Question 6.237E: A cylinder/piston contains carbon dioxide at 150 lbf/in.^2, ...

C.V. Carbon dioxide gas 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 d Q / T +{ }_{1} S _{2 \text { gen }}={ }_{1} Q _{2} / T _{ amb }+{ }_{1} S _{2  gen}

Process:      P = CV ^{3} \text { or } PV ^{-3}=\text { constant } ,  which is polytropic with n = -3

State 1:      P _{1}=150   lbf / in ^{2}, T _{1}=600   F =1060   R , V _{1}=7   ft ^{3}  Ideal gas

Process:    P = C V ^{3} \text { or } PV ^{-3}= const.    polytropic with n = -3.

From Eq.6.37 or 6.39

Notice:
n = -3, k = 1.3
n < k

……………………………………

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.6.39:
\begin{aligned}\left(S_{2}-S_{1}\right)_{\text {total }}=&\left(S_{2}-S_{1}\right)_{A}+\left(S_{2}-S_{1}\right)_{B}+\left(S_{2}-S_{1}\right)_{C} \\=& \int \frac{\delta Q_{a}}{T_{a}}-\int \frac{\delta Q_{b}}{T_{b}}+S_{\operatorname{gen} A}+\int \frac{\delta Q_{b}}{T_{b}}-\int \frac{\delta Q_{c}}{T_{c}}+S_{\operatorname{gen} B} \\&+\int \frac{\delta Q_{c}}{T_{c}}-\int \frac{\delta Q_{a}}{T_{a}}+S_{ gen C} \\=& S_{\operatorname{gen} A}+S_{\operatorname{gen} B}+S_{\operatorname{gen} C} \geq 0\end{aligned}