining the system in thermodynamic equilibrium at every stage, which allows reversible operation and maximizes efficiency. In practice, real cycles approximate this through near-equilibrium steps, minimizing irreversibilities like friction or rapid changes. Such cycles are analyzed using thermodynamic potentials and diagrams to quantify performance metrics like work output and heat transfer.^[56]

Flow Processes

Flow processes in thermodynamics occur in open systems, where mass crosses the system boundaries, allowing for the analysis of energy transfer accompanying fluid flow. These processes are typically analyzed using a control volume, which is a fixed region in space enclosing the system of interest, enabling the application of conservation laws to account for mass inflow and outflow.^[57] Unlike closed systems, open systems in flow processes involve both energy and mass exchange, making enthalpy a key property due to its inclusion of flow work.^[58] In steady flow processes, fluid properties remain constant over time at every point within the control volume, and the mass flow rate is uniform across inlet and outlet sections. The first law of thermodynamics for such processes, applied on a per-unit-mass basis, states that the change in enthalpy plus changes in kinetic and potential energy equals the heat added minus the shaft work done: \Delta h + \Delta \left( \frac{v^2}{2} + gz \right) = q - w_s, where h is specific enthalpy, v is velocity, z is elevation, g is gravitational acceleration, q is specific heat transfer, and w_s is specific shaft work.^[58] This equation highlights how energy is conserved as fluid moves through devices, with negligible accumulation inside the control volume under steady conditions.^[59] The rate form of the steady flow energy equation, suitable for continuous processes, is given by:

\dot{m} \left( h_1 + \frac{v_1^2}{2} + g z_1 \right) + \dot{Q} = \dot{m} \left( h_2 + \frac{v_2^2}{2} + g z_2 \right) + \dot{W}

where \dot{m} is the mass flow rate, subscripts 1 and 2 denote inlet and outlet, \dot{Q} is the heat transfer rate, and \dot{W} is the work rate (typically shaft work).^[58] This formulation assumes one-dimensional flow and steady state, facilitating calculations for engineering applications.^[57] Unsteady flow processes involve time-dependent changes in properties within the control volume, such as during transient operations where mass or energy accumulates. A common example is the filling of a tank, where incoming fluid increases the control volume's mass and internal energy until equilibrium is reached, requiring integration of the general energy balance over time: \Delta E_{CV} = Q - W + \sum \dot{m}_i (h_i + \frac{v_i^2}{2} + g z_i) \Delta t - \sum \dot{m}_e (h_e + \frac{v_e^2}{2} + g z_e) \Delta t.^[59] These processes are analyzed by considering short time intervals to track variations in stored energy.^[59] Flow processes find widespread application in devices such as turbines, which extract work from expanding fluids; nozzles, which accelerate fluids by converting enthalpy to kinetic energy; and pumps, which impart energy to fluids via mechanical input.^[58] For ideal, incompressible, inviscid flows neglecting friction and heat transfer, Bernoulli's equation provides a simplified approximation: P_1 + \frac{1}{2} \rho v_1^2 + \rho g z_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g z_2, where P is pressure and \rho is density.^[60] Many practical flow analyses assume adiabatic conditions to isolate mechanical energy changes.^[58]

Polytropic Processes

A polytropic process is a thermodynamic process for an ideal gas in which the pressure P and volume V are related by the equation PV^n = \text{constant}, where n is the polytropic index, a constant that characterizes the specific path of the process.^[61]^[62] This relation generalizes several common thermodynamic processes and assumes a constant specific heat during the process, making it useful for modeling quasi-equilibrium changes in gas systems.^[61] The work done during a polytropic process, calculated as the work by the system, is given by

W = \frac{P_2 V_2 - P_1 V_1}{1 - n}

for n \neq 1, where subscripts 1 and 2 denote initial and final states, respectively.^[62]^[63] For an ideal gas, this integrates to

W = \frac{ R T_1 \left(1 - \left(\frac{V_1}{V_2}\right)^{n-1}\right)}{n - 1},

where R is the gas constant, T_1 is the initial temperature, and the formula applies per mole (or scaled by the number of moles).^[63] These expressions derive from integrating W = \int P \, dV using the polytropic relation and the ideal gas law PV = RT (per mole).^[62] Special cases of polytropic processes correspond to specific values of the index n: n = 0 for isobaric (constant pressure), n = 1 for isothermal (constant temperature), n = \gamma for adiabatic (no heat transfer, where \gamma = C_P / C_V is the heat capacity ratio), and n = \infty for isochoric (constant volume).^[61]^[64] These cases unify simpler processes under the polytropic framework, with the index n ranging typically from -\infty to \infty depending on the physical conditions.^[64] Polytropic processes find applications in engineering, particularly in modeling real gas compression and expansion in compressors and internal combustion engines, where actual paths deviate from ideal isothermal or adiabatic conditions due to heat losses or inefficiencies.^[61] They approximate non-ideal behaviors in devices like reciprocating compressors, providing a practical way to calculate performance without assuming perfect reversibility.^[61] Heat transfer in a polytropic process follows from the first law of thermodynamics, Q = \Delta U + W, where \Delta U is the change in internal energy. For an ideal gas, this yields Q = m C_n \Delta T, with the polytropic specific heat C_n = \frac{(n - \gamma) R}{(\gamma - 1)(n - 1)} (per mole, or scaled by mass), where \Delta T = T_2 - T_1.^[63] This specific heat can be negative for $1 < n < \gamma, indicating heat removal during compression, which aligns with entropy considerations in such processes.^[61]^[63]

Processes and Conjugate Variables

Pressure-Volume Relations

In thermodynamics, pressure P and volume V form a conjugate pair of variables associated with mechanical work during processes involving volume changes. For a reversible process, the infinitesimal work done by the system is \delta W = P \, dV, where the positive sign indicates work performed by the system on its surroundings as volume increases.^[65] This relation arises because pressure exerts a force over the changing area of the system's boundary, such as in a piston-cylinder assembly, directly coupling the intensive variable P to the extensive variable V.^[66] Pressure-volume (PV) diagrams provide a graphical representation of thermodynamic processes by plotting P against V, allowing visualization of state changes and cycles. The area under the curve on a PV diagram quantifies the net work done by the system for a given path from initial volume V_i to final volume V_f, calculated as W = \int_{V_i}^{V_f} P \, dV.^[67] For expansion processes, where dV > 0, the system performs positive work, as seen in the outward path of a cycle; conversely, compression (dV < 0) requires work input to the system. In reversible cases, the path follows equilibrium states, enabling precise computation via the integral; for example, in a linear pressure-volume relation like P = a + bV, the work simplifies to an algebraic expression derived from the trapezoidal area.^[68] However, irreversible processes, such as rapid expansions against a constant external pressure, do not trace a unique equilibrium curve on the PV diagram—instead, the effective work is often the rectangular area P_{\text{ext}} \Delta V, which is less than the reversible value due to dissipative losses. These relations find practical application in devices like reciprocating pistons and internal combustion engines, where PV diagrams map the work output during strokes of expansion and compression. In a piston, the work during reversible isothermal expansion of an ideal gas follows W = nRT \ln(V_f / V_i), but real engines exhibit path deviations from ideality, reducing efficiency.^[41] For cyclic processes in engines, the enclosed area of the PV loop represents net work per cycle, guiding design optimizations for power output. The PV work term also connects to thermodynamic potentials, such as appearing in the differential form dU = T \, dS - P \, dV, where the negative sign accounts for work done by the system during expansion.^[69]

Temperature-Entropy Relations

In thermodynamics, temperature T and entropy S form a conjugate pair of variables, where temperature acts as the intensive "force" driving heat transfer and entropy as the extensive "displacement."^[70] For reversible processes, the infinitesimal heat transfer \delta Q_{\text{rev}} is related to the change in entropy by the equation

\delta Q_{\text{rev}} = T \, dS,

which originates from the second law and defines entropy changes in equilibrium thermodynamics.^[71] This relation highlights how heat exchange at constant temperature contributes directly to entropy increase, distinguishing thermal processes from mechanical ones. Temperature-entropy (TS) diagrams provide a graphical representation of thermodynamic processes, plotting temperature on the vertical axis and entropy on the horizontal axis. Isentropic processes, such as reversible adiabatic expansions or compressions, appear as vertical lines because entropy remains constant (dS = 0). Isothermal processes, involving heat transfer at fixed temperature, are depicted as horizontal lines, with the length corresponding to the entropy change. The area under a curve on a TS diagram represents the heat transferred during the process, Q = \int T \, dS, offering a visual measure of thermal energy flow.^[72] A prominent example is the Carnot cycle, idealized for heat engines and refrigerators, which forms a rectangle on the TS diagram: two vertical isentropic legs connect two horizontal isothermal legs at high temperature T_h and low temperature T_c.^[73] During the isothermal expansion at T_h, heat Q_h is absorbed, increasing entropy by Q_h / T_h; the isentropic expansion lowers temperature without entropy change; isothermal compression at T_c rejects heat Q_c, decreasing entropy by Q_c / T_c; and isentropic compression returns to the initial state. This rectangular shape underscores the cycle's reversibility and maximum efficiency, given by \eta = 1 - T_c / T_h. In irreversible processes, such as those with finite temperature gradients, entropy generation occurs, causing the total entropy change \Delta S > \int \delta Q / T, leading to paths that deviate upward from reversible ones on the TS diagram and reduced efficiency.^[74] These relations are applied in analyzing heat engines and refrigeration cycles, where TS diagrams quantify efficiency limits and heat transfer requirements; for instance, in vapor-compression refrigeration, the evaporator and condenser processes align with isothermal segments, optimizing coefficient of performance.^[75] The total entropy change for any reversible process is calculated as

\Delta S = \int \frac{\delta Q_{\text{rev}}}{T},

with integration along the process path. For an ideal gas, entropy depends on both temperature and volume, yielding the differential form

dS = \frac{C_V}{T} \, dT + \frac{R}{V} \, dV,

where C_V is the heat capacity at constant volume and R is the gas constant; integration provides explicit changes, such as \Delta S = C_V \ln(T_2 / T_1) + R \ln(V_2 / V_1) for processes between states 1 and 2.^[76] TS diagrams complement pressure-volume representations by emphasizing thermal aspects over mechanical work.

Chemical Potential and Particle Number

In thermodynamic processes involving open systems, the chemical potential \mu serves as the conjugate variable to the particle number N, quantifying the change in the system's internal energy associated with adding or removing particles at constant entropy and volume. This relationship manifests in the chemical work term \delta W_{\text{chem}} = \mu \, dN, which accounts for the energy exchange due to particle transfer. The full differential form of the internal energy for such systems incorporates this term alongside thermal and mechanical contributions, expressed as

dU = T \, dS - P \, dV + \mu \, dN,

where T is temperature, S is entropy, P is pressure, and V is volume. This equation extends the first law of thermodynamics to scenarios where particle number is not conserved, enabling analysis of systems interacting with their surroundings through matter exchange.^[77]^[78]^[79] Processes driven by chemical potential gradients include diffusion, where particles move from regions of higher \mu to lower \mu to equalize potentials and maximize entropy, as seen in the spontaneous mixing of gases or solutes. Chemical reactions also involve changes in N for reacting species, with the reaction proceeding in the direction that reduces the total free energy, guided by differences in \mu for reactants and products. Phase changes accompanied by mass transfer, such as evaporation or dissolution, similarly rely on \mu equalization across phases, where the transferring particles adjust until their chemical potentials match in equilibrium.^[80]^[81]^[82] In the grand canonical ensemble, which models systems in contact with a particle reservoir, the particle number N fluctuates around an average value determined by the fixed chemical potential \mu, temperature, and volume. Equilibrium is achieved when \mu is uniform across connected systems, minimizing the grand potential and stabilizing particle exchange; fluctuations in N arise from probabilistic variations but average to the value set by \mu. This framework is essential for understanding irreversible processes like those in non-equilibrium statistical mechanics.^[83]^[84] Applications of these concepts appear in electrochemistry, where \mu differences drive ion transport in electrochemical cells, contributing to the cell potential via the Nernst equation. In osmosis, water flows across a semipermeable membrane from low to high solute concentration regions until the chemical potentials of the solvent equalize on both sides, generating osmotic pressure. Battery reactions exemplify chemical work, as \mu gradients between electrodes, such as in lithium-ion systems, enable the conversion of chemical energy to electrical work during discharge, with the voltage reflecting the \mu difference per transferred charge.^[85]^[86]^[87]

Thermodynamic Potentials in Processes

Internal Energy and Enthalpy

In thermodynamics, the internal energy U represents the total microscopic energy of a system, encompassing kinetic and potential energies of its constituent particles, excluding macroscopic contributions like bulk kinetic or potential energy. For a closed system with fixed particle number, the differential form of the internal energy is given by the fundamental thermodynamic relation dU = T \, dS - P \, dV, where T is temperature, S is entropy, P is pressure, and V is volume; this expression arises from combining the first and second laws of thermodynamics for reversible processes.^[88]^[89] The enthalpy H, defined as H = U + P V, serves as a thermodynamic potential particularly suited for processes at constant pressure. Its differential form is dH = T \, dS + V \, dP, which follows directly from the definition of H and the fundamental relation for U.^[90]^[91] This form highlights enthalpy's utility, as at constant pressure (dP = 0), dH = T \, dS = \delta q, equating the change in enthalpy to the heat transferred in reversible processes. Enthalpy is obtained via a Legendre transform of the internal energy with respect to volume, replacing V as the independent variable with its conjugate P, which facilitates analysis when pressure is controlled.^[92]^[93] In specific thermodynamic processes, these potentials simplify energy accounting. For an isochoric process (dV = 0), the first law dU = \delta q + \delta w reduces to \Delta U = q_V since no work is done (\delta w = -P \, dV = 0), making internal energy the direct measure of heat capacity at constant volume.^[94] Conversely, in an isobaric process (dP = 0), \Delta H = q_P, so enthalpy captures the heat transfer, accounting for both internal energy change and pressure-volume work. In adiabatic processes (\delta q = 0), conservation principles from the first law imply \Delta U = \delta w (with sign convention for work done by the system), preserving total energy through work alone.^[95] Applications of these potentials extend to practical engineering contexts. In steady-flow processes, such as those in turbines or nozzles, the specific enthalpy h = u + P v (per unit mass) appears in the energy balance equation h_1 + \frac{v_1^2}{2} + g z_1 = h_2 + \frac{v_2^2}{2} + g z_2 + w_s, where flow work P v is incorporated naturally.^[96] For combustion reactions at constant pressure, the standard enthalpy change \Delta H quantifies the heat released or absorbed, as q_P = \Delta H, which is crucial for designing engines and reactors. These primary potentials, internal energy and enthalpy, provide the foundational Legendre transforms for deriving free energies in systems involving temperature or other constraints.^[97]

Free Energies

The Helmholtz free energy, denoted as F, is defined as F = U - TS, where U is the internal energy, T is the temperature, and S is the entropy.^[98] Its differential form is dF = -S \, dT - P \, dV + \mu \, dN, where P is pressure, V is volume, \mu is the chemical potential, and N is the number of particles.^[99] At constant temperature and volume, the Helmholtz free energy reaches a minimum at thermodynamic equilibrium, providing a criterion for stability in such conditions.^[100] The Gibbs free energy, denoted as G, is defined as G = H - TS, where H = U + PV is the enthalpy, equivalently expressed as G = U + PV - TS.^[98] Its differential form is dG = -S \, dT + V \, dP + \mu \, dN.^[100] At constant temperature and pressure, the Gibbs free energy minimizes at equilibrium, making it particularly useful for processes under these constraints.^[100] This form incorporates the chemical potential term \mu \, dN to account for changes in particle number during chemical processes.^[100] In thermodynamic processes at constant temperature, the change in Helmholtz free energy \Delta F equals the maximum non-pressure-volume work that can be extracted from the system, \Delta F = w_{\max, \text{non-PV}}.^[101] For processes at constant temperature and pressure, the change in Gibbs free energy \Delta G determines spontaneity: if \Delta G < 0, the process is spontaneous.^[102] These relations stem from the second law, where free energies decrease during spontaneous processes at the respective constant conditions, reflecting the available work and directionality. Applications of Gibbs free energy include phase transitions, where equilibrium occurs when \Delta G = 0 between phases, such as at the melting or boiling point.^[103] In electrochemistry, the relation \Delta G = -nFE links the free energy change to the cell potential E, with n as the number of electrons transferred and F as Faraday's constant, quantifying the maximum electrical work in electrochemical cells.

Second Law Classifications

Spontaneous Processes

A spontaneous thermodynamic process is one that occurs naturally without external intervention, characterized by an increase in the entropy of the universe, where \Delta S_{\text{universe}} > 0.^[104] This criterion stems from the second law of thermodynamics, which dictates that all spontaneous changes cause such an entropy increase, driving systems toward greater disorder or equilibrium.^[105] Unlike idealized reversible processes, spontaneous processes are inherently irreversible, as reversing them would require work input to decrease universal entropy, which violates the second law.^[106] In isolated systems, spontaneous processes proceed to maximize the total entropy, as no energy or matter exchange occurs with the surroundings, leading to the highest probable state.^[104] For non-isolated systems, such as those in contact with a thermal reservoir, spontaneity is instead governed by minimization of appropriate thermodynamic potentials like free energy, reflecting the entropy-driven tendency under constraints.^[105] Key examples include heat flowing spontaneously from a hotter object to a colder one, as articulated in the Clausius statement of the second law: heat cannot pass from a colder to a hotter body without external work.^[107] Similarly, the mixing of two gases in a container occurs spontaneously due to increased entropy from molecular dispersion, without any separating force needed.^[104] These processes find applications in phenomena like diffusion, where particles spread from high to low concentration to increase entropy, and in chemical reactions that proceed in the forward direction when the overall entropy change is positive, such as the combustion of fuel releasing heat and gases.^[106] In both cases, the directionality aligns with the second law's prohibition on entropy decrease, ensuring that spontaneous events contribute to the universe's thermodynamic arrow. At the boundary, reversible processes occur with \Delta S_{\text{universe}} = 0, representing equilibrium where no net spontaneous change happens without infinitesimal driving forces, allowing maximum efficiency in idealized cycles.^[105]

Non-spontaneous Processes

Non-spontaneous thermodynamic processes are those that lack a natural tendency to occur and would lead to a decrease in the entropy of the universe (\Delta S_{\text{universe}} < 0) if attempted without external aid, effectively representing the reverse of spontaneous processes. The second law of thermodynamics prohibits such isolated occurrences, as it mandates that the entropy of an isolated system cannot decrease, thereby requiring these processes to be driven by external work or coupling to compensating spontaneous reactions elsewhere.^[108] These processes are distinguished by their operation against natural gradients, such as temperature or chemical potential differences, and necessitate continuous energy input to proceed. A classic example is refrigeration, where heat is extracted from a colder region and transferred to a hotter one, defying the natural flow of heat from hot to cold; this requires mechanical work, as stated in the Clausius formulation of the second law.^[109] Another key example is electrolysis, which uses electrical energy to drive non-spontaneous reactions, such as the decomposition of water into hydrogen and oxygen gases, overcoming the positive Gibbs free energy change of the reaction.^[110] Under the second law, non-spontaneous processes cannot occur in isolated systems due to the entropy decrease they imply, but in open systems, they become viable through work input from external sources, ensuring the net \Delta S_{\text{universe}} > 0 when accounting for the broader context. This work often originates from spontaneous processes in other parts of the system or environment, allowing local order to form at the expense of greater disorder elsewhere.^[108] Practical applications encompass uphill chemical reactions, such as electrolytic metal refining or hydrogen production, and devices like heat pumps that move heat against thermal gradients for space heating. The performance of these systems is constrained by the Carnot limit, which defines the theoretical maximum efficiency based on the temperature reservoirs involved, underscoring the second law's role in bounding real-world irreversibilities.^[110]^[111]^[112] Unlike spontaneous processes that align with equilibrium tendencies, non-spontaneous ones are inherently unnatural, sustaining non-equilibrium states by actively countering entropy's drive toward disorder. They often involve engineered cycles to integrate the required work efficiently within larger systems.^[108]

Effectively Reversible Processes

Real thermodynamic processes can approximate the reversible limit through quasistatic changes, where the system passes through near-equilibrium states with infinitesimal driving forces, resulting in negligible entropy production (\Delta S_{\text{universe}} \approx 0).^[113] These are analyzed as reversible for efficiency calculations, such as in heat engines, despite minor irreversibilities, bridging idealized reversibility (\Delta S_{\text{universe}} = 0) and spontaneous irreversibility (\Delta S_{\text{universe}} \gg 0). Examples include slow, frictionless piston expansions or phase changes at exact transition temperatures. Entropy changes are computed along hypothetical reversible paths: \Delta S \approx \int \frac{\delta Q_{\text{rev}}}{T}.^[114]

References

[1]
3.4 Thermodynamic Processes – University Physics Volume 2
The manner in which a state of a system can change from an initial state to a final state is called a thermodynamic process.
[2]
4.5: Thermodynamics processes - Physics LibreTexts
Nov 8, 2022 · Isochoric. When the volume of a system remains constant during a thermodynamic process, the process is called isochoric.
[3]
Thermodynamic Process - an overview | ScienceDirect Topics
A thermodynamic process is the succession of states a system passes through from an initial to a final state, without requiring equilibrium.
[4]
[PDF] Review of thermodynamic processes
Let's define a thermodynamic process to be a sequence of events in which a system of interest evolves from one equilibrium state to another.
[5]
1.2 Definitions and Fundamental Ideas of Thermodynamics - MIT
A thermodynamic system is a quantity of matter of fixed identity, around which we can draw a boundary (see Figure 1.3 for an example).
[6]
Thermodynamic Foundations – Introduction to Aerospace Flight ...
Thermodynamics is the science of energy and the principles governing its transformation from one form to another, such as work and heat, and the production of ...
[7]
[PDF] Thermodynamic processes [tln79]
Quasi-static process: • A quasi-static process involves infinitesimal steps between equilibrium states along a definite path in the space of state variables.
[8]
A SUMMARY OF THERMODYNAMIC FUNDAMENTALS
thermodynamic process - the path of succession of states through which the system passes in moving from an initial state to a final state. polytropic ...
[9]
Thermodynamic Asymmetry in Time
Dec 7, 2016 · By the mid-nineteenth century, Rudolf Clausius in Germany and William Thomson (later Lord Kelvin) in England had developed the theory in great ...
[10]
A History of Thermodynamics: The Missing Manual - PMC
As noted by Gibbs, in 1850, Clausius established the first modern form of thermodynamics, followed by Thomson's 1851 rephrasing of what he called the Second Law ...
[11]
[PDF] Basics of Thermodynamics
Thermodynamics is the study of how heat moves around in 'macroscopic' objects, involving laws and models.
[12]
Entropy – a masterclass | Feature - RSC Education
Entropy is a state function; that is, it has a value that depends only upon the current state of the system and is independent of how that state was prepared.
[13]
None
### Summary of Key Concepts from the Document
[14]
4.1 Reversible and Irreversible Processes - UCF Pressbooks
A reversible process allows both system and environment to return to original states, while an irreversible process does not allow this.
[15]
4.3 Features of reversible processes - MIT
Reversible processes are idealizations or models of real processes. One familiar and widely used example is Bernoulli's equation, which you saw in Unified. They ...
[16]
[DOC] Why a reversible process does the maximum amount of work
In thermodynamics, “reversible” has a technical meaning. A reversible process is one that is infinitely slow! Obviously, no real process is reversible but you ...
[17]
[PDF] T dq ds ≡ T dq ds ≥
a) For a reversible process the entropy of the universe remains constant. b) For an irreversible process the entropy of the universe will increase. Thus, a ...
[18]
[PDF] Chapter 5 - WPI
All actual processes are irreversible. ▻ A process is reversible when no irreversibilities are present within the system and its surroundings. This type of ...
[19]
13.3 The Second and Third Laws of Thermodynamics
The second law of thermodynamics states that a spontaneous process increases the entropy of the universe, Suniv > 0. If Δ Suniv < 0, the process is ...<|control11|><|separator|>
[20]
reversible and irreversible processes, entropy and introduction ... - MIT
Maximum work is achieved during a reversible expansion (or compression). For example, suppose we have a thermally insulated cylinder that holds an ideal gas.
[21]
[PDF] The Thermodynamic Definition of Entropy Can all of the energy in ...
Feb 11, 2021 · The Clausius Inequality Holds for Irreversible Cycles: Entropy is defined using only reversible heat transfers, dS = đqrev/T. But, what ...
[22]
[PDF] 6 Quasistatic thermodynamic processes
6.1 Definition of quasistatic processes. Quasistatic processes are processes that proceed slowly enough that the system is in internal equilibrium.
[23]
UNIFIED ENGINEERING Thermodynamics Chapter 2
Often we will consider processes that change "slowly" -- termed quasi-equilibrium or quasi-static processes. A process is quasi-equilibrium if the time rate of ...
[24]
Thermodynamics of an Irreversible Quasi-Static Process - NASA ADS
Quasi-static processes are not reversible when sliding friction forces are present. An example is considered consisting of a cylinder containing a gas and ...
[25]
[PDF] Thermodynamics
The beginning of the Kelvin temperature scale has a deep physical meaning: At the absolute zero T = 0 the molecules of the ideal gas freeze and stop to fly ...<|control11|><|separator|>
[26]
Quasi-Static Processes - Richard Fitzpatrick
A quasi-static process must be carried out on a timescale that is much longer than the relaxation time of the system.
[27]
Work Done in Basic Thermodynamic Processes - Physics
In an isothermal process, there is no change in the internal energy of an ideal gas. Important: The work done during any thermodynamic process is path ...
[28]
Thermodynamic processes
An adiabatic process is a process during which no heat enters or leaves the system. We then have ΔU = -ΔW, i.e. ΔW equals the change in a physical property of ...
[29]
[PDF] 485 Chapter 13: Entropy and Applications
Notice that the TdS term gives the heat transfer and the –PdV term gives the work available from a reversible process. Both terms have the units of energy, ...
[30]
[PDF] Lecture 9: Phase Transitions
At the point of the phase transition, the heat will go into latent heat of vaporization and the temperature will not change. Eventually, all the liquid is ...Missing: process | Show results with:process
[31]
Isothermal Processes - HyperPhysics
For a constant temperature process involving an ideal gas, pressure can be expressed in terms of the volume: The result of an isothermal heat engine process ...
[32]
4.2 Difference between Free and Isothermal Expansions - MIT
The difference between reversible and irreversible processes is brought out through examination of the isothermal expansion of an ideal gas.
[33]
[PDF] Adiabatic Processes and Heat Engines
An adiabatic process occurs when the system is isolated thermally. No heat can transfer in or out of the system (q = 0). Work is still defined as: dw = ...
[34]
3.6 Adiabatic Processes for an Ideal Gas - UCF Pressbooks
In an adiabatic process, an ideal gas's temperature increases during compression and decreases during expansion, with no heat flow.
[35]
reversible and irreversible processes, entropy and introduction ... - MIT
For a reversible (quasi-static), adiabatic process we can write the First Law as: du = dq - pdv. or. cvdT = -pdv = thus. We can think of the above equation as ...
[36]
[PDF] Irreversible Processes
Examples: • Block sliding on table comes to rest due to friction: KE converted to heat. • Heat flows from hot object to cold object.
[37]
[PDF] Physics Notes Class 11 Chapter 12 Thermodynamics Physics Notes ...
compression of a gas in a diesel engine's cylinder, and even the very fast propagation of sound waves in the air. ... Q: What are some real-world applications of ...
[38]
[PDF] Speed of Sound in Gases: N2O2 <--> 2NO and N2O4 <--> 2NO2
When the medium is disturbed, its pressure, density, particle velocity, and temperature are affected. However, sound propagation is very nearly adiabatic, even ...
[39]
[PDF] Lecture 15 - PHY 171
Feb 20, 2012 · Therefore, an adiabatic P − V curve is steeper than an isothermal P − V curve, as seen in the figure. In contrast, during an adiabatic ...
[40]
15.2 The First Law of Thermodynamics and Some Simple Processes
Describe the processes of a simple heat engine. Explain the differences among the simple thermodynamic processes—isobaric, isochoric, isothermal, and adiabatic.
[41]
[PDF] Chapter 12 - NJIT
▫ Isobaric. ▫ Pressure stays constant. ▫ Horizontal line on the PV diagram. ▫ Isovolumetric. ▫ Volume stays constant. ▫ Vertical line on the PV diagram. ▫ ...
[42]
[PDF] Chapter Two Homework: 2.1,2.3,2.5,2.7 First law of thermodynamics
Aug 27, 2001 · . Thus the enthalpy change during an isobaric process equals to the heat adsorbed or released from the system.
[43]
[PDF] Thermodynamics Molecular Model of a Gas Molar Heat Capacities
Along an isobaric process (constant pressure):. Q = nCP ∆T. (These are both intensive quantities, like specific heat.) Page 13. Molar Specific Heat of an Ideal ...
[44]
15.2 The First Law of Thermodynamics and Some Simple Processes
Among them are the isobaric, isochoric, isothermal and adiabatic processes. These processes differ from one another based on how they affect pressure, volume, ...
[45]
Constant Volume - Physics
Constant Volume. A constant volume process is also known as an isochoric process. An example is when heat is added to a gas in a container with fixed walls.Missing: calculation | Show results with:calculation
[46]
[PDF] PHYS 1220, Engineering Physics, Chapter 19 – The First Law of
For an ideal gas undergoes a isobaric process: dW= p dV=nRdT . dW=nRdT ... - Equation of state for Ideal gas undergoes an Adiabatic process. • dQ=0.
[47]
[PDF] Chapter 16. A Macroscopic Description of Matter
A constant-volume process is called an isochoric process. Consider the gas in a closed, rigid container. Warming the gas with a flame will raise its pressure ...
[48]
[PDF] Chapter 19 The First Law of Thermodynamics
The first law of thermodynamics is an extension of the principle of conservation of energy. It includes the transfer of both mechanical and thermal energy.
[49]
[PDF] 1 Entropy change in the isobaric-isochoric cycle of an ideal gas
... process of an ideal gas, the infinitesimal amount of heat is given by. δQ = dU + P dV = CV dT + PdV. From the equation of state of the ideal gas. PV = νRT.
[50]
7.9: Thermodynamic Cycles - Engineering LibreTexts
Aug 5, 2022 · The closed-loop, steady-state cycle is modeled as a collection of steady-state devices that are connected together to form a closed loop of ...
[51]
Example of a Cyclic Process - Physics
The net work involved is the enclosed area on the P-V diagram. If the cycle goes clockwise, the system does work. A cyclic process is the underlying principle ...
[52]
Energy Balance for Cycles – Thermodynamics - UCF Pressbooks
A thermodynamic cycle is a series of processes that begin and end at the same thermodynamic state. The figure below demonstrates what a cycle may look like on ...Missing: loop | Show results with:loop<|separator|>
[53]
[PDF] Reflections on the motive power of heat and on machines fitted to ...
I. .",0&. THE WORK OF N.-L.-SADI CARNOT. Bg tM 1JJditur, 1. II. THE LIFE OF N.-L.-SADI CARNOT. Bg Mons. H. Carnoe, •. III. REFJ,ECTIONS ON THE MOTIVE POWER OF ...
[54]
3.5 The Internal combustion engine (Otto Cycle) - MIT
The Otto cycle is a set of processes used by spark ignition internal combustion engines (2-stroke or 4-stroke cycles).
[55]
5.2: Heat Engines and the Carnot Cycle - Chemistry LibreTexts
Jan 15, 2023 · The Carnot cycle is a theoretical cyclic heat engine that can used to examine what is possible for an engine for which the job is convert heat into work.
[56]
3.5: Thermodynamic Processes - Physics LibreTexts
Mar 2, 2025 · A quasi-static process refers to an idealized or imagined process where the change in state is made infinitesimally slowly so that at each ...
[57]
[PDF] IV. First Law of Thermodynamics
Analysis of flow processes begins with the selection of an open system. 2. An open system is a region of space called a control volume. (CV). Control volume.
[58]
Steady Flow Energy Equation
This energy is composed of two parts: the internal energy of the fluid (u) and the flow work (pv) associated with pushing the mass of fluid across the system ...
[59]
[PDF] Energy Analysis for Open Systems • Open System Mass Balances
... 8. Energy Conservation for Open Systems me mi. CV. Q. W. Consider mass, work, and heat entering and/or leaving a control volume during a short time, ∆t.
[60]
Energy Equation & Bernoulli's Equation – Introduction to Aerospace ...
In conjunction with the conservation of mass and momentum, the Bernoulli equation can be used to analyze the flow through a propeller operating at low flow ...
[61]
[PDF] All About Polytropic Processes - SMU Physics
Aug 26, 2022 · Because dS = Q/T, a decreasing entropy means that heat is removed. Thus a polytropic compression that falls between an isotherm and an adiabat ...
[62]
[PDF] LECTURE NOTES ON THERMODYNAMICS
May 17, 2025 · These are lecture notes for AME 20231, Thermodynamics, a sophomore-level undergraduate course taught in the Department of Aerospace and ...
[63]
[PDF] Polytropic Processes
Cn = R(γ - n). (n - 1)(γ - 1). = CV. (n - γ). (n - 1) and thus we can write the formula for the heat of a polytropic process as q = CV. (n - γ). (n - 1). (T2 - ...
[64]
[PDF] Polytropic Process - WPI
a) Determine the work for the process, in kJ. b) Determine the heat transfer for the process, in kJ, assuming the specific heat (0.742. kJ/(kgK) is constant ...Missing: definition | Show results with:definition
[65]
lecture2
... conjugate to them. For example pressure $p$ is conjugate to volume $V$ because $p\;dV$ is the work done by pressure $p$ in changing the volume by $dV$ .) We ...
[66]
[PDF] 2 Thermodynamics : Summary - Physics Courses
state variables, which can be taken to be E, V , and N, and writing ¯dW = p dV − µ dN, we have. ¯dQ = dE + p dV − µ dN . If, for example, we want to use ...
[67]
Work, Heat, and the First Law
In an adiabatic process the heat flow is 0 (q=0). In this case the internal energy is equal to work done on or by the system.
[68]
https://physics.bu.edu/~duffy/EssentialPhysics/chapter15/section15dash2.pdf
[69]
[PDF] LECTURE NOTES ON INTERMEDIATE THERMODYNAMICS
Mar 28, 2025 · to 400 kPa, some steam is extracted for heating feedwater in an open feedwater heater. ... In this isobaric process, one always has P = 100 kPa.
[70]
[PDF] Lecture 4: 09.16.05 Temperature, heat, and entropy
o Looking again at the reversible process definition of entropy: ▫ Temperature may be thought of as a thermodynamic force, and entropy its conjugate.
[71]
5.5 Calculation of Entropy Change in Some Basic Processes - MIT
... heat transfer across a finite temperature difference and the heat exchange is reversible. From the definition of entropy ( $ dS = dQ_\textrm{rev}/T$ ),.
[72]
[PDF] Thermodynamics Second Law The Carnot Engine - De Anza College
Thermodynamics. Second Law. The Carnot Engine. Lana ... Carnot's Theorem. Carnot's Theorem. No real ... We can represent the Carnot Cycle on a TS diagram:.
[73]
[PDF] The Second Law of Thermodynamics
T-s Diagram for the Carnot Cycle. Temperature. Entropy. 1. Isentropic compression. Q in. W=Q in. -Q out. Q out. 2. Isothermal expansion. 3. Isentropic expansion.
[74]
https://ronney.usc.edu/AME436/Lecture7/AME436-S19-lecture7.pdf
[75]
https://purdue.edu/freeform/me200/wp-content/uploads/sites/29/2023/03/ME-200-Chapter-6.pdf
[76]
5.4 Entropy Changes in an Ideal Gas - MIT
We thus examine the entropy relations for ideal gas behavior. The starting point is form (a) of the combined first and second law.
[77]
[PDF] dU = TdS – pdV + μdN (1.68) This equation is consistent with: U = TS
Eqs. 1.57-1.60 show that when two solutions are put in thermal contact, energy (dU) will be transferred from the solution with smaller ∂S/∂U (larger T) to ...Missing: source | Show results with:source
[78]
[PDF] 1(a) Basic Ideas of Thermodynamics - UBC Physics
dU = TdS + µ dN. (40). Thermodynamic Potentials: Let us for the time being ... We can immediately solve for dU here, to get. dU = TdS - pdV + µdN. (49).
[79]
Lecture 14: Chemical Potential — Thermodynamic and Statistical ...
The chemical potential quantifies the change in internal energy when one particle is added to a system. The chemical potential is the heat transferred from (or ...
[80]
[PDF] Lecture 11: Entropy and Chemical Potential - UNLV Physics
Such a process due to an exchange of particles is called diffusion. Since diffusion is also a spontaneous process, it must lead to the increase of entropy ...
[81]
[PDF] CHEMICAL REACTIONS AND DIFFUSION “Classical ...
“Network thermodynamics” describes dynamics. ... MULTI-COMPONENT SYSTEM: Applies to reactions, diffusion, phase changes, etc. ... equilibrium chemical potentials ...
[82]
315: Phase Equilibria and Diffusion in Materials
The second law of thermodynamics (equation 5.4) tells us that the change ... chemical potentials and thermodynamic activities are the same in all three phases.
[83]
[PDF] Lecture 7: Ensembles
Chemical potential is higher in more dense regions. It is like potential energy for particle number. Particles move from high to low , until the 's are all ...
[84]
[PDF] Chapter 10 Grand canonical ensemble
– The equilibrium with respect to particle exchange leads to identical chemical poten- tials: µ = µ1 = µ2. Energy and particle conservation. We assume that the ...
[85]
[PDF] Lecture 13: 10.21.05 Electrochemical Equilibria
Nov 2, 2005 · Analysis of a battery provides an example of utilizing the general solution model for the chemical potential, and also introduces a (yet another) ...
[86]
[PDF] Osmotic pressure and mechanics of cell membranes
In thermodynamic equilibrium the Gibbs free energy G is minimized, which means that chemical potentials of water are the same on both sides of the semipermeable ...
[87]
Computational understanding of Li-ion batteries - Nature
Mar 18, 2016 · The driving force for this process is the difference of the lithium chemical potential in the two electrodes. During discharge, the cathode ...
[88]
[PDF] SIO 224 Basic thermodynamics These notes are an abbreviated ...
Internal energy is an extensive quantity and is given the symbol U in Callen. 2. Thermodynamic Equilibrium. It is generally observed that isolated systems ...
[89]
[PDF] Thermodynamics and Statistical Mechanics - Rutgers Physics
dU = T dS − P dV ⇒ dS = 1. T. dU +. P. T. dV. Hence. ( ∂S. ∂U. ) V. = 1. T . This relation would be essential for calculating the temperature from some-.Missing: fundamental | Show results with:fundamental
[90]
lecture 7
Notice that the conjugate variables are always paired up. The sign changes when one changes the independent variable compared to the fundamental relation (42). ...<|control11|><|separator|>
[91]
[PDF] 8.044 Lecture Notes Chapter 5: Thermodynamics, Part 2
enthalpy. dH = dE + P dV + V dP =⇒ dH = T dS + V dP. (3). Note then that the enthalpy H is most naturally H(S, P). It is most useful in analyzing processes at ...
[92]
[PDF] Legendre transforms
Feb 15, 2019 · In thermodynamics, the internal energy U can be Legendre transformed into various thermodynamic potentials, with associated conjugate pairs of ...
[93]
[PDF] the legendre transform and two-dimensional thermodynamics
... internal energy.” The internal energy is the Legendre conjugate of Helmholtz free energy obtained by replacing temperature with entropy as an in- dependent ...
[94]
[PDF] First Law of Thermodynamics - atmo.arizona.edu
Sep 11, 2013 · The change of internal energy during an isochoric process is proportional to the change of temperature alone, and similarly for the change of ...<|control11|><|separator|>
[95]
2.1 First Law of Thermodynamics - MIT
Since this molecular motion is primarily a function of temperature, the internal energy is sometimes called ``thermal energy.'' Figure 2.1: Random motion is the ...
[96]
Steady Flow Energy Equation - MIT
In an open flow system, enthalpy is the amount of energy that is transferred across a system boundary by a moving flow. This energy is composed of two parts: ...
[97]
Heat and Enthalpy - Chemistry 301
The change in enthalpy for a process at constant pressure is exactly equal to the heat that flows between the system and the surroundings for that process.
[98]
4. Free Energy and Equilibrium
Answer: We can combine the mathematical definition of the Helmholtz free energy, A=U−TS with the first law to get A=q+w−TS. The differential of A is then. dA= ...
[99]
Uncover the Power of Helmholtz Free Energy in Thermodynamics ...
Its differential form,: dF = -S dT - P dV + μ dN. where P is pressure, V volume, μ chemical potential, and N the number of particles, articulates how changes ...
[100]
[PDF] Lecture 8: Free energy - Matthew D. Schwartz
The solution to this differential equation is then f(x)=ax for some a. That ... We define the Helmholtz free energy as. F E −TS. (21). Free energy is a ...
[101]
[PDF] 553 Chapter 16. Foundations of Thermodynamics
changes in V. The independent variable and the force are conjugate variables. the change in A when V changes is the thermodynamic force, –P. dA = – S dT – P dV.
[102]
[PDF] Lecture 8 7/09/07 A. The Helmholtz Free Energy and Reversible Work
Jul 9, 2025 · The Helmholtz Free Energy and Reversible Work. The entropy change ... maximum work that can be produced, ∆G measures the upper limit of ...
[103]
[PDF] Ch 17 Free Energy and Thermodynamics - Spontaneity of Reaction
During a phase transition, ΔG=0 since the system is at equilibrium. This transition occurs when T=273K. • ΔG rxn. = ΔH – TΔS. • 0 ...<|control11|><|separator|>
[104]
7.3: Criteria for Spontaneous Change: The Second Law of ...
May 8, 2021 · The second law of thermodynamics states that a spontaneous process increases the entropy of the universe, Suniv > 0. If ΔSuniv < 0, the process ...Definition: The Second Law of... · Gibbs Energy and Changes of...
[105]
19.4: Criteria for Spontaneous Change: The Second Law of ...
Jul 12, 2023 · The second law of thermodynamics states that a spontaneous process increases the entropy of the universe, Suniv > 0. If ΔSuniv < 0, the process ...
[106]
12.3 Second Law of Thermodynamics: Entropy - Physics | OpenStax
Mar 26, 2020 · The second law of thermodynamics states that the total entropy of a system either increases or remains constant in any spontaneous process; it ...
[107]
https://openstax.org/books/university-physics-volume-2/pages/4-5-statements-of-the-second-law-of-thermodynamics
[108]
Second Law of Thermodynamics - HyperPhysics
Second Law of Thermodynamics: It is not possible for heat to flow from a colder body to a warmer body without any work having been done to accomplish this flow.<|control11|><|separator|>
[109]
Heat Pump - HyperPhysics Concepts
A heat pump is subject to the same limitations from the second law of ... heat engine and therefore a maximum efficiency can be calculated from the Carnot cycle.
[110]
Applications of Thermodynamics: Heat Pumps and Refrigerators
The efficiency of a perfect, or Carnot, engine is E f f C = 1 − ( T c T h ) ; thus, the smaller the temperature difference, the smaller the efficiency and the ...
[111]
[PDF] Thermodynamic reversibility - University of Pittsburgh
Sep 3, 2016 · Thermodynamically reversible processes are non-equilibrium states that can change easily in either direction, with balanced driving forces, and ...
[112]
[PDF] REVERSIBLE AND IRREVERSIBLE PROCESS
Examples: Some examples of nearly reversible processes are: (i) Frictionless relative motion. (ii) Expansion and compression of spring. (iii) Frictionless ...
[113]
4.6 Entropy – University Physics Volume 2 - UCF Pressbooks
Because the process is slow, we can approximate it as a reversible process. The temperature is a constant, and we can therefore use Equation 4.8 in the ...
[114]
https://chem.libretexts.org/Bookshelves/General_Chemistry/Map%3A_Principles_of_Modern_Chemistry_(Oxtoby_et_al.)/Unit_4%3A_Equilibrium_in_Chemical_Reactions/13%3A_Spontaneous_Processes_and_Thermodynamic_Equilibrium/13.4%3A_Entropy_Changes_in_Reversible_Processes
[115]
Entropy and the Second Law of Thermodynamics: Disorder and the ...
The entropy of a system can in fact be shown to be a measure of its disorder and of the unavailability of energy to do work.<|control11|><|separator|>