Isothermal process

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

<math>T = \text{constant}</math>
<math>\Delta T = 0</math>
<math>dT = 0</math>
For ideal gases only, internal energy <math>\Delta U = 0</math>

while in adiabatic processes:

<math>Q = 0.</math>

Etymology

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

Examples

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). 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. Phase changes, such as melting or evaporation, are also isothermal processes when, as is usually the case, they occur at constant pressure. 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. 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.

In the isothermal compression of a gas there is work done on the system to decrease the volume and increase the pressure.

:<math>W_{A\to B} = -\int_{V_A}^{V_B}p\,dV</math>

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 =  applies and with T being constant (as this is an isothermal process), the expression for work becomes:

:<math>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</math>

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

thumb|upright=1.80|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 and change along an isotherm with a constant 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 ). 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 ). For any change in state 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 = 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 relationship for = 2 [atm·m3] for isothermal expansion from 2 atm (state ) to 1 atm (state ).

The work done (designated <math>W_{A\to B}</math>) has two components. First, expansion work against the surrounding atmosphere pressure (designated as ), and second, usable mechanical work (designated as ). The output 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.

:<math>W_{A\to B} = -p\,V\left(\ln\frac{V_B}{V_A}\right) = -W_{p \Delta V} -W_{\rm mech}</math>

The system attains state ( = 2 [atm·m3] with = 1 atm and = 2 m3) when the applied force reaches zero. At that point, <math>W_{A\to B}</math> equals –140.5 kJ, and is –101.3 kJ. By difference, = –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 is a function of and , and approaches 100% as approaches zero.

To pursue the nature of isothermal expansion further, note the red line on Figure 3. The fixed value of 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

:<math>\Delta S = \frac{Q_\text{rev{T}</math>

where Qrev is the heat transferred (internally reversible) to the system and T is absolute temperature. 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.