Consider one mole of an ideal gas enclosed in a cylinder fitted with movable frictionless piston.

Let the gas be heated at constant volume first. Let the temperature of the gas increase by dT when dQ quantity of heat is supplied.


Consider one mole of an ideal gas enclosed in a cylinder fitted with

= dU + PdV
= dU space space space space space space open square brackets because space d V equals 0 close square brackets

therefore space space space space space space space dQ apostrophe equals 1 cross times straight C subscript straight p cross times dT

By first law of thermodynamics,

dQ = dU + dW = dU + PdV

or CpdT = Cv x dT + RdT

open square brackets because space at space constant space pressure space PdV space equals space RdT close square brackets
therefore space space space space space straight C subscript straight p equals straight C subscript straight v space plus space straight R
or space straight C subscript straight p minus straight C subscript straight v equals straight R


where Cp, Cv and R are in same units.

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