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Two cylinders $A$ and $B$ fitted with pistons contain equal amounts of an ideal diatomic gas at $300\,K$ . the position of $A$ is free to move while that of $B$ is held fixed. The same amount of heat is given to the gas in each cylinder. If the rise in temperature of the gas in $A$ is $30\,K$ , then the rise in temperature of the gas in $B$ is .... $K$
$30$
$18$
$50$
$42$
Solution
In both cylinders $A$ and $B$ the gases are diatomic $(\gamma=1.4) .$ Piston $\mathrm{A}$ is free to move i.e. it is isobaric process. Piston $\mathrm{B}$ is fixed i.e. it is isocahoric process. If same amount of heat $\Delta \mathrm{Q}$ is given to both then
$(\Delta \mathrm{Q})_{\text {isobaric }}=(\Delta \mathrm{Q})_{\text {isochoric }} \Rightarrow \mu \mathrm{C}_{\mathrm{P}}(\Delta \mathrm{T})_{\mathrm{A}}=\mu \mathrm{C}_{\mathrm{v}}(\Delta \mathrm{T})_{\mathrm{B}}$
$\Rightarrow(\Delta \mathrm{T})_{\mathrm{B}}=\frac{\mathrm{C}_{\mathrm{P}}}{\mathrm{C}_{\mathrm{V}}}(\Delta \mathrm{T})_{\mathrm{A}}=\gamma(\Delta \mathrm{T})_{\mathrm{A}}=1.4 \times 30=42 \mathrm{K}$
