Is This Stoichiometry Problem Truly Unsolvable?

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SUMMARY

The stoichiometry problem involving the hydrogenation of toluene (C6H5CH3) is solvable through a systematic approach using molar balances and reaction extents. The two simultaneous reactions yield benzene (C6H6) and biphenyl ((C6H5)2) while consuming hydrogen (H2). Initial conditions of 40% toluene and 60% H2 lead to final conditions of 10% toluene and 30% CH4, allowing for the calculation of mole fractions and conversion rates. The discussion confirms that the problem can be approached by assuming an initial mole amount and solving the resulting equations.

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Homework Statement



Toluene (C6H5CH3) is hydrogenated according to the two simultaneous
chemical reactions
C6H5CH3 + H2 → C6H6 + CH4
2C6H5CH3 + H2 → (C6H5)2 + 2CH4
Initially, the reactor contains 40% toluene and 60% H2 (% mole). At the end of the operation, the reactor contains 10% toluene and 30% CH4 (% mole). Find:

a. The mole fraction of the diphenyl, (C6H5)2, at the end of the operation
b. The conversion of H2

Homework Equations



I came up with the following equations from the molar balances(a,b and c are the final mole fractions of H2, Benzene and Biphenyl, respectively, ε1 and ε2 are the extents of reactions):

0.1=(0.4*n1-ε1-2*ε2)/n2
a=(0.6*n1-ε1-ε2)/n2
b=ε1/n2
0.3=(ε1+2*ε2)/n2
c=ε2/n2

The Attempt at a Solution



The final amount of moles n2 is related to the initial amount by the following equation:

n2=n1=0.4*n1-ε1-2*ε2+0.6*n1-ε1-ε2+ε1+ε1+2*ε2+ε2

Now I have five equations with six unknowns but I can suppose an initial amount of moles of 10 and then solve the equations with five unknowns. However this system of equations is unsolvable. I'm studying stoichiometry and found this problem inside a reactor design book. Any help would be welcome. Thanks.
 
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I agree. Both reactions convert 1 toluene molecule to 1 CH4 molecule and the total number of molecules is conserved - the given concentration of CH4 is redundant.

The two possible extreme cases are:
Just reaction one, 30% H2, 30% C6H6.
Just reaction two, 45% H2, 15% (C6H5)2
And everything in between is possible, too.
 

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