3D1 :  Calculating and Using the Heat Capacities of Ideal Gas Mixtures  4 pts 

Three ideal gases, Nitric Oxide (NO), Carbon Monoxide (CO), and Oxygen (O_{2}), at 220 kPa and 350^{o}C are held in a tank with three chambers, as shown below.  


The dividers between the chambers are removed and the three gases are allowed to mix.The mixture contains 30 mole% NO, 50 mole% CO, and 20 mole% O_{2}. The mixture is then heated to 735^{o}C.  
Calculate the ΔU, in J/mole, of the mixture for the heating process. Assume the mixture is an ideal gas.  
Read :  The key to this problem is that enthalpy does not depend on pressure for an ideal gas. So, the initial and final pressures are not relevant. We want to determine the change in the internal energy, but only the constant pressure heat capacities are tabulated. We can either use C_{v} = C_{p}  R and then integrate C_{v} with respect to T to get DU or we can integrate C_{p} with respect to T to get DH and then use the definiition of enthalpy to get DU. The final aspect of the problem is that the system contains a mixture. We can either use the mole fractions to determine the constants of the heat capacity polynomial for the mixture and then integrate C_{p} with respect to T one time, or we can integrate C_{p} for each chemical component with respect to T and sum the resulting DH values to get DH for the mixture. Either way, once we have DH, we use the definition of enthalpy to determine DU.  
Diagram:  The figure given in the problem statement is adequate. Just include the initial and final temperatures.  


Given:  P_{1} =  220  kPa  y_{NH3 }=  0.30  mol NO/mol  
T_{1} =  350  ^{o}C =  623.15  K  y_{CH4 }=  0.50  mol CO/mol  
T_{2} =  735  ^{o}C =  1008.15  K  y_{O2 }=  0.20  mol O_{2}/mol  
Find:  DU =  ???  J/mole  
Assumptions:  1  The initial state and the final state are equilibrium states.  
2  There is no change in internal energy or enthalpy due to mixing of the gases.  
3  The pure components and the mixture behave as ideal gases.  
Equations / Data / Solve:  
The internal energy of an ideal gas does not depend on pressure, only on temperature.  
Therefore, the question becomes, what is the change in internal energy from T_{1} = 400 ^{o}C, to T_{2} = 600 ^{o}C.  

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  

T in Kelvin !  Eqn 6  
Nitric Oxide 
Carbon Monoxide 
Oxygen  
Heat Capacity Constants from the NIST WebBook:  298.  1200.  298.  1300.  298.  6000.  
A  23.83491  25.56759  29.659  
B  12.58878  6.09613  6.137261  
R =  8.314  J/mol K  C  1.139011  4.054656  1.186521  
D  1.497459  2.671301  0.09578  
E  0.214194  0.131021  0.219663  
Method #1:  Calculate the constants for the heat capacity polynomial for the gas mixture and then integrate to determine DH for the mixture.  
Mixture  

25.86607  

8.05215  

1.44832  

1.765732  DH_{mix} =  12528  J/mol  

0.085836  DU_{mix} =  9327  J/mol  
Method #2:  Calculate DH and then DU for EACH gas and then compute the molar average DU and DH using the following equations:  

Eqn 7 

Eqn 8  
NO  CO  O_{2}  Mixture  
DH =  12633  12307  12923  12528  J/mol  
DU =  9433  9106  9722  9327  J/mol  
Verify:  Assumptions 1 & 2 cannot be verified from the data given in the problem.  
The ideal gas assumption needs to be verified.  
We need to determine the specific volume and check if : 



Eqn 9  
V_{1}  23.55  L/mol  V_{2}  38.10  L/mol  
The ideal gas assumption is valid because V > 20 L/mole For both the initial and final states.  
Answers :  DU_{mix} =  9327  J/mol 