Problem 2.5.5

 

The equilibrium constant, measured in fugacity, for ammonia synthesis reaction at 450 degrees Centigrade and 200 atmospheres is found to be 6.56 times 10-3.  Determine the maximum yield of ammonia in a 1:30 nitrogen-hydrogen mixture.  Critical properties are given as:

 

Critical property

Nitrogen

Hydrogen

Ammonia

Temperature, degrees K

126

41.2

406

Pressure, atm

33.6

20.8

111.6

 

Solution:

 

In the absence Pitzer acentric factors we would use only generalized correlation charts for reading the fugacity coefficients. 

It has been recommended that, we should add 8 atmospheres to the critical pressure of hydrogen and helium to obtain pseudo-critical pressure that if used with pseudo critical temperature, give a better estimate of the fugacity coefficient and compressibility factor.  Thus reduced pressures and reduced temperature values are:

 

Reduced property

Nitrogen

Hydrogen

Ammonia

Temperature

5.74

17.6

1.78

Pressure

5.95

9.62

1.79

 

Fugacity coefficient values can be read as 1.14, 1.09, and 0.91 for nitrogen, hydrogen, and ammonia, respectively.

 

It should be noted that 0.5 moles of nitrogen react with 1.5 moles of hydrogen to produce 1 mole of ammonia.  Thus, the equilibrium constant, expressed in fugacities is equal to (phiNH3) divided by [phiH2]1.5 times [phiN2]0.5.  This results in KPHI of 0.909.

 

As fugacity is equal to phi time pressure, Equilibrium constant expressed in partial pressures, KP, is equal to KF over KPHI. Thus KP is equal to 6.56 times 10-3 over 0.909 or 0.007215.  As partial pressure, P, is equal to mole fraction, y, times total pressure, pT, KP is also equal to KY times pTDELTA N.  Delta N is equal to 1 minus 0.5 minus 1.5 or minus 1.  So, the term pTDELTA N is equal to 200-1 or 0.005.  This gives KY equal to 0.007215 over 0.005 or 1.443.

 

If X moles of ammonia are produced, then, nitrogen is 0.5 minus 0.5 X, and hydrogen is 1.5 minus 1.5 X.  And total number of moles is 2 minus X.  Mole fraction of ammonia is equal to X over (2 minus X).  Plugging the mole fraction values for hydrogen and nitrogen, we obtain X equal to 0.41, or a mole fraction of 0.256.