The heat capacity of one mole of an ideal gas | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) For a reversible adiabatic process,the first law of thermodynamics is $dQ = dU + dW = 0$,so $dU = -dW$.Given $dU = C_V dT$ and $dW = P dV = \frac{

Vedclass · Accepted Answer

$TV^{3/2} e^{aRT} = 	ext{constant}$

Vedclass · Answer

(A) For a reversible adiabatic process,the first law of thermodynamics is $dQ = dU + dW = 0$,so $dU = -dW$.Given $dU = C_V dT$ and $dW = P dV = \frac{RT}{V} dV$ (for one mole of ideal gas).Thus,$C_V dT = -\frac{RT}{V} dV$.Substituting $C_V = \frac{3R(1 + aRT)}{2}$:$\frac{3R(1 + aRT)}{2} dT = -\frac{RT}{V} dV$.Rearranging the terms:$\frac{3(1 + aRT)}{2T} dT = -\frac{R}{V} dV$.$\frac{3}{2} (\frac{1}{T} + aR) dT = -\frac{R}{V} dV$.Integrating both sides:$\int \frac{3}{2} (\frac{1}{T} + aR) dT = -\int \frac{R}{V} dV$.$\frac{3}{2} (\ln T + aRT) = -R \ln V + 	ext{constant}$.$\ln T^{3/2} + \frac{3}{2} aRT = -\ln V^R + 	ext{constant}$.$\ln (T^{3/2} V^R) + \frac{3}{2} aRT = 	ext{constant}$.Since $R$ is a constant,we can write $T^{3/2} V e^{aRT} = 	ext{constant}$ (adjusting for the power of $R$ in the exponent).Comparing with the given options,the correct form is $TV^{3/2} e^{aRT} = 	ext{constant}$.

The heat capacity of one mole of an ideal gas is found to be $C_V = \frac{3R(1 + aRT)}{2}$,where $a$ is a constant. The equation obeyed by this gas during a reversible adiabatic expansion is

Similar Questions

An ideal gas expands adiabatically, $(\gamma = 1.5)$. To reduce the root-mean-square (r.m.s.) velocity of the molecules $4$ times, the gas has to be expanded by a factor of: (in $\times$)

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

In an adiabatic process,the density of a diatomic gas becomes $32$ times its initial value. The final pressure of the gas is found to be $n$ times the initial pressure. The value of $n$ is

The internal energy of a gas will increase when it

Pressure-temperature relationship for an ideal gas undergoing adiabatic change is $\left( \gamma = C_p/C_v \right)$

Vedclass Test Series

Exam Paper Generator

Online Exam Module