Question 766: Thermodynamics & KTG

Question

Initial pressure and volume of a gas are ' P ' and ' $V$ ' respectively. First its volume is expanded to ' 4 V ' by isothermal process and then again its volume is reduced to ' V ' by adiabatic process then its final pressure if $\left(\gamma=\frac{3}{2}\right)$

Accepted Answer

To determine the final pressure after the given processes, we can follow these steps:
Isothermal Expansion:
Initial pressure and volume of the gas are $ P $ and $ V $, respectively.
The gas undergoes an isothermal expansion to a volume $ 4V $.
For an isothermal process, the relationship between pressure and volume is given by:
$    P_1 V_1 = P_2 V_2    $
Substituting the given values:
$    P \cdot V = P_2 \cdot 4V    $
Solving for $ P_2 $, we get:
$    P_2 = \frac{P}{4}    $
Adiabatic Compression:
Next, the gas undergoes adiabatic compression back to the original volume $ V $.
For an adiabatic process, the equation is:
$    P_2 V_2^\gamma = P_3 V_3^\gamma    $
Given that $ V_2 = 4V $, $ V_3 = V $, and $\gamma = \frac{3}{2}$:
$    \frac{P}{4} \cdot (4V)^\gamma = P_3 \cdot V^\gamma    $
Substituting $\gamma = \frac{3}{2}$, this becomes:
$    \frac{P}{4} \cdot (4V)^{3/2} = P_3 \cdot V^{3/2}    $
Simplifying further:
$    \frac{P}{4} \cdot 4^{3/2} = P_3    $
Since $4^{3/2} = 8$, we have:
$    P_3 = \frac{P}{4} 	imes 8 = 2P    $
Thus, the final pressure after the entire process is $2P$.

Answer

P

Answer

2P

Answer

3P

Answer

4P

Thermodynamics & KTG

Initial pressure and volume of a gas are ' P ' and ' V ' respectively. First its volume is expanded to ' 4 V ' by isothermal process and then again its volume is reduced to ' V ' by adiabatic process then its final pressure if \left(\gamma=\frac{3}{2}\right)

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