Question

Air is contained in a piston - cylinder arrangement as shown in Fig. with a cross - sectional area of $4$ cm${}^2$ and an initial volume of $20cc$. The air is initially at a pressure of $1atm$ and temperature of $20^{\circ }C$. The piston is connected to a spring whose spring constant is $k=10^{4}N/m$, and the spring is initially undeformed. How much heat (in joule) must be added to the air to increase the pressure to $3$ atm. (For air, $C_V—718J/k{g}^{\circ }C$, molecular mass of air $28.97$)

Answer 1

Answer: 12.74

When pressure changes from 1 atm to 3 atm , the change in pressure $P=2$ atm
$=2\times 1\times 10^{5}N/m^2$
The force exerted on the piston
$F=PA=2\times 10^5\times 4\times 10^{-4}$
$=80N$
The compression of the spring
$x=\frac{F}{k}=\frac{80}{10^4}=0.008$ m
The change in volume of the air due to displacement of piston by $x$
$\Delta V=Ax=4\times 10^{-4}\times 0.008$
$=3.2\times 10^{-6}{m}^3$
$\therefore$ Final volume $V_2=V_1+\Delta V$
$=20\times 10^{-6}+3.2\times 10^{-6}$
$=23.2\times 10^{-6}{m}^3$
By equation of state
$\frac{P_1{V}_1}{T_1}=\frac{P_2{V}_2}{T_2}$
$T_2=\frac{P_2{V}_2{T}_1}{P_1{V}_1}$
$=\left(\frac{3}{1}\right)\times \frac{\left(23.2\times 10^{-6}\right)}{\left(20\right)\times 10^{-6}}\times \left(273+20\right)$
$=1020K$
The change in internal energy of air
$\Delta U=m{C}_V\Delta T$
$=\left(2.38\times 10^{-5}\right)\times 718\times \left(1020-293\right)$
$=12.42J$ Work done in compressing the spring by $x$
$W=\frac{1}{2}k{x}^2=\frac{10^4}{2}\times {\left(0.008\right)}^2=0.32J$
From first law of thermodynamics
$Q=\Delta U+W=12.42+0.32=12.74J$