# Example Problem with Complete Solution

7D-1 : ΔS of H2 in a Compression Process 6 pts
In a piston-and-cylinder device, carbon dioxide (CO2) gas is compressed from 110 kPa and 300K to 1.4 MPa and 640K. Determine the change in the specific entropy of the CO2 assuming it behaves as an ideal gas. Use
a.) The Shomate Heat Capacity Equation
b.) The Ideal Gas Entropy Function
c.) The NIST Webbook

Read : This problem is an application of the 2nd Gibbs Equation.  In part (a) we must evaluate the integal of CP / T dT ourselves using the Shomate Equation.  The results we obtain should very closely match the results we get in part (b) when we use the the Ideal Gas Entropy Function (values obtained form the Ideal Gas Property Tables).  Finally, in part (c) we can determine how much error was introduced by our ideal gas assumption using the NIST Webbook to evaluate the specific entropy change.
Given: T1 300 K T2 640 K
P1 110 kPa P2 1400 kPa
Find: DS ??? kJ/kg-K
Diagram: Assumptions: 1 - The carbon dioxide behaves as an ideal gas.
Equations / Data / Solve:
Part a.) The 2nd Gibbs Equation is the one best suited to this problem because we know the inlet and outlet pressures.  The 2nd Gibbs Equation for ideal gases is: Eqn 1
The heat capacity is determined from the Shomate Equation. Eqn 2
The values of the constants in the Shomate Equation for carbon dioxide are obtained from the NIST WebBook: T (K) 298. - 1200.
A 24.99735 B
55.18696
C -33.69137
D 7.948387
E -0.136638
Substituting the Eqn 2 into Eqn 1 and integrating yields: Eqn 3
We will need the value of the Universal Gas Constant and the molecular weight to determine the change in the specific entropy.
R 8.314 J/mole-K MW 44.010 g/mol
Now, we can substitute values into Eqn 3 to complete part (a) : 32.351 J/mole-K 21.149 J/mole-K
Now, we can plug values into Eqn 3 : DS 11.202 J/mole-K
DS 0.2545 kJ/kg-K
Part b.) In this part of the problem, we use the 2nd Gibbs Equation in terms of the Ideal Gas Entropy Function: Eqn 4
Properties are determined from Ideal Gas Property Tables:
At T1: SoT1 0.0052249 kJ/kg-K
At T2: SoT2 0.74030 kJ/kg-K
Now, we can plug values into Eqn 4 : DS 0.2545 kJ/kg-K
Part c.) The NIST Webbook yields the following values for the specific entropy of carbon dioxide:
S1 2.7264 kJ/kg-K
S2 2.9796 kJ/kg-K
Therefore : DS 0.2532 kJ/kg-K
Verify: The ideal gas assumption needs to be verified.
We need to determine the specific volume at each state and check if : Solving the Ideal Gas EOS for molar volume yields : Plugging in values gives us : V1 22.7 L/mol
V2 3.80 L/mol
The specific volume at state 2 is much less than 20 L/mol, so the ideal gas assumption is questionable at best.
Answers : a.) DS 0.255 kJ/kg-K
b.) DS 0.255 kJ/kg-K
c.) DS 0.253 kJ/kg-K
Comparison:

The results in parts (a) and (b) are identical.  This is not a surprise, assuming we integrated the Shomate Equation correctly !

The error in DS associated with the
ideal gas assumption in this problem is:

0.53%

We expected the error to be greater than 1% since the molar volume is much less than 20 L/mole. 