(B) For an adiabatic process,the relationship between pressure $P$ and temperature $T$ is given by $P^{1-\gamma} T^\gamma = \text{constant}$,which can

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(B) For an adiabatic process,the relationship between pressure $P$ and temperature $T$ is given by $P^{1-\gamma} T^\gamma = \text{constant}$,which can be rewritten as $P T^{\frac{\gamma}{1-\gamma}} = \text{constant}$.Given that $P \propto T^3$,we have $P T^{-3} = \text{constant}$.Comparing the exponents of $T$,we get $\frac{\gamma}{1-\gamma} = -3$.Solving for $\gamma$:$\gamma = -3(1-\gamma)$$\gamma = -3 + 3\gamma$$2\gamma = 3$$\gamma = \frac{3}{2}$.Thus,the ratio $\frac{C_p}{C_v} = \gamma = \frac{3}{2}$.

Class 11 Physics Thermodynamics Adiabatic Process

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During an adiabatic process,the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio of $\frac{C_p}{C_v}$ for the gas is:

JEE MAIN 2024 All JEE MAIN

A
$\frac{5}{3}$
B
$\frac{3}{2}$
C
$\frac{7}{5}$
D
$\frac{9}{7}$

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Thermodynamics

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Adiabatic Process

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During an adiabatic process,the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio of $\frac{C_p}{C_v}$ for the gas is:

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$A$ given ideal gas with $\gamma = \frac{C_p}{C_v} = 1.5$ is at a temperature $T$. If the gas is compressed adiabatically to one-fourth of its initial volume,the final temperature will be ..... $T$.

$Assertion :$ Adiabatic expansion is always accompanied by fall in temperature.$Reason :$ In adiabatic process, volume is inversely proportional to temperature.

The pressure and density of a diatomic gas $\left(\gamma=\frac{7}{5}\right)$ changes adiabatically from $(P, d)$ to $(P^{\prime}, d^{\prime})$. If $\frac{d^{\prime}}{d}=32$, then $\frac{P^{\prime}}{P}$ is:

$200 \ cc$ of an ideal gas $(\gamma = 1.5)$ expands adiabatically. If the rms speed of the gas molecules becomes half of the initial value,the final volume of the gas is (in $cc$)

An ideal gas $(\gamma = 1.5)$ is expanded adiabatically. To reduce the root mean square velocity of molecules two times, the gas should be expanded (in $\times$)

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$Assertion :$ Adiabatic expansion is always accompanied by fall in temperature.
$Reason :$ In adiabatic process, volume is inversely proportional to temperature.