prove that Cp - Cv =R

here,

Cv - molar specific heat at constant volume

Cp - molar specific heat at constant pressure

where, according to the first law of thermodynamics,

Cp = (1/n)(dU/dT) + R

or for ideal monoatomic gases

Cp = (5/2)R

and

Cv = (1/n)(dU/dT)

or Cv = (3/2)R

thus Cp - Cv = (5/2)R - (3/2)R

or

Cp - Cv = R
hope it helps!!