Question

An ideal gas has molar heat capacity at constant pressure $C_p=\frac{5R}{2}$. The gas is kept in a cylindrical vessel fitted with a piston which is free to move. Mass of the frictionless piston is 9 kg. Initial volume of the gas is $0.0027m^3$ and cross-section area of the piston is $0.09m^2$. The initial temperature of the gas is 300 K. Atmospheric pressure $P_0=1.05\times {10}^{5}N/m^2$. An amount of $2.5\times {10}^{4}J$ of heat energy is supplied to the gas such that piston move without acceleration, then

• A. Initial pressure of the gas is $1.06\times {10}^{5}N/m^2$
• B. Work done by the gas is 1000 $J$
• C. Final pressure of the gas is $1.06\times {10}^{5}N/m^2$
• D. Work done by gas is $9.94\times {10}^{3}J$

Answer 1

Answer: (A), (C), (D)

The correct options are
A Initial pressure of the gas is $1.06\times {10}^{5}N/m^2$
C Final pressure of the gas is $1.06\times {10}^{5}N/m^2$
D Work done by gas is $9.94\times {10}^{3}J$

$PV=nRT\dots \dots \left(1\right)$
Since, pressure of gas is balancing outside atmospheric pressure and pressure due to weight of piston.
Mass of piston is 9 kg and area of piston is $0.09m^2$

So, $P=P_0+\frac{9g}{0.09}$

$P=1.06\times {10}^{5}N/m^2$
So, Initial and final pressure of the gas is $1.06\times {10}^{5}N/m^2$

As mass of piston is not changed and piston is frictionless. So, process is isobaric.

$P_f=P_1=1.06\times {10}^{5}N/m^2$

$dQ=n{C}_{p}dT\dots \dots \left(2\right)$

or $P_2{V}_2-P_1{V}_1=nRdT\dots \dots \left(3\right)$

(3) $÷$ (2) $\frac{P(V_2-V_1)}{dQ}=\frac{2}{5}$

$\Rightarrow P(V_2-V_1)=\frac{2}{5}\times 2.5\times {10}^4$

$W=P\Delta V={10}^{4}J$