 Different type of processes
Question

# A gas undergoes a process in which the pressure P and volume V are related as ${\mathrm{VP}}^{2\mathrm{n}-1}=\mathrm{constant}.$  Find the bulk modulus for the gas.

Moderate
Solution

## $\mathrm{V}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\mathrm{P}}^{2\mathrm{n}-1}=\mathrm{constant}$$⇒\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{V}\left(2\mathrm{n}-1\right)\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{P}}^{2\mathrm{n}-2}\text{\hspace{0.17em}}\mathrm{dP}\text{\hspace{0.17em}\hspace{0.17em}}+\text{\hspace{0.17em}}{\mathrm{P}}^{2\mathrm{n}-1}\mathrm{dV}=0$$⇒\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{V}\left(2\mathrm{n}-1\right)\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{P}}^{2\mathrm{n}-2}\text{\hspace{0.17em}}\mathrm{dP}\text{\hspace{0.17em}\hspace{0.17em}}-\text{\hspace{0.17em}}{\mathrm{P}}^{2\mathrm{n}-1}\mathrm{dV}$$⇒\mathrm{ }-\frac{\mathrm{dP}}{\frac{\mathrm{dV}}{\mathrm{V}}}=\mathrm{ }\frac{{\mathrm{P}}^{2\mathrm{n}-1}}{\left(2\mathrm{n}-1\right)\mathrm{ }{\mathrm{P}}^{2\mathrm{n}-2}}$$⇒\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\frac{{\mathrm{P}}^{2\mathrm{n}-1-2\mathrm{n}+2}}{\left(2\mathrm{n}-1\right)\text{\hspace{0.17em}}}=\left(\frac{\mathrm{P}}{2\mathrm{n}-1}\right)$

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