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An isothermal process is a type of thermodynamic process in which the temperature T of a system remains constant: ΔT = 0. This typically occurs when a system is in contact with an outside thermal reservoir, and a change in the system occurs slowly enough to allow the system to be continuously adjusted to the temperature of the reservoir through heat exchange (see quasi-equilibrium). In contrast, an adiabatic process is where a system exchanges no heat with its surroundings (Q = 0).

Simply, we can say that in an isothermal process

  • For ideal gases only, internal energy

while in adiabatic processes:

Etymology

[edit]

The noun isotherm is derived from the Ancient Greek words ἴσος (ísos), meaning "equal", and θέρμη (thérmē), meaning "heat".

Examples

[edit]

Isothermal processes can occur in any kind of system that has some means of regulating the temperature, including highly structured machines, and even living cells. Some parts of the cycles of some heat engines are carried out isothermally (for example, in the Carnot cycle).[1] In the thermodynamic analysis of chemical reactions, it is usual to first analyze what happens under isothermal conditions and then consider the effect of temperature.[2] Phase changes, such as melting or evaporation, are also isothermal processes when, as is usually the case, they occur at constant pressure.[3] Isothermal processes are often used as a starting point in analyzing more complex, non-isothermal processes.

Isothermal processes are of special interest for ideal gases. This is a consequence of Joule's second law which states that the internal energy of a fixed amount of an ideal gas depends only on its temperature.[4] Thus, in an isothermal process the internal energy of an ideal gas is constant. This is a result of the fact that in an ideal gas there are no intermolecular forces.[4] Note that this is true only for ideal gases; the internal energy depends on pressure as well as on temperature for liquids, solids, and real gases.[5]

In the isothermal compression of a gas there is work done on the system to decrease the volume and increase the pressure.[4] Doing work on the gas increases the internal energy and will tend to increase the temperature. To maintain the constant temperature energy must leave the system as heat and enter the environment. If the gas is ideal, the amount of energy entering the environment is equal to the work done on the gas, because internal energy does not change. For isothermal expansion, the energy supplied to the system does work on the surroundings. In either case, with the aid of a suitable linkage the change in gas volume can perform useful mechanical work. For details of the calculations, see calculation of work.

For an adiabatic process, in which no heat flows into or out of the gas because its container is well insulated, Q = 0. If there is also no work done, i.e. a free expansion, there is no change in internal energy. For an ideal gas, this means that the process is also isothermal.[4] Thus, specifying that a process is isothermal is not sufficient to specify a unique process.

Details for an ideal gas

[edit]
Figure 1. Several isotherms of an ideal gas on a p-V diagram, where p for pressure and V the volume.

For the special case of a gas to which Boyle's law[4] applies, the product pV (p for gas pressure and V for gas volume) is a constant if the gas is kept at isothermal conditions. The value of the constant is nRT, where n is the number of moles of the present gas and R is the ideal gas constant. In other words, the ideal gas law pV = nRT applies.[4] Therefore:

holds. The family of curves generated by this equation is shown in the graph in Figure 1. Each curve is called an isotherm, meaning a curve at a same temperature T. Such graphs are termed indicator diagrams and were first used by James Watt and others to monitor the efficiency of engines. The temperature corresponding to each curve in the figure increases from the lower left to the upper right.

Calculation of work

[edit]
Figure 2. The purple area represents the work for this isothermal change.

In thermodynamics, the reversible work involved when a gas changes from state A to state B is[6]

where p for gas pressure and V for gas volume. For an isothermal (constant temperature T), reversible process, this integral equals the area under the relevant PV (pressure-volume) isotherm, and is indicated in purple in Figure 2 for an ideal gas. Again, p = ⁠nRT/V⁠ applies and with T being constant (as this is an isothermal process), the expression for work becomes:

In IUPAC convention, work is defined as work on a system by its surroundings. If, for example, the system is compressed, then the work is done on the system by the surrounding so the work is positive and the internal energy of the system increases. Conversely, if the system expands (i.e., system surrounding expansion, so free expansions not the case), then the work is negative as the system does work on the surroundings.

It is also worth noting that for ideal gases, if the temperature is held constant, the internal energy of the system U also is constant, and so ΔU = 0. Since the first law of thermodynamics states that ΔU = Q + W in IUPAC convention, it follows that Q = −W for the isothermal compression or expansion of ideal gases.

Example of an isothermal process

[edit]
Figure 3. Isothermal expansion of an ideal gas. Black line indicates continuously reversible expansion, while the red line indicates stepwise and nearly reversible expansion at each incremental drop in pressure of 0.1 atm of the working gas.

The reversible expansion of an ideal gas can be used as an example of work produced by an isothermal process. Of particular interest is the extent to which heat is converted to usable work, and the relationship between the confining force and the extent of expansion.

During isothermal expansion of an ideal gas, both p and V change along an isotherm with a constant pV product (i.e., constant T). Consider a working gas in a cylindrical chamber 1 m high and 1 m2 area (so 1m3 volume) at 400 K in static equilibrium. The surroundings consist of air at 300 K and 1 atm pressure (designated as psurr). The working gas is confined by a piston connected to a mechanical device that exerts a force sufficient to create a working gas pressure of 2 atm (state A). For any change in state A that causes a force decrease, the gas will expand and perform work on the surroundings. Isothermal expansion continues as long as the applied force decreases and appropriate heat is added to keep pV = 2 [atm·m3] (= 2 atm × 1 m3). The expansion is said to be internally reversible if the piston motion is sufficiently slow such that at each instant during the expansion the gas temperature and pressure is uniform and conform to the ideal gas law. Figure 3 shows the p–V relationship for pV = 2 [atm·m3] for isothermal expansion from 2 atm (state A) to 1 atm (state B).

The work done (designated ) has two components. First, expansion work against the surrounding atmosphere pressure (designated as WpΔV), and second, usable mechanical work (designated as Wmech). The output Wmech here could be movement of the piston used to turn a crank-arm, which would then turn a pulley capable of lifting water out of flooded salt mines.

The system attains state B (pV = 2 [atm·m3] with p = 1 atm and V = 2 m3) when the applied force reaches zero. At that point, equals –140.5 kJ, and WpΔV is –101.3 kJ. By difference, Wmech = –39.1 kJ, which is 27.9% of the heat supplied to the process (- 39.1 kJ / - 140.5 kJ). This is the maximum amount of usable mechanical work obtainable from the process at the stated conditions. The percentage of Wmech is a function of pV and psurr, and approaches 100% as psurr approaches zero.

To pursue the nature of isothermal expansion further, note the red line on Figure 3. The fixed value of pV causes an exponential increase in piston rise vs. pressure decrease. For example, a pressure decrease from 2 to 1.9 atm causes a piston rise of 0.0526 m. In comparison, a pressure decrease from 1.1 to 1 atm causes a piston rise of 0.1818 m.

Entropy changes

[edit]

Isothermal processes are especially convenient for calculating changes in entropy since, in this case, the formula for the entropy change, ΔS, is simply

where Qrev is the heat transferred (internally reversible) to the system and T is absolute temperature.[7] This formula is valid only for a hypothetical reversible process; that is, a process in which equilibrium is maintained at all times.

A simple example is an equilibrium phase transition (such as melting or evaporation) taking place at constant temperature and pressure. For a phase transition at constant pressure, the heat transferred to the system is equal to the enthalpy of transformation, ΔHtr, thus Q = ΔHtr.[3] At any given pressure, there will be a transition temperature, Ttr, for which the two phases are in equilibrium (for example, the normal boiling point for vaporization of a liquid at one atmosphere pressure). If the transition takes place under such equilibrium conditions, the formula above may be used to directly calculate the entropy change[7]

.

Another example is the reversible isothermal expansion (or compression) of an ideal gas from an initial volume VA and pressure PA to a final volume VB and pressure PB. As shown in Calculation of work, the heat transferred to the gas is

.

This result is for a reversible process, so it may be substituted in the formula for the entropy change to obtain[7]

.

Since an ideal gas obeys Boyle's law, this can be rewritten, if desired, as

.

Once obtained, these formulas can be applied to an irreversible process, such as the free expansion of an ideal gas. Such an expansion is also isothermal and may have the same initial and final states as in the reversible expansion. Since entropy is a state function (that depends on an equilibrium state, not depending on a path that the system takes to reach that state), the change in entropy of the system is the same as in the reversible process and is given by the formulas above. Note that the result Q = 0 for the free expansion can not be used in the formula for the entropy change since the process is not reversible.

The difference between the reversible and irreversible is found in the entropy of the surroundings. In both cases, the surroundings are at a constant temperature, T, so that ΔSsur = −⁠Q/T⁠; the minus sign is used since the heat transferred to the surroundings is equal in magnitude and opposite in sign to the heat Q transferred to the system. In the reversible case, the change in entropy of the surroundings is equal and opposite to the change in the system, so the change in entropy of the universe is zero. In the irreversible, Q = 0, so the entropy of the surroundings does not change and the change in entropy of the universe is equal to ΔS for the system.

See also

[edit]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Isothermal process

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from Grokipedia
An isothermal process is a in which the of the remains constant throughout the transition from one equilibrium state to another. This constancy is maintained by coupling the to a or heat bath at the same , enabling to counteract any potential changes due to work or other interactions. Reversible isothermal processes are quasi-static, occurring slowly enough to preserve at every stage. For an , an isothermal process adheres to the equation of state PV=nRTPV = nRT, where and are inversely related since TT is fixed, resulting in a hyperbolic curve on a - diagram. The change in ΔU\Delta U is zero because for an depends solely on . By of , ΔU=Q−W\Delta U = Q - W, this implies that the absorbed QQ equals the work done by the system WW. In a reversible isothermal expansion, the work is calculated as W=nRTln⁡(Vf/Vi)W = nRT \ln(V_f / V_i), where VfV_f and ViV_i are the final and initial , respectively. Isothermal processes play a central role in idealized thermodynamic cycles, such as the , which achieves the highest possible efficiency for a operating between two temperatures through alternating isothermal and adiabatic steps. In the , the isothermal expansion absorbs from a high-temperature , while the isothermal compression rejects to a low-temperature . These processes highlight the principles of reversibility and maximum work extraction in .

Fundamentals

Definition

An isothermal process is a type of thermodynamic process in which the temperature of the system remains constant throughout the change, denoted mathematically as ΔT=0\Delta T = 0 or dT=0dT = 0, meaning TT is held constant. Thermodynamics, the branch of physics that studies the relationships between , work, and transformations in systems, provides the foundational framework for understanding such processes. In an isothermal process, any energy changes occur while maintaining , distinguishing it from other thermodynamic processes. Unlike an , where no is exchanged with the surroundings (Q=0Q = 0), an isothermal process typically involves heat transfer to or from the system to keep the temperature steady. It also differs from an , which maintains constant (ΔP=0\Delta P = 0) but allows temperature variations. This constant-temperature condition is a core concept in analyzing energy balances in closed systems.

Etymology

The term "isothermal" derives from roots isos (ἴσος), meaning "equal," and thermos (θέρμος), meaning "hot" or "heat," literally translating to "equal heat." This linguistic construction underscores the concept of a maintaining uniform . The word entered English around , borrowed from the French isotherme, which was initially applied in to describe lines connecting points of equal on maps. In the realm of , the term "isothermal process" gained prominence in the early amid studies of heat engines. Émile Clapeyron utilized "isothermal" in his 1834 memoir, where he graphically represented Sadi Carnot's cycle using isotherms—curves of constant temperature in pressure-volume diagrams—to analyze engine efficiency. This adoption marked a key step in formalizing thermodynamic nomenclature, building on Carnot's 1824 of constant-temperature operations without initially employing the specific term. The terminology evolved to distinguish isothermal processes from analogous ones, such as "isobaric" (from Greek isos + baros, "weight," denoting constant pressure, first attested in scientific contexts around 1877) and "isochoric" (from Greek isos + chōra, "," indicating constant volume). These parallel formations, emerging in the mid-19th century, standardized descriptions of thermodynamic changes during the field's rapid development by figures like William Rankine and .

Characteristics and Examples

Key Properties

In an isothermal process, the of the remains constant throughout the transformation, which necessitates specific interdependencies among other thermodynamic variables. For systems like ideal gases, this constancy implies that and adjust inversely to one another, ensuring the balance required by the underlying physical laws without altering the state. The absence of net change further means that any transferred to or from the is precisely balanced by the work performed by or on the , maintaining thermal stability. A key implication of this temperature invariance is the potential for the to undergo expansion or compression while avoiding net or cooling, particularly in reversible scenarios where the process proceeds infinitely slowly. In such reversible isothermal processes, the stays in continuous with its surroundings, allowing exchange to occur without gradients. This equilibrium condition underscores the process's reliance on controlled environmental interaction to sustain the constant . Isothermal processes are generally defined for closed systems, where no crosses the system boundary, focusing the analysis on energy transfers via and work alone. Moreover, these processes highlight a fundamental distinction in : state functions, such as for ideal gases, remain unchanged due to their dependence solely on the system's state (here, fixed ), whereas path-dependent quantities like and work vary according to the specific trajectory between initial and final states. This separation emphasizes how isothermal conditions constrain state variables while allowing flexibility in process-dependent aspects.

Real-World Examples

One prominent natural example of an approximate isothermal process is the slow of from a surface at constant , where absorbed during is supplied by the surrounding environment, thereby maintaining the temperature of the nearly constant. This phenomenon occurs in processes like the of wet clothes or the from lakes under calm conditions, illustrating how phase changes can proceed isothermally when coupled with a large . In everyday scenarios, the compression or expansion of air in a bicycle pump can approximate an isothermal process if performed slowly enough to allow heat dissipation to the surroundings, keeping the gas temperature roughly constant despite pressure changes. Here, the inverse relationship between volume and pressure helps sustain thermal equilibrium with the ambient air, though rapid pumping typically deviates toward adiabatic behavior due to limited heat transfer time. Engineered systems often incorporate isothermal stages to optimize efficiency, such as the reversible isothermal expansion and compression in the ideal , which serves as a benchmark for heat engines and refrigerators. In practical cycles, like the vapor-compression system, evaporators and condensers approximate isothermal conditions by facilitating heat exchange at constant temperatures through immersion in large fluid baths. True isothermal processes are idealized in , as real systems achieve them only approximately through sufficiently slow rates or contact with extensive heat reservoirs to counteract any temperature fluctuations. These approximations are essential for modeling in devices like air conditioners, where deviations can reduce performance but still align closely with theoretical predictions under controlled conditions.

Thermodynamic Analysis for Ideal Gases

Internal Energy and Heat

For an ideal gas undergoing an isothermal process, the internal energy UU remains constant because it depends solely on the temperature TT, which is held fixed throughout the process. This temperature dependence arises from the absence of intermolecular forces in an ideal gas, where the internal energy is purely kinetic and proportional to the average molecular kinetic energy. Experimental confirmation comes from Joule's free expansion experiment, in which an ideal gas expands into a vacuum without temperature change, demonstrating that ΔU=0\Delta U = 0 when work and heat transfer are zero, thus establishing U=U(T)U = U(T) only. From the , the of an is given by U=f2n[R](/page/R)TU = \frac{f}{2} n [R](/page/R) T, where ff is the number of per molecule, nn is the number of moles, RR is the , and TT is the absolute temperature. For a monatomic , f=3f = 3 (translational degrees only), yielding U=32n[R](/page/R)TU = \frac{3}{2} n [R](/page/R) T; for diatomic gases like oxygen or at , f=5f = 5 (3 translational + 2 rotational), giving U=52n[R](/page/R)TU = \frac{5}{2} n [R](/page/R) T. In both cases, the molar specific heat at constant volume is Cv=f2[R](/page/R)C_v = \frac{f}{2} [R](/page/R), so the differential change in is dU=nCv dTdU = n C_v \, dT. Since dT=0dT = 0 in an isothermal process, it follows that dU=0dU = 0 regardless of whether the gas is monatomic or diatomic, as CvC_v does not affect the result when temperature is constant. Applying of , which states that the change in equals the added to the system minus the work done by the system (ΔU=Q−W\Delta U = Q - W), yields 0=Q−W0 = Q - W for the isothermal case. Thus, the absorbed by the gas QQ equals the work done by the gas WW, meaning the magnitude of balances the work exactly to maintain constant . This relation holds under the assumptions of non-interacting point particles with no contributions beyond kinetic motion.

Work Calculation

In a reversible isothermal process involving an , the work done by the can be calculated using of , where the change in is (ΔU=0\Delta U = 0) since internal energy depends solely on for an . Thus, the absorbed equals the work done by the (Q=WQ = W), and the work is obtained by integrating the pressure-volume relation along the reversible path. The infinitesimal work done by the system is dW=P dVdW = P \, dV, so for the full process, W=∫ViVfP dVW = \int_{V_i}^{V_f} P \, dV. Substituting the ideal gas law P=nRTVP = \frac{nRT}{V} (with constant TT) yields: W=∫ViVfnRTV dV=nRTln⁡(VfVi).W = \int_{V_i}^{V_f} \frac{nRT}{V} \, dV = nRT \ln\left(\frac{V_f}{V_i}\right). This logarithmic form arises from the integration of 1V\frac{1}{V}, reflecting the hyperbolic shape of the isotherm on a PP-VV diagram. Equivalently, since PiVi=PfVfP_i V_i = P_f V_f for an isothermal process, VfVi=PiPf\frac{V_f}{V_i} = \frac{P_i}{P_f}, so the work can be expressed as: W=nRTln⁡(PiPf).W = nRT \ln\left(\frac{P_i}{P_f}\right). This form is useful when pressure changes are emphasized, such as in compression scenarios. Physically, the work in an isothermal process is path-dependent, meaning its value varies with the specific trajectory on the PP-VV plane between initial and final states; the reversible path, however, yields the maximum work output during expansion (or requires the minimum work input during compression) because the external pressure matches the gas pressure at every step, maximizing the area under the curve. For expansion (Vf>ViV_f > V_i), W>0W > 0, indicating work done by the system on the surroundings; for compression (Vf<ViV_f < V_i), W<0W < 0, indicating work done on the system. The work is expressed in joules (J) in SI units when nn is in moles, R=8.314 J/mol\cdotpKR = 8.314 \, \text{J/mol·K}, and TT is in kelvin. For small volume changes where ΔV/Vi≪1\Delta V / V_i \ll 1, the formula approximates to W≈nRT(ΔVVi)W \approx nRT \left( \frac{\Delta V}{V_i} \right), derived from the Taylor expansion ln⁡(1+x)≈x\ln(1 + x) \approx x for small x=ΔV/Vix = \Delta V / V_i; this linear approximation highlights the process's similarity to near equilibrium.

Entropy and Reversibility

Entropy Changes

In a reversible isothermal process, the change in entropy of the is determined from the Clausius definition, where the differential entropy change is dS=dQrevTdS = \frac{dQ_{\text{rev}}}{T}. With temperature TT held constant throughout the process, integration yields ΔSsystem=QrevT\Delta S_{\text{system}} = \frac{Q_{\text{rev}}}{T}, where QrevQ_{\text{rev}} is the total reversible heat transfer to the . For an ideal gas, the first law of thermodynamics implies that ΔU=0\Delta U = 0 since internal energy depends only on temperature, so Qrev=−WQ_{\text{rev}} = -W, where WW is the work done on the system. This gives ΔSsystem=−WT\Delta S_{\text{system}} = \frac{-W}{T}. The reversible work for such a process is W=nRTln⁡(ViVf)W = nRT \ln\left(\frac{V_i}{V_f}\right), leading to the specific expression ΔSsystem=nRln⁡(VfVi),\Delta S_{\text{system}} = nR \ln\left(\frac{V_f}{V_i}\right), where nn is the number of moles, RR is the gas constant, and ViV_i, VfV_f are the initial and final volumes, respectively. To derive this, start from dS=dQrevTdS = \frac{dQ_{\text{rev}}}{T}. For the reversible path, dQrev=−dW=P dVdQ_{\text{rev}} = -dW = P \, dV (using the convention where dWdW is work on the system), and substituting the ideal gas law P=nRTVP = \frac{nRT}{V} gives dS=nRdVVdS = nR \frac{dV}{V}. Integrating from ViV_i to VfV_f at constant TT confirms the logarithmic form. This entropy change has a clear physical interpretation: for expansion (Vf>ViV_f > V_i), ΔSsystem>0\Delta S_{\text{system}} > 0, indicating an increase in the system's microscopic disorder as molecules access a larger with more possible configurations or microstates; conversely, compression (Vf<ViV_f < V_i) decreases entropy, reflecting reduced disorder. For a reversible process, the entropy change of the universe is zero, ΔSuniverse=0\Delta S_{\text{universe}} = 0, as the system's gain is exactly balanced by the loss in the surroundings. Assuming the surroundings act as a large thermal reservoir at temperature TT, their entropy change is ΔSsurroundings=−QrevT\Delta S_{\text{surroundings}} = -\frac{Q_{\text{rev}}}{T}, ensuring no net production of entropy.

Reversible vs. Irreversible Processes

In a reversible isothermal process, the system remains in thermodynamic equilibrium at every stage through quasi-static changes executed in infinitely slow, infinitesimal steps, enabling the extraction of the maximum possible work. For an ideal gas undergoing expansion from an initial volume ViV_i to a final volume VfV_f at constant temperature TT, the magnitude of the work done by the system is given by ∣Wrev∣=nRTln⁡(Vf/Vi)|W_{\text{rev}}| = nRT \ln(V_f / V_i), where nn is the number of moles and RR is the gas constant. The heat absorbed by the system equals this work output to maintain constant internal energy, and the total entropy change of the universe is zero, satisfying the conditions for reversibility. Irreversible isothermal processes, by contrast, involve non-equilibrium conditions such as sudden expansions, exemplified by free expansion where a gas is released into a vacuum with no opposing external pressure. In such cases, no work is performed (W=0W = 0) and no heat is transferred (Q=0Q = 0), yet the system's internal energy remains unchanged for an ideal gas at constant temperature. The system's entropy increases by ΔSsystem=nRln⁡(Vf/Vi)>0\Delta S_{\text{system}} = nR \ln(V_f / V_i) > 0, resulting in a net positive entropy change for the universe (ΔSuniverse>0\Delta S_{\text{universe}} > 0), as required by the second law of thermodynamics. The primary differences between these processes lie in their and : irreversible isothermal expansions yield less work than their reversible counterparts, reflecting dissipative losses due to finite gradients in or other forces. This aligns with the second law, which mandates ΔSuniverse≥0\Delta S_{\text{universe}} \geq 0 for all spontaneous processes, with equality holding only for reversible ones. Illustrative examples include throttling, an irreversible isothermal process for ideal gases involving sudden drops with minimal work, versus a slow compression or expansion that closely approximates reversibility. In practice, all real isothermal processes are irreversible owing to unavoidable non-equilibrium effects like and finite rates of change, but they are frequently modeled as reversible to calculate upper bounds on work or efficiency in thermodynamic analyses.

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