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Specific Heat Capacities of Gases — Physics STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 SciencePhysicsSpecific Heat Capacities of Gases5 Marks
Question
A sample of an ideal gas $(\gamma=1.5)$ is compressed adiabatically from a volume of $150cm^3$ to $50cm^3$. The initial pressure and the initial temperature are $150kPa$ and $300K$. Find,
  1. The number of moles of the gas in the sample.
  2. The molar heat capacity at constant volume.
  3. The final pressure and temperature.
  4. The work done by the gas in the process.
  5. The change in internal energy of the gas.
✓

Answer

PV = nRT
Given,
$P = 150KPa = 150 \times 10^3Pa, V = 150cm^3= 150 \times 10^{-6}m^3, T = 300k$
  1. $\text{n}=\frac{\text{PV}}{\text{RT}}=\frac{150\times10^3\times150\times10^{-6}}{8.3\times300}$
$=9.036\times10^{-3}=0.009\ \text{moles}.$
  1. $\frac{\text{C}_\text{p}}{\text{C}_\text{v}}=\gamma$
$\Rightarrow\frac{\gamma\text{R}}{(\gamma-1)\text{C}_\text{v}}=\gamma$ $\Big[\therefore\ \text{C}_\text{p}=\frac{\gamma\text{R}}{\gamma-1}\Big]$
$\Rightarrow\text{C}_\text{v}=\frac{\text{R}}{\gamma-1}=\frac{8.3}{1.5-1}=\frac{8.3}{0.5}$
$=2\text{R}=16.6\text{J/mole}$
  1. Given,
$P_1 = 150KPa = 150 \times 10^3Pa, P_2 = ?$
$V_1 = 150cm^3 = 150 \times 10^{-6}m^3, \gamma=1.5$
$V_2 = 50cm^3= 50 \times 10^{-6}m^3, T_1 = 300k, T_2= ?$
Since the process is adiabatic. Hence,
$-\text{P}_1\text{V}_1^\gamma=\text{P}_2\text{V}_2^\gamma$
$\Rightarrow150\times10^3(150\times10^{-6})^\gamma=\text{P}_2\times(50\times10^{-6})^\gamma$
$\Rightarrow\text{P}_2=150\times10^3\times\Big(\frac{150\times10^{-6}}{50\times10^{-6}}\Big)^{1.5}$
$=150000\times3^{1.5}=779.422\times10^3\text{Pa}\approx780\text{KPa}$
  1. $\triangle\text{Q}=\text{W+}\triangle\text{U}$
$\text{W}=-\triangle\text{U}$ $[\therefore\ \triangle\text{U}=0,\ \text{in adiabatic}]$
$=-\text{nC}_\text{v}\text{dT}=-0.009\times16.6\times(520-300)$
$=-0.009\times16.6\times200=-32.8\text{J}\approx-33\text{J}$
  1. $\triangle\text{U}=\text{nC}_\text{v}\text{dT}=0.009\times16.6\times220\approx33\text{J}$

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