Given the same liquid-methane and gaseous oxygen rocket engine as in SP17, the oxygen flow
increases from $T_1 = 170 K$ to $T_2 = 172 K$ and the specific volume on a molar basis increases
from $\bar{v}_1 = 0.055 m^3/kmol$ to $\bar{v}_2 = 0.057 m^3/kmol$.
(a) Find the change in pressure (bar) using the Van der Waals equation of state and Eqn. 11.13a,
which states that for $z = function(x,y)$, then $dz = (\partial z/\partial x)_y dx + (\partial z/\partial y)_x dy$. In this case treat
pressure as z, temperature as x, and specific volume on a molar basis as y.
(b) Compare this with the change in pressure (bar) that you would compute directly from $p_2 - p_1$
using the Van der Waals equation of state.
(c) Now the oxygen goes through a constant temperature expansion process from $\bar{v}_2 = 0.057$
m$^3$/kmol to $\bar{v}_3 = 0.059 m^3/kmol$. Again using the Van der Waals equation of state, derive an
expression for $(\partial P/\partial T)_{\bar{v}} = function(\bar{v})$. Then using the third Maxwell equation, $(\partial s/\partial \bar{v})_T =$
$(\partial P/\partial T)_{\bar{v}}$, find $s_3 - s_2$ in kJ/kmol-K (Hint: at constant T, $(\partial s/\partial \bar{v})_T$ becomes $ds/d\bar{v}$ such that
you can integrate the function ($\bar{v}$) you found earlier.
