Question 2.50: An insulated vessel containing air has two compartments sepa...
An insulated vessel containing air has two compartments separated by a conducting diaphram. Compartment A is of capacity 0.030 m³ at 2 bar and 300K. Compartment B is of capacity 0.025 m³ at 4.5 bar and 450 K. Find (a) equilibrium temperature, and (b) change in entropy of the system. Take C_ν = 0.718 kJ/kg.K and R = 0.287 kJ/kg.K.
m_A=\frac{p_1 V_1}{R T_1}=\frac{2 \times 10^5 \times 0.030}{287 \times 300}=0.0697 \mathrm{~kg}
m_B=\frac{p_2 V_2}{R T_2}=\frac{4.5 \times 10^5 \times 0.025}{287 \times 450}=0.0871 \mathrm{~kg}
Let T = equilibrium temperature.
\begin{gathered}m_A c_ν\left(T-T_1\right)=m_B c_ν\left(T_2-T\right) \\0.0697(T-300)=0.0871(450-T) \\0.1568 T=60.105 \\T=383.3 \mathrm{~K}\end{gathered}
\begin{aligned}\Delta S=& m_A c_ν \ln \left\lgroup\frac{T}{T_1}\right\rgroup +m_B c_ν \ln \left\lgroup \frac{T}{T_2}\right\rgroup \\&=0.0697 \times 0.718 \ln\left\lgroup \frac{383.3}{300}\right\rgroup +0.0871 \times 0.718 \ln \left\lgroup\frac{383.3}{450} \right\rgroup \\&=0.01226-0.01003=0.00227 \mathrm{~kJ} / \mathrm{K}\end{aligned}