Problem $3(10$ pts$)$

A closed system consisting of $5$ kg of a gas undergoes a process during which the relationship

between pressure and specific volume is $P{v}^{1\mathrm{.}3}=C,$ where C is a constant. The process begins

with $P_1=2$ bar, $v_1=0\mathrm{.}5\frac{m^3}{k}g$ and ends with $P_2=0\mathrm{.}5$ bar. Please answer the following:

a$.$ Determine the value of the constant, C $,$ and its SI units.

b$.$ Compute the final volume, in $m^3.$

c$.$ Plot the process on a graph of pressure versus specific volume $($you may use MATLAB,

Excel or any other tool$).$ Indicate the pressure values on the vertical axis and the specific

volume values on the horizontal axis.

d$.$ The area between the curve and the specific volume axis has a special meaning in

thermodynamics. Using your Calculus knowledge, evaluate the integral shown below

and verify your result by using another method $($i$.$e$.$ numerical integration$).$ Include the

units in your result.

Integration value $={\displaystyle \underset{v_1}{\overset{v_2}{}}}Pdv$