PX154 - D4d - interlude

interlude

often, the the first law is written as:

d U = d Q - d W

eg: have used $d W = p d V$
from the heat capacity of the ideal gas:

C_{V} = \frac{1}{n} \frac{d Q}{d T} |_{V}

if $V$ is constant, then it is an isochore, and

W = 0

ie: $d U = d Q - d W$ , with $d W = 0$ , so:

d U = d Q

hence:

C_{V} = \frac{1}{n} \frac{d U}{d T} |_{V}

for constant pressure:

C_{P} = \frac{1}{n} \frac{d Q}{d T} |_{p}

here, volume can change, so, $d W \neq 0$
$C_{P}$ and $C_{V}$ can be related for the ideal gas:

\begin{aligned} d U & = d Q - d W \\ d Q & = d U + d W \\ d Q & = d U + p d V \\ n C_{p} d T & = n C_{V} d T + n R d T \\ ∴ C_{P} & = C_{V} + R \end{aligned} $ $ [Y F e q n 19.7] - i t i s a l s o u s e f u l t o d e f i n e : $ $ γ = \frac{C_{P}}{C_{V}}

- eg: for a monatomic gas
- $C_{V} = \frac{3}{2} R$ , so, $C_{P} = \frac{5}{2} R$ , and $γ = \frac{5}{3}$
- eg: for a diatomic gas
- $C_{V} = \frac{5}{2} R$ , so, $C_{P} = \frac{7}{2} R$ , and $γ = \frac{7}{5}$