Question

For the synthesis of ammonia: $$ \mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \rightleftarrows 2 \mathrm{NH}_{3}(g) $$ the equilibrium constant \(K_{\mathrm{c}}\) at \(375^{\circ} \mathrm{C}\) is \(1.2 .\) Starting $$ \text { with }\left[\mathrm{H}_{2}\right]_{0}=0.76 M,\left[\mathrm{~N}_{2}\right]_{0}=0.60 M, \text { and }\left[\mathrm{NH}_{3}\right]_{0}=0.48 $$ \(M\), which gases will have increased in concentration and which will have decreased in concentration when the mixture comes to equilibrium?

Short answer

NH₃ increases, N₂ and H₂ decrease in concentration.

Step-by-step solution

Write the equilibrium expression

For the reaction \( \mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \rightleftarrows 2 \mathrm{NH}_{3}(g) \), the equilibrium constant expression \( K_c \) is given by:\[ K_c = \frac{[\mathrm{NH}_{3}]^2}{[\mathrm{N}_{2}][\mathrm{H}_{2}]^3} \]

Calculate initial reaction quotient (Q)

Substitute the initial concentrations into the reaction quotient \( Q \):\[ Q = \frac{(0.48)^2}{(0.60)(0.76)^3} \]Calculate \( Q \):\[ Q \approx \frac{0.2304}{0.20736} \approx 1.111 \]

Compare Q with Kc

Compare the value of \( Q = 1.111 \) with \( K_c = 1.2 \):\( Q < K_c \), indicates that the reaction will shift to the right (towards products) to reach equilibrium.

Predict changes in concentration

Since the reaction shifts to the right to reach equilibrium, the concentration of \( \mathrm{NH}_{3} \) will increase, while the concentrations of \( \mathrm{N}_{2} \) and \( \mathrm{H}_{2} \) will decrease.

Key concepts

Equilibrium Constant

The equilibrium constant, often represented as \( K_c \), is a vital aspect in understanding chemical equilibrium. It provides us a snapshot of the ratio of product concentrations to reactant concentrations when a reaction has reached equilibrium. For a balanced chemical equation, the equilibrium constant expression depends on the concentrations of the reactants and products. In the case of the synthesis of ammonia:\[ \mathrm{N}_{2}(g) + 3 \mathrm{H}_{2}(g) \rightleftarrows 2 \mathrm{NH}_{3}(g) \]The equilibrium constant \( K_c \) is given by:\[ K_c = \frac{[\mathrm{NH}_{3}]^2}{[\mathrm{N}_{2}][\mathrm{H}_{2}]^3} \]The value of \( K_c \) indicates whether the products or reactants are favored at equilibrium:

• If \( K_c \) is greater than 1, products are favored.
• If \( K_c \) is less than 1, reactants are favored.

In our scenario at \( 375^{\circ} \text{C} \), \( K_c \) is 1.2, showing a moderate favor towards the products.

Reaction Quotient

The reaction quotient, denoted by \( Q \), is used to predict the direction in which a reaction is likely to proceed to reach equilibrium. It is calculated with the same formula as the equilibrium constant, but using the initial concentrations of the reactants and products instead of the equilibrium concentrations.For the ammonia synthesis:\[ Q = \frac{(0.48)^2}{(0.60)(0.76)^3} \]Calculating this gives us a \( Q \approx 1.111 \). By comparing \( Q \) to \( K_c \), we can determine the reaction direction:

• If \( Q \) equals \( K_c \), the system is at equilibrium.
• If \( Q < K_c \), the reaction proceeds in the forward direction (toward products).
• If \( Q > K_c \), the reaction shifts in the reverse direction (toward reactants).

Since \( Q = 1.111 \) is less than \( K_c = 1.2 \), the reaction will shift to the right, moving towards more products.

Le Chatelier's Principle

Le Chatelier's Principle helps us understand how a system at equilibrium responds to external stresses, such as changes in concentration, temperature, or pressure. This principle states that if an equilibrium system is subjected to a change, the system adjusts to partially counteract that change and re-establish equilibrium.Let's apply this to the synthesis of ammonia. When the initial reaction quotient \( Q \) is found to be less than the equilibrium constant \( K_c \), the reaction will shift towards the right to favor the production of ammonia \( \mathrm{NH}_{3} \). This shift is the system's way of reducing the stress induced by having a lower product concentration compared to products at equilibrium.

In essence, by increasing the concentration of \( \mathrm{NH}_{3} \) and decreasing concentrations of \( \mathrm{N}_{2} \) and \( \mathrm{H}_{2} \), the system balances itself again.

Concentration Changes

Understanding concentration changes during a reaction reaching equilibrium is crucial for predicting the outcome. Concentration changes occur as the reaction strives to reach equilibrium from the initial state, where the reaction quotient \( Q \) differs from the equilibrium constant \( K_c \).In our ammonia synthesis example:

• Because \( Q = 1.111 \) is less than \( K_c = 1.2 \), nitrogen \( (\mathrm{N}_{2}) \) and hydrogen \( (\mathrm{H}_{2}) \) will decrease in concentration as they are consumed to produce more ammonia \( (\mathrm{NH}_{3}) \).
• The ammonia concentration will increase until the concentrations adjust to give a \( Q \) equal to the \( K_c \), thus reaching the new equilibrium state.

Analyzing these concentration changes allows us to predict how the mixture of gases will adjust during the reaction course, ensuring precise control in industrial and laboratory settings.