Question

Derive an expression for the work done during an isothermal process.

Obtain an expression for the workdone by a gas in an isothermal process.

Obtain an expression for work done during isothermal process.

Answer 1

Consider n moles of an ideal gas contained in a cylinder equipped with a movable, massless, and frictionless piston. Let P_i, V_i, and T be the gas's initial pressure, volume, and absolute temperature. Consider its isothermal expansion (or compression) at temperature T in terms of final pressure P_f and volume V_f.

Assuming the equation of state of the gas is:

PV = nRT = constant    ...[Boyle’s law]

as T = constant, R is the universal gas constant.

The work done by the gas,

W = `int_{V_i}^{V_f} pdV`

∴ W = `int_(V_i)^(V_f) (nRT)/VdV    ...("for an ideal gas", p = "nRT"/V)`

∴ W = `nRT int_(V_i)^(V_f)(dV)/V`

∴ W = nRT(ln V_f − ln V_i)

∴ W = nRT ln `V_f/V_i` 

= `2.303 nRT log(V_f/V_i)`