Question

(a) Calculate the reaction Gibbs free energy of \(\mathrm{N}_{2}\) (g) \(+\) \(3 \mathrm{H}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{NH}_{3}(\mathrm{~g})\) when the partial pressures of \(\mathrm{N}_{2}, \mathrm{H}_{2}\), and \(\mathrm{NH}_{3}\) are \(4.2\) bar, \(1.8\) bar, and 21 bar, respectively, and the temperature is 400 . K. For this reaction, \(K=41\) at 400 . K. (b) Indicate whether this reaction mixture is likely to form reactants, is likely to form products, or is at equilibrium.

Short answer

\(\Delta G\) is calculated using \(\Delta G^{\circ}\) and \(Q\). If \(\Delta G < 0\), the reaction forms products; if \(\Delta G > 0\), it forms reactants; if \(\Delta G \approx 0\), it is at equilibrium.

Step-by-step solution

Write the Reaction and Expression for Reaction Gibbs Free Energy

The reaction under consideration is \(\mathrm{N}_{2}(g) + 3\mathrm{H}_{2}(g) \longrightarrow 2\mathrm{NH}_{3}(g)\). The Gibbs free energy change (\(\Delta G\)) for this reaction at any point in time can be given by the equation \(\Delta G = \Delta G^{\circ} + RT\ln Q\), where \(Q\) is the reaction quotient and \(\Delta G^{\circ}\) is the standard Gibbs free energy change, which is related to the equilibrium constant \(K\) at temperature \(T\) by \(\Delta G^{\circ} = -RT\ln K\).

Calculate Standard Gibbs Free Energy Change (\(\Delta G^{\circ}\))

Using the given equilibrium constant (\(K = 41\)) and the temperature (\(T = 400\) K), calculate the standard Gibbs free energy change using the formula \(\Delta G^{\circ} = -RT\ln K\). The gas constant (\(R\)) is \(8.314 J/(mol\cdot K)\).

Calculate the Reaction Quotient (\(Q\))

The reaction quotient \(Q\) is calculated using the partial pressures of the reactants and products at any given time: \(Q = \frac{(P_{\mathrm{NH}_{3}})^2}{(P_{\mathrm{N}_{2}})(P_{\mathrm{H}_{2}})^3}\). Substitute the given partial pressures into this expression to find the value of \(Q\).

Calculate the Gibbs Free Energy Change (\(\Delta G\))

Now you can determine the Gibbs free energy change \(\Delta G\) for the reaction using the equation \(\Delta G = \Delta G^{\circ} + RT\ln Q\) with \(\Delta G^{\circ}\) from Step 2 and \(Q\) from Step 3.

Analyze the Result

If \(\Delta G\) is negative, the reaction will proceed towards the products. If it is positive, the reaction will proceed towards the reactants. If \(\Delta G\) is approximately zero, the reaction is at equilibrium.

Key concepts

Chemical Equilibrium

Understanding chemical equilibrium is essential when dealing with reactions similar to the synthesis of ammonia from nitrogen and hydrogen. At chemical equilibrium, the rate at which the reactants are converted into products is equal to the rate at which products revert to reactants. This dynamic state does not imply that the reactants and products are present in equal amounts, but instead that their concentrations remain constant over time.

To visualize this, imagine a busy road where the number of cars entering is exactly the same as the number leaving. The number of cars on the road stays constant, not because there's no movement, but because there's a balance. The same goes for a chemical reaction at equilibrium; the reactants and products are constantly interconverting, but their amounts don’t change.

Equilibrium Constant

The equilibrium constant (\( K \)) is a number that quantifies the balance between products and reactants in a reaction at equilibrium. It is determined at a specific temperature and remains unchanged unless the temperature changes.

For the given reaction, the value of the equilibrium constant is 41 at 400 K. To interpret this, let's think of the equilibrium constant like the score in a sports game. A higher 'score' for products over reactants means the game is leaning towards a winning side; here, it means the formation of ammonia is favored in this reaction when it is at equilibrium.

Reaction Quotient

The reaction quotient (\( Q \)) is a snapshot of a reaction’s progress before it reaches equilibrium. Like the equilibrium constant, it involves concentrations or partial pressures of reactants and products, but those don't have to be at equilibrium values. It serves as a predictor of which direction the reaction will shift to reach equilibrium.

If we compare this to a race, the reaction quotient tells us whether a reaction is ahead, behind, or right at the finish line when compared to its 'pace' at equilibrium. For our ammonia synthesis, if the reaction quotient is different from the equilibrium constant, it tells us if the reaction will proceed toward making more products or revert to reactants to achieve equilibrium.

Gibbs Free Energy Change

Lastly, the Gibbs free energy change (\( \triangle G \)) is a predictor of the spontaneity of the reaction. It's like a 'force' that drives a chemical reaction towards equilibrium. A negative value for \( \triangle G \) indicates that a reaction will proceed forward, meaning it favors the formation of products. Conversely, a positive \( \triangle G \) suggests the reaction will shift towards reactants. If \( \triangle G \) is zero, the reaction mixture is at equilibrium, and no net change will occur.

By calculating \( \triangle G \) and considering \( K \), we can determine not only the direction a reaction will spontaneously proceed but also gain insights into the degree of favorability of the products or reactants under the given conditions.