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Thermodynamic Processes

A gas is unde...

Question

A gas is undergoing an adiabatic process. At a certain stage A, the values of volume and temperature $\equiv (V_{0}, T_{0})$ and the magnitude of the slope of V-T curve is m. Find the value of $C_{P}$ and $C_{V}$ .

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Solution

For an adiabatic process : $T V^{γ - 1} = K = c o n s t a n t$
Differentiating, we get $T (γ - 1) V^{γ - 2} d V + V^{γ - 1} d T = 0$
$∴$ $\frac{d V}{d T} = - \frac{V}{(γ - 1) T}$

Now $∣ ∣ ∣ \frac{d V}{d T} {∣ ∣ ∣}_{V_{o}, T_{o}} = m$

$∴$ $\frac{V_{o}}{(γ - 1) T_{o}} = m$ $⟹$ $γ - 1 = \frac{V_{o}}{m T_{o}}$ OR $γ = 1 + \frac{V_{o}}{m T_{o}}$ ........(1)

$∴$ $C_{v} = \frac{f}{2} R T_{o} = \frac{R T_{o}}{γ - 1}$ $(∵ f / 2 = (γ - 1)^{- 1})$

$⟹$ $C_{v} = R T_{o} \times \frac{V_{o}}{m T_{o}} = \frac{R V_{o}}{m}$

Also $C_{p} = γ C_{v} = (1 + \frac{V_{o}}{m T_{o}}) \frac{R T_{o}}{m}$

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