# Isothermal Reversible Expansion

An isothermal process is a process which is conducted in a manner such that the temperature remains constant during the entire operation.

Consider an ideal gas confined in a cylinder with a frictionless piston.

Suppose it expands reversibly from volume $V_1 \text{to} V_2$ at a constant temperature, the pressure of the gas is successfully reduced from $P_1 \text{to} P_2$.

Isothermal reversible expansion

The reversibly expansion of the gas takes place in a finite number of infinitesimally small intermediate steps. To start with the external pressure, $P_{gas}$ is arranged equal to the internal pressure of the gas $P_{ext}$ and the piston remains stationary.

If $P_{ext}$ is decreased by an infinitesimal amount dP the gas expands reversibly and the piston moves through a distance dl.

Since dP is so small, for all practical purposes hence, $P_{ext} = P_{gas} = P$

The work done by gas in one infinitesimal step (dw), can be expressed as:

$dw = P \times A \times dl$ (A = cross-section area of piston)

$P \times dV$

where dV is the increase in volume. The total amount of work done by the isothermal reversible expansion of the ideal gas from $V_1 \text{to} V_2$ is, therefore,

$w = \int^{_2}_{V_1}P. dV$…..(vii)

By the ideal gas equation,

$P = \dfrac{nRT}{V}$

$w = \int^{V_2}_{V_1} \dfrac{nRT}{V} dV$

$nRT \int^{V_2}_{V_1} \dfrac{dV}{V}$

On integration, we get

$\text{Since} P_1V_1 = P_2V_2$

$\dfrac{V_2}{V_1} = \dfrac{P_1}{P_2}$

$w = n R t \hspace{2mm} ln \dfrac{P_1}{P_2}$

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