Question 590: Thermodynamics & KTG

Question

An ideal gas expands adiabatically, $(\gamma=1.5)$. To reduce the r.m.s. velocity of the molecules 4 times, the gas has to be expanded

Accepted Answer

Since r.m.s. velocity $\mathrm{v} \propto \sqrt{\mathrm{T}}$,
$$ 	herefore \quad \frac{\mathrm{v}_2}{\mathrm{v}_1}=\sqrt{\frac{\mathrm{T}_2}{\mathrm{~T}_1}} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(i)$$
Given the r.m.s. velocity is reduced four times.
$$ \Rightarrow \quad \mathrm{v}_2=\frac{\mathrm{v}_1}{4} $$
Substituting the above result in (i),
$$ \Rightarrow \quad \frac{1}{4}=\sqrt{\frac{T_2}{T_1}} \Rightarrow \frac{T_1}{T_2}=16 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...(ii)$$
For adiabatic expansion, $\mathrm{T}_1 \mathrm{~V}_1^{\gamma-1}=\mathrm{T}_2 \mathrm{~V}_2^{\gamma-1}$
$$ \begin{aligned} & 	herefore \quad\left(\frac{V_2}{V_1}ight)^{\gamma-1}=\frac{T_1}{T_2}=16 \quad \ldots[	ext { From(ii) }] \ & \Rightarrow \quad\left(\frac{V_2}{V_1}ight)^{1.5-1}=16 \ & 	herefore \quad\left(\frac{V_2}{V_1}ight)^{0.5}=16 \Rightarrow \frac{V_2}{V_i}=256 \end{aligned} $$
The gas has to be expanded 256 times.

Answer

256 times

Answer

128 times

Answer

64 times

Answer

8 times

Thermodynamics & KTG

An ideal gas expands adiabatically, (\gamma=1.5). To reduce the r.m.s. velocity of the molecules 4 times, the gas has to be expanded

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