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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!

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    $\begingroup$ 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. $\endgroup$
    – Andrew Steane
    Sep 29 at 14:26

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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.

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