First slide

Different type of processes

Question

A gas is undergoing an adiabatic process. At a certain stage A, the values of volume and temperature $(V_{0}, T_{0})$ . From the details given in the graph, find the value of adiabatic constant $γ$

Moderate

A

$\frac{V_{0}}{T_{0} \tan θ} + 1$
B

$\frac{V_{0} \tan θ}{T_{0}} + 1$
C

$\frac{V_{0} \tan^{2} θ}{T_{0}} + 1$
D

$\frac{V_{0}}{T_{0}} + \tan θ$

Solution

We have, for an adiabatic process, in relation with temperature and volume
${TV}^{γ - 1}$ = constant
Differentiating with respect to T.
$V^{γ - 1} . 1 + T . (γ - 1) V^{γ - 2} \frac{dV}{dT} = 0$
$\frac{dV}{dT} = \frac{V}{T (γ - 1)}$
But from the graph ${|\frac{dV}{dT}|}_{(T_{0}, V_{0})} = \tan (π - θ) = - \tan θ$

$\frac{V_{0}}{T_{0} (γ - 1)} = \tan θ \Rightarrow γ - 1 = \frac{V_{0}}{T_{0} \tan θ}$
$γ = \frac{V_{0}}{T_{0} \tan θ} + 1$

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