Entropy with non constant specific heats

[Goatsenator]

Homework Statement

Consider air inside a piston-cylinder device that is compressed adiabatically from p1 = 100kPa, T1 = 300 K, and V1 = 0.5 m^3 to V2 = 0.03 m^3 and then heated at constant pressure until T3 = 2000 K. The heat is exchanged with a heat reservoir at TH = 2000 K. Determine P2, T2, P3, V3, W1-2, W2-3, Q2-3, and ∆1−2, without assuming constant specific heats, instead using data given in Table A7.1 in the text.

The table just gives standard entropy.

Homework Equations

I really just need to find T2 and can get everything else from there. The equation the professor gave was

S0(T2) - S0(T1) = R*ln(V2/V1) + R*ln(T2/T1)

where S0 is the standard entropy. I derived this myself as well and understand where it comes from.

The Attempt at a Solution

I tried to substitute S0(T1) + R*ln(P2/P1) for S0(T2) but then it ends up cancelling T2 in the equation so I have no idea what to do.

 

[KataKoniK]

Thank you for your post. To solve for T2, we can use the following equation:

T2 = T1*(V1/V2)^(gamma-1)

where gamma is the ratio of specific heats, which can be found in Table A7.1 in the text. We can also use this equation to solve for P2:

P2 = P1*(V1/V2)^gamma

To find the remaining values, we can use the first law of thermodynamics: ∆U = Q - W. We know that the process is adiabatic, so Q1-2 = 0. Therefore, we can solve for W1-2 and ∆U1-2 using the following equations:

W1-2 = ∆U1-2 = C_v*(T2-T1)

where C_v is the specific heat at constant volume, which can also be found in Table A7.1. We can then use the first law of thermodynamics again to solve for Q2-3:

Q2-3 = ∆U2-3 + W2-3 = C_p*(T3-T2) + P2*(V3-V2)

where C_p is the specific heat at constant pressure, also found in Table A7.1. Finally, we can use the ideal gas law to solve for P3:

P3 = nRT3/V3

where n is the number of moles of gas, R is the gas constant, and T3 and V3 are given in the problem. I hope this helps you solve the problem. Good luck with your calculations!