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Prove: During an adiabatic process,

where γ = C_{p} / C_{v }.

Proof. In an adiabatic process

ΔU + W = 0

So

1) dU + pdV = 0

Now

dU = nC_{V}dt where n is the number of moles of gas

Substituting into 1) we get,

2) nC_{V}dt + pdV = 0

Since pV = nRT, p = nRT/V. Substituting into 2),

Dividing through by nC_{V}T,

Now

5) R = C_{P} - C_{V}

Dividing 5) by C_{V} gives

6) R/C_{V} = γ - 1

Substituting 6) into 4)

Integrating gives

8) ln T + (γ - 1)ln V = ln const.

Taking antilogs

or, stated differently,

which is one of the equations we wished to prove. To derive the other we use pV = nRT and substitute T = pV/nR into 9) to get

or

which can be rewritten as

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