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  • An ideal gas has pressure P, volume V and temperature T. The ratio <img alt="\mathrm{C_p / C_v=\gamma}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7BC_p%20/%20C_v%3D%5Cgamma%7D"

An ideal gas has pressure P, volume V and temperature T. The ratio \mathrm{C_p / C_v=\gamma}and U is the internal energy. If R is the gas constant, then which one is incorrect

Option: 1

\mathrm{C_v=\frac{R}{\gamma-1}}


Option: 2

\mathrm{U=n C_v T}


Option: 3

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


Option: 4

\mathrm{U=n C_p T}


Answers (1)

best_answer

The internal energy of n moles of an ideal gas at absolute temperature T is given by

\mathrm{U=n C_v T}            ....[1]

Where\mathrm{C_v} is the molar specific heat at constant volume. We know that

\mathrm{\begin{aligned} & C_p-C_v=R \text { or } \frac{C_p}{C_v}-1=\frac{R}{C_v} \\ & \text { Or } \gamma-1=\frac{R}{C_v} \text { or } C_v=\frac{R}{\gamma-1} \end{aligned}}          ....[2]

Now, the ideal gas equation for n nodes is

\mathrm{P V=n R T \text { or } n \frac{P V}{R T}}

Using (2) and (3) in (1), we have

\mathrm{U=\frac{P V}{R T} \times \frac{R}{\gamma-1} \times T=\frac{P V}{(\gamma-1)}}

Posted by

Suraj Bhandari

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