$4.(20$ points$)($Variable Specific Heat of Air$)$: As shown in class, the specific heat of air is not completely constant with temperature. As a first approximation, the constant volume specific heat can be given by:

$\backslash [$

c$\_\{$v$\}($T$)=0\mathrm{.}0001$ T$+0\mathrm{.}6821$

$\backslash ]$

where T is in K and $\backslash (\backslash$mathrm$\{$c$\}\_\{\backslash$mathrm$\{$v$\}\}\backslash )$ is in $\backslash (\backslash$mathrm$\{$kJ$\}/\backslash$mathrm$\{$kg$\}*\backslash$mathrm$\{$~K$\}\backslash ).$ Suppose $0\mathrm{.}1$ kg of air is contained in a rigid vessel at atmospheric conditions of $300$ K and $100$ kPa $.$ Then, $1500$ W of heat is added to the air until its temperature reaches $900$ K $.$ Heat losses to the environment can be neglected.

$($a$)(3$ points$)$ Determine the volume of the vessel.

$($b$)(12$ points$)$ Determine the time that this process will take. You will need to set up a transient energy balance equation and then perform an integration. HINT: Be careful with units when calculating.

$($c$)(5$ points$)$ How would your answer to part $($b$)$ compare if you used a constant volume specific heat at the averaged temperature during the process $($i$.$e$.600$ K $)?$ HINT: This is similar to what you had to do in a HW $\backslash$#$2$ problem.