Hydrogen
(H_{2}) gas is
compressed from 4.8 bar and 320K to 15.4
bar and 1300K. Determine the change in the specific entropy of the H_{2}, in kJ/kg, assuming the H_{2} behaves as an ideal gas. Use…





a.) The Shomate Heat Capacity Equation 














b.) The Ideal Gas Entropy Function 














c.) with constant Heat
Capacity, C_{P}, determined at 810K and 10.1
bar. 

































Read : 
All equations given
are for the molar change in entropy. Make sure and divide your final answer by the molecular weight of hydrogen to obtain a final answer
as the change in specific entropy, in kJ/kgK. 












Given: 

m 
1 
kg 


Hydrogen 





T_{1} 
320 
K 


T_{2} 
1300 
K 



P_{1} 
4.8 
bar 


P_{2} 
15.4 
bar 













c.) 
T_{C} 
810 
K 


P_{C} 
10.1 
bar 












Find: 
Part (a)  (c) 
DS 
??? 
kJ/(kg K) 
^{} 










^{} 




Diagram: 















Assumptions: 
1  
The system consists of one kg of hydrogen, which behaves as an ideal gas. 












Equations
/ Data / Solve: 



















Part a.) 
Here, we use the
equation given
in the problem statement: 

Eqn 1 












The heat capacity is
determined from the Shomate Equation. 





















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 










11.6093 









D 
2.8444 









E 
0.15967 












Substituting 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 
2.016 
g/mol 













Now, we can substitute
values into Eqn 3 to complete part (a) : 



















41.564 
J/moleK 
















9.6921 
J/moleK 












Now, we can plug
values into Eqn 3 : 


DS 
31.872 
J/moleK 








DS 
15.810 
kJ/kgK 









Part b.) 
In this part of the
problem, we use the equation given in the problem statement: 



















Eqn 4 













Properties are
determined from Ideal Gas Entropy Tables: 
At T_{1}: 
S^{o}_{T1} 
1.0139 
kJ/kgK 







At T_{2}: 
S^{o}_{T2} 
21.631 
kJ/kgK 









Now, we can plug
values into Eqn 4 : 


DS 
15.809 
kJ/kgK 











Part c.) 
Once again, we will
use the equation given in the problem statement: 

















Eqn 5 












Heat
capacity is determined from NIST WebBook: 
At 810 K: 
C^{o}_{P} 
29.679 
J/(mol K) 
















41.604 
J/moleK 













Now, we can plug
values into Eqn 5 : 


DS 
31.912 
J/moleK 








DS 
15.830 
kJ/kgK 







Verify: 
The ideal gas
assumption needs to be verified. 





We need to determine
the specific volume at each state and check if : 










(hydrogen is a diatomic
gas). 


Solving the Ideal Gas EOS for molar volume yields : 

















Conversion Factors: 




1 L = 
0.001 
m^{3} 








1 bar = 
100000 
N/m^{2} 








1 J = 
1 
Nm 













Plugging in values
gives us : 



V_{1} 
5.54 
L/mol 










V_{2} 
7.02 
L/mol 

















The specific volume at each
state is greater than 5 L/mol and therefore the ideal gas assumption is reasonable. 














Answers : 
a.) 
DS 
15.81 
kJ/kgK 





b.) 
DS 
15.81 
kJ/kgK 













c.) 
DS 
15.83 
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: 
0.13% 




















We expected the error to be less than 1% since the molar volumes are greater than 5 L/mole. 































