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  • An ideal diatomic gas undergoes a process in which its internal energy relates to the volume as <img alt="\mathrm{U=\alpha \sqrt{V}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%

An ideal diatomic gas undergoes a process in which its internal energy relates to the volume as \mathrm{U=\alpha \sqrt{V}}, where \mathrm{\alpha} is a constant. Then the molar specific heat of the gas is:
 

Option: 1

2.5 \mathrm{R}


 


Option: 2

\mathrm{3.5 R}
 


Option: 3

\mathrm{4 R}
 


Option: 4

4.5 \mathrm{R}


Answers (1)

best_answer

\mathrm{U}=\frac{\mathrm{RT}}{(\gamma-1)}

Using equation \mathrm{\mathrm{pV}=\mathrm{RT}, \quad \mathrm{U}=\frac{\mathrm{pV}}{\gamma-1}}

\mathrm{\therefore \mathrm{W}=\int \mathrm{pdV}=\int \frac{\mathrm{U}(\gamma-1)}{\mathrm{V}} \mathrm{dV}=\alpha(\gamma-1) \int_{\mathrm{V}_1}^{\mathrm{V}_2} \mathrm{~V}^{-1 / 2} \mathrm{dV}\quad or \quad \mathrm{W}=2 \alpha(\gamma-1)}

\mathrm{\left[\sqrt{\mathrm{V}_2}-\sqrt{\mathrm{V}_1}\right]}

Since \mathrm{\quad \Delta \mathrm{U}=\alpha\left[\sqrt{\mathrm{V}_2}-\sqrt{\mathrm{V}_1}\right]}

\mathrm{\therefore \mathrm{W}=2(\gamma-1) \Delta \mathrm{U}}

\mathrm{\mathrm{Q}=\mathrm{C} \Delta \mathrm{T}=\Delta \mathrm{U}[1+2(\gamma-1)]}

\mathrm{or \: \mathrm{C} \Delta \mathrm{T}=\frac{\mathrm{R}}{\gamma-1} \Delta \mathrm{T}[1+2(\gamma-1)] \quad \: or \: \quad \mathrm{C}=\frac{\mathrm{R}}{\gamma-1}+2 \mathrm{R}=4.5 \mathrm{R} (for \quad \gamma= 1.4)}




 

Posted by

Pankaj Sanodiya

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