Entropy with non constant specific heats

[Goatsenator]

1. 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 ∆S1−2, without assuming constant specific heats, instead using data given in Table A7.1 in the text.

The table just gives standard entropy. 2. Homework Equations 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. 3. The Attempt at a Solution

I tried to substitute S0(T1) + R*ln(P2/P1) for S0(T2) but that doesn't seem to help.

 

[KataKoniK]

Hello,

Thank you for sharing your attempt at a solution. I can see that you are on the right track in using the equation relating entropy to pressure and temperature. However, in this problem, we are not assuming constant specific heats, so the equation you have used may not be applicable. Instead, we can use the First Law of Thermodynamics to solve for the various parameters.

First, let's consider the adiabatic compression from state 1 to state 2. Since the process is adiabatic, we know that Q1-2 = 0. Therefore, we can use the First Law of Thermodynamics to write:

W1-2 = ΔU1-2 = m(Cv1)(T2-T1)

where m is the mass of the air, Cv1 is the specific heat at constant volume for air at state 1, and T2-T1 is the change in temperature during the adiabatic compression.

Next, let's consider the heating at constant pressure from state 2 to state 3. Since the process is at constant pressure, we know that:

W2-3 = PΔV2-3 = P(V3-V2)

where P is the constant pressure during this process and V3-V2 is the change in volume during the heating.

We can also use the First Law of Thermodynamics to write:

Q2-3 = ΔU2-3 + W2-3 = m(Cp2)(T3-T2) + P(V3-V2)

where Cp2 is the specific heat at constant pressure for air at state 2, and T3-T2 is the change in temperature during the heating process.

Now, we can use the given data in Table A7.1 to determine the specific heats at constant volume and pressure for air at states 1 and 2. From the table, we can see that:

Cv1 = 0.718 kJ/kg-K
Cp2 = 1.005 kJ/kg-K

We can also use the ideal gas law to relate pressure, temperature, and volume at each state:

P1V1/T1 = P2V2/T2 = P3V3/T3

Using this, we can solve for the unknown parameters in terms of the given parameters. I will leave it to you to plug in the numbers and solve for the values.

I hope this helps. Good luck with your problem