Isothermal Process

The Meaning of an Isothermal Process

An isothermal process is a thermodynamic change that takes place at a constant temperature. The word itself comes from the Greek: iso (equal) and therme (heat). For a system to change its volume or pressure while its temperature stays fixed, two conditions must hold simultaneously.

First, the process must be carried out slowly — so slowly that the system has time to exchange heat with its surroundings and remain in thermal equilibrium at every instant. If you compress a gas rapidly, its temperature rises; if you expand it rapidly, its temperature falls. Only a quasi-static, infinitely slow change keeps the temperature constant.

Second, the system must be in contact with a heat reservoir — a body so large that its own temperature does not change measurably when it gives or takes heat from the system. The reservoir acts as a thermal anchor, absorbing any heat generated by compression or supplying any heat needed during expansion.

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The Equation for an Isothermal Process

For $n$ moles of an ideal gas at constant temperature $T$, the ideal gas law gives

Since $n$, $R$, and $T$ are all constant, the product $PV$ is constant. If the gas changes from an initial state $(P_1, V_1)$ to a final state $(P_2, V_2)$ isothermally,

This is the defining relation. On a $P$–$V$ diagram, the curve representing an isothermal process is a rectangular hyperbola — the isotherm. Different temperatures give different hyperbolas; a higher temperature corresponds to a curve lying farther from the origin.

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Work Done in an Isothermal Process

The work done by the gas during any reversible process is

For an isothermal process, we need $P$ as a function of $V$ at constant $T$. From $PV = nRT$,

Substituting into the work integral,

Since $n$, $R$, and $T$ are constants, they come out of the integral:

The integral of $dV/V$ is $\ln V$. Therefore,

This is the work done by the gas during an isothermal expansion ($V_2 > V_1$, so $W > 0$). For an isothermal compression ($V_2 < V_1$), the work done by the gas is negative; the work done on the gas is the positive quantity $-W$.

Using $P_1 V_1 = P_2 V_2$, we can also write the result in terms of pressure ratios:

so

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Properties of an Isothermal Process (as listed in the textbook)

The textbook lists three key properties of an isothermal process for an ideal gas. Each is derived directly from the ideal gas law and the definition of constant temperature.

›Proof

Property (I): For an isothermal process, $PV = \text{constant}$.

Start from the ideal gas law: $PV = nRT$. In an isothermal process, $T$ is constant. Since $n$ and $R$ are also constants, the product $nRT$ is a fixed number. Hence $PV$ is constant. This is simply Boyle’s law.

›Proof

Property (II): For an isothermal process, $P \propto 1/V$.

From Property (I), $PV = \text{constant} = C$. Rearranging, $P = C/V$. Since $C$ is a constant, pressure is inversely proportional to volume. Doubling the volume halves the pressure, and so on.

›Proof

Property (III): For an isothermal process, the $P$–$V$ graph is a rectangular hyperbola.

The equation $P = C/V$ is the equation of a rectangular hyperbola. On a $P$–$V$ diagram, this curve is symmetric about the line $P = V$ (when plotted on equal scales). Different constant temperatures give different hyperbolas — higher $T$ means larger $C = nRT$, so the curve shifts outward (up and to the right).

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Change in Internal Energy and Heat Exchange

For an ideal gas, the internal energy $U$ depends only on temperature. In an isothermal process, temperature does not change, so

From the first law of thermodynamics,

With $\Delta U = 0$, we get

$Q = W$ …