Final answer:
The final temperature of air that is isentropically compressed from 4 MPa and 300 K to 9 MPa cannot be determined without knowing the specific heat ratio (\gamma) at those conditions. To accurately compute it, thermodynamic tables or specialized software that accommodate for variable specific heats of air must be used.
Step-by-step explanation:
To determine the final temperature of air compressed isentropically from 4 MPa and 300 K to 9 MPa, when air is assumed as an ideal gas with variable specific heats, we use the isentropic relations for ideal gases. An isentropic process means that entropy is constant, and for an ideal gas with variable specific heats, the relation between temperature and pressure during an isentropic process can be described by:
T_2 / T_1 = (P_2 / P_1)^{(\gamma-1)/\gamma}
Where
T_1 = initial temperature,
T_2 = final temperature,
P_1 = initial pressure,
P_2 = final pressure,
\gamma = ratio of specific heats, which varies with temperature for an ideal gas with variable specific heats.
To find T_2, we need to know the specific heats of air at different temperatures, or we can use tables or specialized software to find the exact value of \gamma at the given conditions and calculate the final temperature using the above relation. With 300 K as our T_1, 4 MPa as our P_1, and 9 MPa as our P_2, the only missing variable is \gamma. Without this value, we cannot complete the calculation correctly.
Since the initial temperature and final pressure values are known, the final temperature can be found using appropriate thermodynamic tables or software that take into account the variable specific heats of the gas.