I want to know how to prove Cp-Cv=R

I don't know if you'll understand, but I'll try.

PV = RT\ (for\ one\ mole\ of\ gas)

P\Delta V = R\Delta T... 1

Law of conservation of energy;

\Delta Q = \Delta U + \Delta W

\Delta U = Q_1

\Delta W = P\Delta V

Hence,

\Delta Q = Q_1+ P\Delta V... 2

From 1 and 2;

Q_2 = Q_1+ R\Delta T

Where Q1 is the energy involved during isovolumetric change.
And let's call delta Q, Q2, where this is the energy involved during isobarric change.

The respective formulae are:

Q_1 = c_v \Delta T

Q_2 = c_p \Delta T

c_p \Delta T = c_v \Delta T+ R\Delta T

Remove the delta T;

c_p = c_v + R

c_p - c_v = R

There you are. I don't quite get it myself, but this is what my teacher gave me.