$(35$ points$)$ Air initially at $100$ kPa and $300$ K is used in an IC engine to produce useful energy. The engine may be modeled using an Otto cycle with $0\mathrm{.}5$ kg of air. In the Otto cycle, the air undergoes the following processes:

$12$ : Adiabatic compression to $\frac{1}{12}$ of the initial volume

$23$ : Isochoric heat addition to a maximum temperature of $1100$ K

$34$ : Adiabatic expansion to the initial volume

$41$ : Isochoric heat rejection to the initial state

You may assume that the air behaves like an ideal gas, but at higher temperatures, the initial specific heat values cannot be relied upon.

a$.(5$ points$)$ Select appropriate specific heat values for the entire cycle to minimize errors to due temperature variations. Write your answers in the space below to at least $2$ decimal places and briefly state why these values were chosen.

$C_P=,k\frac{J}{k}g**K,CV=,k\frac{J}{k}{g}^{**}K,k=$

b$.(12$ points$)$ Using the values you determined in part $($a$)$ for the entire cycle, determine temperature, pressure, and volume at each state. Fill out the values in the chart below.

$1$

$\backslash$table$[[$State$,$Temperature $(K),$Pressure $(kPa),$Volume $(m3)$