How to derive Mayer's formula i.e. Cp-Cv = R

You know the first law of thermodynamics?

\Delta U = \Delta Q + \Delta W

\Delta U = c_v \Delta T (when pressure constant)

\Delta Q = c_p \Delta T (when pressure constant)

\Delta W = -P\Delta V (negative since work is done by the gas)

Now, what is -P\Delta V ?

PV = RT for one mole of gas

Since pressure is constant, and R is a constant, a change in V will cause a change in T.

P\Delta V = R\Delta T

Substitute in the first law of thermodynamics:

c_v \Delta T = c_p \Delta T- R\Delta T

Divide by delta T:

c_v = c_p - R

Make R the subject:

c_p - c_v = R