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In the book of Chemical, biochemical and engineering thermodynamics 4th edition from Stanley I. Sandler (Wiley, 2006), there is a problem about the ammonia producing reaction at page 770.

13.17 The simple statement of the LeChatelier-Braun principle given in Sec. 13.1 leads one to expect that if the concentration of a reactant were increased, the reaction would proceed so as to consume the added reactant. This, however, is not always true. Consider the gas-phase reaction Show that if the mole fraction of nitrogen is less than 0.5, the addition of a small amount of nitrogen to the system at constant temperature and pressure results in the reaction of nitrogen and hydrogen to form ammonia whereas if the mole fraction of nitrogen is greater than 0.5, the addition of a small amount of nitrogen leads to the dissociation of some ammonia to form more nitrogen and hydrogen. Why does this occur?

So I approached this problem the following way:

\begin{align} \ce{N_2 + 3H_2<=>2NH_3} \end{align}

The equilibrium constant:

$$K_c=\frac{[NH_3]^2}{[N_2][H_2]^3}$$

Now when we write this in terms of reactant consumption X:

$$K_c=\frac{(2x)^2}{([N_2]-x)([H_2]-3x)^3}$$

Now, when I use some values to try to prove this. I never get a negative X value, so I am not getting ammonia dissociation? Mind that I am not an expert on chemical thermodynamics and this question was also proposed to us in an industrial chemical engineering class where we discussed the Haber-Bosch process.

Can someone help me out here? Why is this not working? Or what is the right strategy to approach this problem.

Thank you in advance

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    $\begingroup$ In effect, your $x$ is ${1\over2}\ce{[NH3]}$. Well, of course it can never be negative, because the negative concentration of ammonia (or any other substance, for that matter) is physically impossible. $\endgroup$ – Ivan Neretin Jun 7 '16 at 13:57
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    $\begingroup$ I think you should write a mass balance for the initial gas so that you can express $[\ce{N2}]$ and $[\ce{H2}]$ as mole fractions. It is also seems as if the question is asking you to start with an assumed equilibrium mixture of gas and then calculate the change in this equilibrium composition as $y_{\ce{N2}}$ is changed. In other words, you need to find $\frac{d x}{d y_{\ce{N2}}}$. $\endgroup$ – Curt F. Jun 7 '16 at 14:02
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    $\begingroup$ But that is rather easy, you just fill in some values, you get the equilibrium values. Then you take Kc constant. Iteration 1 --> you find the equilibrium, then you adjust the N2 value and look how the equilibrium shifts. Then with the new values, you can adjust N2 again and look how the equilibrium shifts. $\endgroup$ – Thomas Jun 7 '16 at 15:04

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