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Q.

A certain amount of mono-atomic ideal gas undergoes a process ${ρu}^{η} =C$ , where $ρ$ is the density of the gas and u is the internal energy of the gas. It was found that the ratio $r= \frac{ΔW}{ΔQ}$ for the process was r=2/3. What is the value of $η$ ? ( $Δ W$ is work done by the gas and $ΔQ$ is heat given to gas)

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answer is 3.

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Detailed Solution

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$ρ u^{η} = C \Rightarrow V^{- 1} T^{η} = C \Rightarrow T V^{- 1 / η} = C \Rightarrow P V^{1 - (1 / η)} = C$

Compare with $P V^{x} = C$

$\Rightarrow x = 1 - \frac{1}{η} N o w, Δ Q α (\frac{R}{r - 1} + \frac{R}{1 - x}) a n d Δ W α (\frac{R}{1 - x}) \Rightarrow \frac{Δ W}{Δ Q} = \frac{(\frac{R}{1 - x})}{\frac{R}{r - 1} + \frac{R}{1 - x}} = (\frac{r - 1}{1 - x}) = \frac{2}{3} = \frac{\frac{5}{3} - 1}{\frac{5}{3} - x} = \frac{2}{3} \Rightarrow x = 2 / 3 = 1 - \frac{1}{η} \Rightarrow η = 3$

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A certain amount of mono-atomic ideal gas undergoes a process ρuη=C, where ρ is the density of the gas and u is the internal energy of the gas. It was found that the ratio r=ΔWΔQ for the process was r=2/3. What is the value of η ? (ΔW is work done by the gas and ΔQ is heat given to gas)