PX154 - D4e - adiabat

adiabatic expansion of an ideal gas

Q = 0

d U = - d W = - p d V

n C_{V} d T = \frac{- n R T}{V} d V

rearrange:

\frac{d T}{T} = - \frac{R}{C_{V}} \frac{d V}{V}

- note: for $\frac{R}{C_{V}} = \frac{C_{P} - C_{V}}{C_{V}} = γ - 1$
$\frac{d T}{T} = - (γ - 1) \frac{d V}{V}$
$\frac{d T}{T} + (γ - 1) \frac{d V}{V} = 0$
- integrate: $\ln (T V^{γ - 1}) = c o n s t a n t$
$$TV^{\gamma-1}=constant$$
- using ideal gas law:
$$pV^{\gamma}=constant$$

compare with the isotherm:
- here, $p V = n R T$ with $T$ constant
- so, for isotherm: $p V = c o n s t a n t$

work done along the adiabat

we have, $W = - Δ U$ , since $Q = 0$
take $d W = d U$
$W = - \int n C_{V} d T$ , here, ( $C_{V} = \frac{1}{n} \frac{d U}{d T}$ )
$= - n C_{V} (T_{2} - T_{1})$
$$\therefore W = -n C_{V} \Delta T$$
- can also be written as:
$W = - \frac{1}{γ - 1} (p_{2} V_{2} - p_{1} V_{1})$
note:
whether we go $A \to B$ via adiabat or $A \to C \to B$ , $Δ T$ is the same $∵ Δ U$ is the same
for $A \to C \to B$ :
- isotherm: $Δ U = 0$
- isochore: $Δ U = n C_{V} Δ T$
  $∴ Δ U_{t o t a l} = 0 + n C_{V} Δ T$
for adiabat: $W = Δ U$
- $∴ W = - n C_{V} Δ T$
  - $C_{V} =$ heat capacity at constant volume, even though, V is not constant