# Isothermal Processes

## An isothermal process is a change of a thermodynamic system, in which the temperature remains constant.

#### Key Points

• For an ideal, the product of pressure and volume (PV) is a constant if the gas is kept at isothermal conditions.

• For an ideal gas, the work involved when a gas changes from state A to state B through an isothermal process is given as <equation contenteditable="false">$W_{A\to B} = nRT\ln{\frac{V_B}{V_A}}$.

• For many systems, if the temperature is held constant, the internal energy of the system also is constant. It follows that Q =-W in this case.

#### Terms

• The observation that the pressure of an ideal gas is inversely proportional to its volume at constant temperature.

• A hypothetical gas whose molecules exhibit no interaction and undergo elastic collision with each other and with the walls of the container.

• Capable of returning to the original state without consumption of free energy and increase of entropy.

#### Figures

1. ##### Isotherms of an Ideal Gas

Several isotherms of an ideal gas on a PV diagram.

2. ##### Work Done by Gas During Expansion

The blue area represents "work" done by the gas during expansion for this isothermal change.

An isothermal process is a change of a system, in which the temperature remains constant: ΔT = 0. This typically occurs when a system is in contact with an outside thermal reservoir (heat bath), and the change occurs slowly enough to allow the system to continually adjust to the temperature of the reservoir through heat exchange. In contrast, an adiabatic process is where a system exchanges no heat with its surroundings (Q = 0). (See our atom on "Adiabatic Process.") In other words, in an isothermal process, the value ΔT = 0 but Q ≠ 0, while in an adiabatic process, ΔT ≠ 0 but Q = 0.

### Ideal Gas in an Isothermal Process

For an ideal, the product of pressure and volume (PV) is a constant if the gas is kept at isothermal conditions. (This is historically called Boyle's law.) However, the cases where the product PV is an exponential term, does not comply. The value of the constant is nRT, where n is the number of moles of gas present and R is the ideal gas constant. In other words, the ideal gas law PV = nRT applies. This means that

$P = \frac{nRT}{V} = \frac{constant}{V}$

holds. The family of curves generated by this equation is shown in Figure 1. Each curve is called an isotherm.

### Calculation of Work

In thermodynamics, the work involved when a gas changes from state A to state B is simply

$W_{A\to B} = \int_{V_A}^{V_B}P\,dV$

For an isothermal, reversible process, this integral equals the area under the relevant pressure-volume isotherm, and is indicated in blue in Figure 2 for an ideal gas. Again, P = nRT / V applies and with T being constant (as this is an isothermal process), we have

$W_{A\to B} = nRT\int_{V_A}^{V_B}\frac{1}{V}dV = nRT\ln{\frac{V_B}{V_A}}$

It is also worth noting that, for many systems, if the temperature is held constant, the internal energy of the system also is constant, and so $\Delta U = 0$. From the first law of thermodynamics, it follows that $Q =-W$ for this same isothermal process.

#### Key Term Glossary

Occurring without gain or loss of heat.
##### Appears in these related concepts:
atom
The smallest possible amount of matter which still retains its identity as a chemical element, now known to consist of a nucleus surrounded by electrons.
##### Appears in these related concepts:
contrast
A difference in lightness, brightness and/or hue between two colors that makes them more or less distinguishable
##### Appears in these related concepts:
energy
A quantity that denotes the ability to do work and is measured in a unit dimensioned in mass × distance²/time² (ML²/T²) or the equivalent.
##### Appears in these related concepts:
equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equal sign; from which one is to determine a particular quantity.
##### Appears in these related concepts:
gas
Matter in a state intermediate between liquid and plasma that can be contained only if it is fully surrounded by a solid (or held together by gravitational pull); it can condense into a liquid, or can (rarely) become a solid directly.
##### Appears in these related concepts:
gas constant
A universal constant, R, that appears in the ideal gas law, (PV = nRT), derived from two fundamental constants, the Boltzman constant and Avogadro's number, (R = NAk).
##### Appears in these related concepts:
heat
energy transferred from one body to another by thermal interactions
##### Appears in these related concepts:
ideal gas
A hypothetical gas whose molecules exhibit no interaction and undergo elastic collision with each other and with the walls of the container.
##### Appears in these related concepts:
Ideal Gas
Theoretical gas characterized by random motion whose individual molecules do not interact with one another and are chemically inert.
##### Appears in these related concepts:
internal energy
The sum of all energy present in the system, including kinetic and potential energy; equivalently, the energy needed to create a system, excluding the energy necessary to displace its surroundings.
##### Appears in these related concepts:
isothermal
Of or pertaining to a process that takes place at constant temperature.
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Law
A concise description, usually in the form of a mathematical equation, used to describe a pattern in nature
##### Appears in these related concepts:
mole
In the International System of Units, the base unit of amount of substance; the amount of substance of a system which contains as many elementary entities as there are atoms in 12 g of carbon-12. Symbol: mol.
##### Appears in these related concepts:
pressure
the amount of force that is applied over a given area divided by the size of that area
##### Appears in these related concepts:
reversible
Capable of returning to the original state without consumption of free energy and increase of entropy.
##### Appears in these related concepts:
thermodynamic
Relating to the conversion of heat into other forms of energy.
##### Appears in these related concepts:
thermodynamics
a branch of natural science concerned with heat and its relation to energy and work
##### Appears in these related concepts:
work
A measure of energy expended in moving an object; most commonly, force times displacement. No work is done if the object does not move.