Consider one mole of an ideal gas enclosed in a cylinder with perfectly conducting walls and fitted with a perfectly frictionless and conducting piston. Let P_{1}, V_{1} and T be the initial pressure, volume and temperature of the gas. Let the gas expand to a volume V_{2} when pressure reduces to P_{2}, at constant temperature T. At any instant during expansion let the pressure of the gas be P. If A is the area of cross section of the piston, then force F = P x A.

Let assume that the pressure of the gas remains constant during an infinitesimally small outward displacement dx of the piston.

Work done, **dW = Fdx = PAcbc = PdV**

Total work done by the gas in expansion from initial volume V_{1} to final volume V_{2} is

**W = ^{v2}∫_{v1} PdV**

We know, PV = RT, so, P = RV

Then, **W = ^{v2}∫_{v1} (RT/V) dV = RT ^{v2}∫_{v1} (1/V) dV**

So, **W = RT [log _{e}V]^{v2}_{v1}**

W = RT [log_{e}V_{2 }– log_{e}V_{1}]

= RT log_{e} (V_{2}/V_{1})

**W = 2.3026 RT log _{10} (V_{2}/V_{1})**

This is the equation for **workdone during an isothermal expansion.**

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