Fyneman exercise I ch39-1

Q.

If an ideal gas is compressed adiabatically, we have found (Eq. 39-14) that $PV^\gamma = constant$. On the other hand, under all conditions, $\frac{PV}{T} = constant$. Combine these to deduce how $P$ and $T$, or $V$ and $T$, are connected during an adiabatic compression.

A.

P & T

Putting $V^\gamma = \frac{constant}{P}$ to $\frac{P^\gamma V^\gamma}{T^\gamma} = constant$ gives

$P^{1-\gamma}T^\gamma = constant$

V & T

Putting $P=\frac{constant}{V^\gamma}$ to $\frac{PV}{T} = constant$ gives

$V^{\gamma -1}T=constant$