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Question
Derive an expression for the work done during an isothermal process.
Solution
Consider the isothermal expansion of an ideal gas. Let its initial volume be Vi and the final volumes be Vf. The work done in an infinitesimally small isothermal expansion is given by
dW = pdV
The total work done in bringing out the expansion from the initial volume Vi to the final volume Vf is given by,
W = `int_{V_i}^{V_f} pdV`
∴ W = `int_{V_i}^{V_f} (nRT)/V`dV .............`("for an ideal gas", p = "nRT"/V)`
∴ W = `nRTint_{V_i}^{V_f}(dV)/V`
∴ W = nRT(In Vf - In Vi)
∴ W = nRT In `V_f/V_i`
This is the required expression for the work done during an isothermal process.
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