Isothermal process

Thermodynamics
The classical Carnot heat engine
Branches Classical Statistical Chemical Quantum thermodynamics Equilibrium / Non-equilibrium
Laws Zeroth First Second Third
Systems Closed system Open system Isolated system State Equation of state Ideal gas Real gas State of matter Phase (matter) Equilibrium Control volume Instruments Processes Isobaric Isochoric Isothermal Adiabatic Isentropic Isenthalpic Quasistatic Polytropic Free expansion Reversibility Irreversibility Endoreversibility Cycles Heat engines Heat pumps Thermal efficiency
System properties Note: Conjugate variables in italics Property diagrams Intensive and extensive properties Process functions Work Heat Functions of state Temperature / Entropy (introduction) Pressure / Volume Chemical potential / Particle number Vapor quality Reduced properties
Material properties Property databases Specific heat capacity  ${\displaystyle c=}$ ${\displaystyle T}$ ${\displaystyle \partial S}$ ${\displaystyle N}$ ${\displaystyle \partial T}$ Compressibility  ${\displaystyle \beta =-}$ ${\displaystyle 1}$ ${\displaystyle \partial V}$ ${\displaystyle V}$ ${\displaystyle \partial p}$ Thermal expansion  ${\displaystyle \alpha =}$ ${\displaystyle 1}$ ${\displaystyle \partial V}$ ${\displaystyle V}$ ${\displaystyle \partial T}$
Equations Carnot's theorem Clausius theorem Fundamental relation Ideal gas law Maxwell relations Onsager reciprocal relations Bridgman's equations Table of thermodynamic equations
Potentials Free energy Free entropy Internal energy ${\displaystyle U(S,V)}$ Enthalpy ${\displaystyle H(S,p)=U+pV}$ Helmholtz free energy ${\displaystyle A(T,V)=U-TS}$ Gibbs free energy ${\displaystyle G(T,p)=H-TS}$
History Culture History General Entropy Gas laws "Perpetual motion" machines Philosophy Entropy and time Entropy and life Brownian ratchet Maxwell's demon Heat death paradox Loschmidt's paradox Synergetics Theories Caloric theory Vis viva ("living force") Mechanical equivalent of heat Motive power Key publications An Experimental Enquiry Concerning ... Heat On the Equilibrium of Heterogeneous Substances Reflections on the Motive Power of Fire Timelines Thermodynamics Heat engines Art Education Maxwell's thermodynamic surface Entropy as energy dispersal
Scientists Bernoulli Boltzmann Bridgman Carathéodory Carnot Clapeyron Clausius de Donder Duhem Gibbs von Helmholtz Joule Kelvin Lewis Massieu Maxwell von Mayer Nernst Onsager Planck Rankine Smeaton Stahl Tait Thompson van der Waals Waterston
Other Nucleation Self-assembly Self-organization Order and disorder
Category
v t e

An isothermal process is a type of

Simply, we can say that in an isothermal process

• ${\displaystyle T={\text{constant}}}$
• ${\displaystyle \Delta T=0}$
• ${\displaystyle dT=0}$
• For ideal gases only, internal energy ${\displaystyle \Delta U=0}$

while in adiabatic processes:

• ${\displaystyle Q=0.}$

Etymology

The adjective "isothermal" is derived from the Greek words "ἴσος" ("isos") meaning "equal" and "θέρμη" ("therme") meaning "heat".

Examples

Isothermal processes can occur in any kind of system that has some means of regulating the temperature, including highly structured

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

For the special case of a gas to which

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

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

${\displaystyle W_{A\to B}=-\int _{V_{A}}^{V_{B}}p\,dV}$

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:

${\displaystyle W_{A\to B}=-\int _{V_{A}}^{V_{B}}p\,dV=-\int _{V_{A}}^{V_{B}}{\frac {nRT}{V}}dV=-nRT\int _{V_{A}}^{V_{B}}{\frac {1}{V}}dV=-nRT\ln {\frac {V_{B}}{V_{A}}}}$

In

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

Example of an isothermal process

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

The work done (designated ${\displaystyle W_{A\to B}}$) has two components. First, expansion work against the surrounding atmosphere pressure (designated as W_pΔV), and second, usable mechanical work (designated as W_mech). The output W_mech 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.

${\displaystyle W_{A\to B}=-p\,V\left(\ln {\frac {V_{B}}{V_{A}}}\right)=-W_{p\Delta V}-W_{\rm {mech}}}$

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, ${\displaystyle W_{A\to B}}$ equals –140.5 kJ, and W_pΔV is –101.3 kJ. By difference, W_mech = –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 W_mech is a function of pV and p_surr, and approaches 100% as p_surr 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

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

${\displaystyle \Delta S={\frac {Q_{\text{rev}}}{T}}}$

where Q_rev 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, ΔH_tr, thus Q = ΔH_tr.^[3] At any given pressure, there will be a transition temperature, T_tr, 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]

${\displaystyle \Delta S_{\text{tr}}={\frac {\Delta H_{\text{tr}}}{T_{\text{tr}}}}}$.

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

${\displaystyle Q=-W=nRT\ln {\frac {V_{\text{B}}}{V_{\text{A}}}}}$.

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

${\displaystyle \Delta S=nR\ln {\frac {V_{\text{B}}}{V_{\text{A}}}}}$.

Since an ideal gas obeys

${\displaystyle \Delta S=nR\ln {\frac {P_{\text{A}}}{P_{\text{B}}}}}$.

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 ΔS_sur = −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

• Joule–Thomson effect
• free expansion
  )
• Adiabatic process
• Cyclic process
• Isobaric process
• Isochoric process
• Polytropic process
• Spontaneous process

References