Question
Prove for an adiabatic process:
- $\text{TV}^{\gamma-1}=\text{constant}.$
- $\text{P}^{1-\gamma}\text{T}^\gamma=\text{constant.}$
✓
Answer
- We know $\text{PV}^{\gamma}=$ constant for adiabatic process.
Also, $\text{PV}=\text{nRT}$
$\therefore\text{P}=\frac{\text{nRT}}{\text{V}}$
Replacing P, we have $\frac{\text{nRT}}{\text{RT}}.\text{V}^{\gamma}=$ constant (or) $\text{TV}^{\gamma-1}=$ constant.
- From PV = nRT, we have $\text{V}=\frac{\text{nRT}}{\text{P}}$
Replacing V, we have, $\text{P}\Big(\frac{\text{nRT}}{\text{P}}\Big)^{\gamma}=$ constant.
$\therefore\text{T}^{\gamma}\text{P}^{1-\gamma}=$ constant.
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