Example Problem with Complete Solution

7C-1 : ΔS of H2 in a Compression Process 6 pts
Hydrogen gas is compressed from 100 kPa and 300 K to 1 MPa and 400 K.  Assuming that the hydrogen behaves as an ideal gas, determine the change in the specific entropy of the hydrogen for this process using :
               
a.) The Shomate Heat Capacity Equation.
b.) The Ideal Gas Entropy Function.
c.) The NIST Webbook.  Compare your answers from parts (a) and (b) to your answer in part (c).
 
               
Read: Figure 1
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 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 cahnge.
Given: T1 300 K T2 400 K
P1 100 kPa P2 1000 kPa
Find: ΔS ??? kJ/kg-K
Diagram: Equation 1
Assumptions:
1-  The hydrogen 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:
Equation 2
Eqn 1
The heat capacity is determined from the Shomate Equation.
Equation 3 Eqn 2
The values of the constants in the Shomate equation for hydrogen are obtained from the NIST WebBook: T (K) 298. - 1500.
A 33.1078
B -11.508
C 11.6093
D -2.8444
E -0.15967
Substituting the Eqn 2 into Eqn 1 and integrating yields:
Equation 4 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 2.02 g/mol
Now, we can substitue values into Eqn 3 to complete part (a) :
Equation 5
8.3569 J/mole-K
Equation 6
19.1437 J/mole-K
Now, we can plug values into Eqn 3 : ΔS -10.7868 J/mole-K
ΔS -5.3506 kJ/kg-K
Part b.) In this part of the problem, we use the 2nd Gibbs Equationin terms of the Ideal Gas Entropy Function:
Equation 7
Eqn 4
Properties are determined from Ideal Gas Property Tables:
At T1: SoT1 0.08850 kJ/kg-K
At T2: SoT2 4.23378 kJ/kg-K
Now, we can plug values into Eqn 4 : ΔS -5.3506 kJ/kg-K
Part c.) The NIST Webbook yields the following values for the specific entropy of hydrogen:
S1 53.525 kJ/kg-K
S2 48.172 kJ/kg-K
Therefore : ΔS -5.3530 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 : Equation 8
(hydrogen is a diatomic gas).
Solving the Ideal Gas EOS for molar volume yields : Equation 9
Plugging in values gives us : V1 24.9 L/mol
V2 3.33 L/mol
The specific volume at state 2 is less than 5 L/mol, so the ideal gas assumption is questionable.
Answers : a.) ΔS -5.351 kJ/kg-K
b.) ΔS -5.351 kJ/kg-K
c.) ΔS -5.353 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 ΔS associated with the
ideal gas assumption in this problem is:		    
0.045%  
             
We expected the error to be greater than 1% since the molar volume is less than 5 L/mole.