7D1 :  ΔS of H2 in a Compression Process  6 pts 

In a pistonandcylinder device, carbon dioxide (CO_{2}) gas is compressed from 110 kPa and 300K to 1.4
MPa and 640K. Determine the change in the specific entropy of the CO_{2} assuming it behaves as
an ideal gas.
Use… 

a.) The Shomate Heat Capacity Equation b.) The Ideal Gas Entropy Function c.) The NIST Webbook d.) Compare your answers from parts (a) and (b) to your answer in part (c). 

Read :  This problem is an application of the 2nd Gibbs Equation. In part (a) we must evaluate the integal of C_{P} / 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:  T_{1}  300  K  T_{2}  640  K  
P_{1}  110  kPa  P_{2}  1400  kPa  
Find:  DS  ???  kJ/kgK  
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  

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/moleK  MW  44.010  g/mol  
Now, we can substitute values into Eqn 3 to complete part (a) :  

32.351  J/moleK  

21.149  J/moleK  
Now, we can plug values into Eqn 3 :  DS  11.202  J/moleK  
DS  0.2545  kJ/kgK  
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 T_{1}:  S^{o}_{T1}  0.0052249  kJ/kgK  
At T_{2}:  S^{o}_{T2}  0.74030  kJ/kgK  
Now, we can plug values into Eqn 4 :  DS  0.2545  kJ/kgK  
Part c.)  The NIST Webbook yields the following values for the specific entropy of carbon dioxide:  
S_{1}  2.7264  kJ/kgK  
S_{2}  2.9796  kJ/kgK  
Therefore :  DS  0.2532  kJ/kgK  
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 :  V_{1}  22.7  L/mol  
V_{2}  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/kgK  
b.)  DS  0.255  kJ/kgK  
c.)  DS  0.253  kJ/kgK  
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.  