Question 6.238E: A cylinder/piston contains 4 ft^3 of air at 16 lbf/in.^2, 77...

C.V. Air of constant mass m _{2}= m _{1}= m out to ambient.

Energy Eq.3.5:            m \left( u _{2}- u _{1}\right)={ }_{1} Q _{2}-{ }_{1} W _{2}

Entropy Eq.6.37:        m \left( s _{2}- s _{1}\right)=\int dQ / T +{ }_{1} S _{2  \text { gen }} ={ }_{1} Q _{2} / T _{0}+{ }_{1} S _{2  \text { gen }}

Process:      {Pv _{1}}^{ n }= {P _{2} v _{2}}^{ n }          Eq.6.27

State 1:    \left( T _{1}, P _{1}\right)           State 2:  \left( T _{2}, P _{2}\right)

Thus the unknown is the exponent n.

The relation from the process and ideal gas is in Eq.6.28

n = 1.305,  V_{2}=V_{1}\left(P_{1} / P_{2}\right)^{1 / n}=4 \times(16 / 20) 1 / 1.305=0.854   ft ^{3}

…………………………………

Eq.3.5: E_{2}-E_{1}={ }_{1} Q_{2}-{ }_{1} W_{2}

Eq.6.37 : S_{2}-S_{1}=\int_{1}^{2} d S=\int_{1}^{2} \frac{\delta Q}{T}+{ }_{1} S_{2 gen}

Eq.6.27 : P V^{n}=\text { constant }=P_{1} V_{1}^{n}=P_{2} V_{2}^{n}

Eq.6.28 :
\begin{array}{l}\frac{P_{2}}{P_{1}}=\left(\frac{V_{1}}{V_{2}}\right)^{n} \\\frac{T_{2}}{T_{1}}=\left(\frac{P_{2}}{P_{1}}\right)^{(n-1) / n}=\left(\frac{V_{1}}{V_{2}}\right)^{n-1}\end{array}