Question

An ideal gas expands from 100 cm³ to 200 cm³ at a constant pressure of 2.0 × 10⁵ Pa when 50 J of heat is supplied to it. Calculate (a) the change in internal energy of the gas (b) the number of moles in the gas if the initial temperature is 300 K (c) the molar heat capacity C_p at constant pressure and (d) the molar heat capacity C_v at constant volume.

Answer 1

Initial volume of the gas, V₁ = 100 cm³
Final volume = V₂ = 200 cm³
Pressure = 2 × 10⁵ Pa
Heat supplied, dQ = 50 J

(a) According to the first law of thermodynamics,
dQ = dU + dW
dW = P$∆V=2\times {10}^5\times (200-100)\times {10}^{-6}=20$
⇒ 50 = dU + 2 × 10
⇒ dU = 30 J

(b) For a monatomic gas,
U = $\frac{3}{2}n$RT
30 = n × $\frac{3}{2}$ × 8.3 × 300

$\Rightarrow n=\frac{2}{(83\times 3)}=\frac{2}{249}=0.008$

(c) Also,
dU = nC_vdT
$\Rightarrow C_{\mathit{v}}\mathit{=}\frac{\mathit{d}\mathit{U}}{\mathit{n}\mathit{d}\mathit{T}}=\frac{30}{0.008\times 300}=12.5$
C_p = C_v + R = 12.5 + 8.3 = 20.8 J/mol-K

(d) C_v = 12.5 J/mol-K