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  • An ideal gas undergoes a process such that <img alt="\mathrm{P} \propto \frac{1}{\mathrm{~T}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7BP%7D%20%5Cpropto%20%5Cfrac%7B1%7D%7B%

An ideal gas undergoes a process such that \mathrm{P} \propto \frac{1}{\mathrm{~T}}. The molar heat capacity of this process is 33.24 \mathrm{~J} / \mathrm{mol} \mathrm{K}.

Option: 1

The work done by the gas is 2 \mathrm{R} \Delta \mathrm{T}


Option: 2

Degree of freedom of the gas is 4 .


Option: 3

\gamma\left(\frac{C_{P}}{C_{V}}\right) for the gas is 1.5.


Option: 4

All of the above


Answers (1)

best_answer

Given PT = Constant.
\mathrm{ \text { and } \mathrm{PV}=\mathrm{nRT} }

\mathrm{ \Rightarrow \mathrm{P}^{2} \mathrm{~V}=\mathrm{constant}}
\mathrm{ \Rightarrow \mathrm{PV}^{1 / 2}=\mathrm{K}}.

From first law of thermodynamics

\mathrm{ \Delta \mathrm{Q}=\Delta \mathrm{U}+\Delta \mathrm{W}}
\mathrm{ \mathrm{C} \Delta \mathrm{T}=\mathrm{C}_{\mathrm{V}} \mathrm{T}+\left(\frac{\mathrm{P}_{\mathrm{f}} \mathrm{V}_{\mathrm{f}}-\mathrm{P}_{\mathrm{i}} \mathrm{V}_{\mathrm{i}}}{1-\frac{1}{2}}\right)}
\mathrm{ \mathrm{C}=\mathrm{C}_{\mathrm{V}}+2 \mathrm{R}}


\mathrm{ 33.24=\frac{\mathrm{R}}{\gamma-1}+2 \mathrm{R}}
\mathrm{ \Rightarrow \gamma=1.5}
\mathrm{ and \, \gamma=1+\frac{2}{\mathrm{f}}}
\mathrm{ \therefore \mathrm{f}=4}

COMPREHENSION:
One mole of an ideal gas has an internal energy given by \mathrm{ \mathrm{U}=\mathrm{U}_{0}+2 \mathrm{PV}}, where \mathrm{ \mathrm{P}} is the pressure and \mathrm{ \mathrm{V}}the volume of the gas.\mathrm{\mathrm{U}_{0}}  is a constant. This gas undergoes the quasi-static cyclic process \mathrm{A B C D} as shown in the \mathrm{U-V} diagram.

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seema garhwal

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