3C4 :  Enthalpy Change of N2 Using the IG Heat Capacity  5 pts 

Nitrogen
gas is heated from 450 K to 1120
K. Determine ΔU and ΔH in kJ/kg
by: a.) Integrating the Shomate Equation 

b.) Treating the C_{P} value as a constant, determined using the Shomate
Equation at the average
temperature, 800 K c.) Treating the C_{P} value as a constant, determined using the Shomate Equation at room temperature, 25^{oC} 

Read :  The Shomate Equation will yield the most accurate estimate of the enthalpy change. Assuming a constant value of C_{p} determined at the average temperature should yield a reasonable estimate of DH as well. Using the C_{p} value at room temperature should not be very accurate. We can compare this result to the value we get in part (a).  
Diagram:  A diagram is not necessary for this problem.  
Given:  T_{1}  600  K  T_{2}  1000  K  
Find:  DH_{12} =  ???  kJ/kg  
a.)  Nitrogen behaves as an ideal gas  
b.)  Nitrogen behaves as an ideal gas with a linear relationship between C_{p} and T.  
This is equivalent to using a constant value of C_{p} that is equal to the average of C_{p}(T_{1}) and C_{p}(T_{2}).  
c.)  Nitrogen behaves as an ideal gas with a constant heat capacity.  
Assumptions:  The assumptions are part of the problem statement.  
Equations / Data / Solve:  
Let's begin by collecting the data we will need from the NIST Webbook :  Temp (K)  298.  6000.  
A  26.092  
MW  28.01  g/mole  B  8.218801  
C  1.976141  
D  0.159274  
E  0.044434  
Part a.)  The enthalpy change associated with a temperature change for an ideal gas can be determined from :  

Eqn 1  
The Shomate Equation for the ideal gas heat capacity is :  

Eqn 2  
where : 

Eqn 3  
and : 

Eqn 4  
Combining Eqns 1, 2 and 3 and integrating yields :  


Eqn 5  
Plug in values for the temperatures and the constants to get :  DH  12615  J/mol  

Eqn 6  
DH  450.4  kJ/kg  
We can determine DU using the definition of enthalpy : 

Eqn 7  
For ideal gases, Eqn 7 becomes : 

Eqn 8  
We can then solve Eqn 8 for DU : 

Eqn 9  
R  8.314  J/molK  DU  9289  J/mol  

Eqn 10  
DU  331.6  kJ/kg  
Part b.)  First we need to use the Shomate Equation, Eqns 2 & 3, to evaluate C_{p}(T_{1}) and C_{p}(T_{2}) :  
t_{1}  0.6  C_{p}(T_{1})  30.470  J/molK  
t_{2}  1  C_{p}(T_{2})  32.538  J/molK  
Therefore, the average value of C_{p} is :  C_{p,avg}  31.504  J/molK  
When the heat capacity is a constant, Eqn 1 simplifies to : 

Eqn 11  
DH  12602  J/mol  
DH  449.9  kJ/kg  
DU  331.2  kJ/kg  
This amounts to about 0.1% error relative to the result in part (a).  
Part c.)  We can use Eqns 2 & 3 to evaluate the heat capacity at 25^{o}C or 298.15 K :  
t_{1}  0.29815  C_{p}(298.15K)  28.871  J/molK  
DH  11548  J/mol  
DH  412.3  kJ/kg  
DU  293.6  kJ/kg  
This amounts to almost 9% error relative to the result in part (a).  
That is not acceptable.  
Verify:  No assumptions were made other than the ones in the problem statement.  
Answers :  a.)  DU  332  kJ/kg  DH  450  kJ/kg  
b.)  DU  331  kJ/kg  DH  450  kJ/kg  
c.)  DU  294  kJ/kg  DH  412  kJ/kg 