# Work done by adiabatic expansion derivation

I know that $$W=-C_v(T_2-T_1)$$ for an adiabatic expansion, and I know how to derive it. However, in this video (https://youtu.be/gaZmZjBtgAM?si=Px3v2qDG3CIdupgi&t=358) it mentions the formula $$W=-\frac{C_v}{R}(P_fV_f-P_oV_o)$$ and I am confused on its derivation.

Some of my guesses are to substitute $$\frac{PV}{R} = nT$$, but no idea where the $$n$$ disappears to. Any help is appreciated!

• Could be that in the first formula $C_v$ means the heat capacity of the whole gas, and in the second formula it refers to heat capacity per mole. Sep 29 at 14:26

Some of my guesses are to substitute $$\frac{PV}{R} = nT$$, but no idea where the $$n$$ disappears to. Any help is appreciated!

The equation

$$W=-C_{v}(T_{2}-T_{1})\tag{1}$$

Assumes one mole of an ideal gas, i.e., $$n=1$$ and that $$C_{v}$$ is the molar heat capacity.

Then, since for 1 mole of an ideal gas,

$$PV=RT$$ $$T_{2}=\frac{P_{2}V_{2}}{R}$$ $$T_{1}=\frac{P_{1}V_{1}}{R}$$

Substituting into eq (1)

$$W=-\frac{C_v}{R}(P_{2}V_{2}-P_{1}V_{1})$$

Hope this helps.