Cp - Cv = R

 Cp and Cv are molar heat capacities representing constant pressure and constant volume. R is the Universal gas constant. To derive relation between them consider one mole of an ideal gas taken in a cylinder fitted with frictionless piston. Let the gas is heated at a constant volume so that its temperature increases by dt and no work is done. the amount of heat is used to increase only the internal energy. i.e

dQ= dU= 1*CvdT ----- (i)

where Cv is the molar specific heat capacity at constant volume. Again, When the gas is heated at constant pressure, so that temperature increases by dT, let dQ be the heat supplied then,

dQ = 1*Cp*dT = CpdT ------(ii)

Where Cp is the molar specific heat at constant pressure Then from first law of thermodynamics, we can write

dQ = dU + Pdv -------- (iii)

Where dU is the increase in internal energy, P is the constant pressure at which the gas is heated and dV is increase in volume.

Using equation (i) and (ii) in (iii), we get,

dQ = dU + pdv -----(iv)

For 1 mole gas, ideal gas equation is PV = RT.

Differentiating above equation keeping pressure constant pressure constant,

d(PV)/dt = d(RT)/dt

Pdv = Rdt

Substituting the value of PdV from equation (v) to equation (iv), we get,

CpdT = CvdT + RdT

CpdT = (Cv + R)dt

Cp - Cv = R