Answer
a) $T_{2}=568.24K$
b) $T_{2}=572.97K$
Work Step by Step
a) Based on the ideal gas equation:
$\upsilon_{1}=\frac{RT_{1}}{P_{1}}=\frac{0.1889\frac{kPam^3}{kgK}*(200+273.15)K}{1000kPa}=0.08938\frac{m^3}{kg}$
Knowing that is a polytropic process:
$\upsilon_{2}=\upsilon_{1}(\frac{P_{1}}{P_{2}})^{\frac{1}{n}}=0.08938\frac{m^3}{kg}*(\frac{1000kPa}{3000kPa})^{\frac{1}{1.2}}=0.03578\frac{m^3}{kg}$
$T_{2}=\frac{P_{2}\upsilon_{2}}{R}=\frac{3000kPa*0.03578\frac{m^3}{kg}}{0.1889\frac{kPam^3}{kgK}}=568.24K$
b) Using the Van Der Waals equation:
$a=\frac{27R^2T_{cr}^2}{64P_{cr}}=\frac{27*(0.1889\frac{kPam^3}{kgK})^2*(304.2K)^2}{64*7390kPa}=0.1885\frac{m^6kPa}{kg^2}$
$b=\frac{RT_{cr}}{8P_{cr}}=\frac{0.1889\frac{kPam^3}{kgK}*304.2K}{8*7390kPa}=0.000972\frac{m^3}{kg}$
$(P_{1}+\frac{a}{\upsilon_{1}^2})(\upsilon_{1}-b)=RT_{1}$
$(1000+\frac{0.1885}{\upsilon_{1}^2})(\upsilon_{1}-0.000972)=0.1889*473.15$
Solving for $\upsilon_{1}$
$\upsilon_{1}=0.08824\frac{m^3}{kg}$
$\upsilon_{2}=\upsilon_{1}(\frac{P_{1}}{P_{2}})^{\frac{1}{n}}=0.08824\frac{m^3}{kg}(\frac{1000}{3000})^{\frac{1}{1.2}}=0.03532\frac{m^3}{kg}$
$T_{2}=\frac{(P_{2}+\frac{a}{\upsilon_{2}^2})(\upsilon_{2}-b)}{R}=\frac{(3000+\frac{0.1885}{0.03532^2})(0.03532-0.000972)}{0.1889}$
$T_{2}=572.97K$
