Question

Consider the following statements: "Consider the reaction $\mathrm{A}(g)+\mathrm{B}(g)\rightleftharpoons \mathrm{C}(g)$, for which at equilibrium $[\mathrm{A}]=2M$ $[\mathrm{B}]=1M$, and $[\mathrm{C}]=4M.$ To a 1 -L container of the system at equilibrium, you add 3 moles of $B$. A possible equilibrium condition is $[\mathrm{A}]=1M,[\mathrm{B}]=3M$, and $[\mathrm{C}]=6M$ because in both cases $K=2.$ " Indicate everything that is correct in these statements and everything that is incorrect. Correct the incorrect statements, and explain.

Short answer

In summary, both initial and final equilibrium conditions and K values in the statements are consistent and correct. Initially, with $[\mathrm{A}]=2M$, $[\mathrm{B}]=1M$, and $[\mathrm{C}]=4M$, K = 2. After adding 3 moles of B, a possible new equilibrium condition is $[\mathrm{A}]=1M$, $[\mathrm{B}]=3M$, and $[\mathrm{C}]=6M$, and K remains 2. There is no incorrect statement in the exercise.

Step-by-step solution

Verify the initial equilibrium condition and K value

We are given the equilibrium concentrations for $A$, $B$, and $C$: $[\mathrm{A}]=2M$, $[\mathrm{B}]=1M$, and $[\mathrm{C}]=4M$. We can calculate the K value using the expression: $K={\displaystyle \frac{[\mathrm{C}]}{[\mathrm{A}][\mathrm{B}]}}$ Plugging in the given values: $K={\displaystyle \frac{4}{2\times 1}}=2$ The initial equilibrium condition and K value are consistent.

Analyze the impact of adding 3 moles of B

Now, let's consider the effect of adding 3 moles of B to the 1-L container. The new concentration of B will be: $[\mathrm{B}]=1+3=4M$ Since $K$ is supposed to remain constant, we should be able to achieve a new equilibrium condition with the same value of K. Recalling: $K={\displaystyle \frac{[\mathrm{C}]}{[\mathrm{A}][\mathrm{B}]}}$ Now we need to verify the proposed new equilibrium condition: $K={\displaystyle \frac{6}{1\times 3}}=2$ The final equilibrium condition ($[\mathrm{A}]=1M$, $[\mathrm{B}]=3M$, and $[\mathrm{C}]=6M$) and K value are also consistent.

Identify the correct and incorrect statements and explain

Given that both initial and final equilibrium conditions as well as K values are consistent, all statements are correct: 1. The initial equilibrium condition is $[\mathrm{A}]=2M$, $[\mathrm{B}]=1M$, and $[\mathrm{C}]=4M$, and K = 2. 2. After adding 3 moles of B to the container, a possible new equilibrium condition is $[\mathrm{A}]=1M$, $[\mathrm{B}]=3M$, and $[\mathrm{C}]=6M$, and K remains 2. There's no incorrect statement in the exercise.

Key concepts

Equilibrium Constant

Understanding the concept of equilibrium constant (K) is crucial in studying chemical reactions that reach a state of equilibrium. In such a state, the rate of the forward reaction equals the rate of the reverse reaction, leading to constant concentrations of reactants and products over time.

The equilibrium constant is a mathematical expression that quantifies the ratio of the concentrations of products to reactants, each raised to the power of their stoichiometric coefficients in the balanced chemical equation. For a generic reaction:
$$aA+bB\rightleftharpoons cC+dD$$
The equilibrium constant ($K$) can be expressed as:
$$K=\frac{[C{]}^c[D{]}^d}{[A{]}^a[B{]}^b}$$
where $[A]$,$[B]$, $[C]$, and $[D]$ represent the molar concentrations of the respective substances.

In the given exercise, the reaction: $$A(g)+B(g)\rightleftharpoons C(g)$$has an equilibrium constant calculated using the initial concentrations provided. By comparing the initial and new equilibrium conditions, we can see the equilibrium constant remains unchanged, indicating that the system has reached equilibrium under both conditions.

Le Chatelier's Principle

Le Chatelier's principle is a fundamental concept in chemistry that predicts how a system at equilibrium responds to changes in concentration, pressure, or temperature. According to this principle, if an external stress is applied to a system at equilibrium, the system will adjust itself to minimize that stress.

For example, increasing the concentration of a reactant or product will shift the equilibrium position to favor the reaction that consumes the increased substance. Conversely, decreasing the concentration of a substance will shift the equilibrium to produce more of it. The principle provides a qualitative understanding of the direction in which a reaction will shift, but it does not quantify the extent of the shift, which is where equilibrium calculations come into play.

When 3 moles of reactant $B$ were added to the reaction in the exercise, Le Chatelier's principle predicted that the reaction would shift to the right to reduce the concentration of $B$ by forming more product $C$. This shift is confirmed by the equilibrium calculations showing new concentrations that result in the same equilibrium constant value.

Concentration Changes in Equilibrium

The impact of concentration changes on a system at equilibrium is not just a theoretical consideration, but a very practical aspect of chemical reactions. When the concentration of a reactant or product in an equilibrium system is changed, the position of equilibrium shifts to counteract this change, as per Le Chatelier's principle.

In the context of our exercise, the addition of 3 moles of $B$ to a 1-L container increased its concentration. This disturbed the equilibrium, and as predicted by Le Chatelier's principle, resulted in a shift towards the products. The system adjusted by converting $A$ and the added $B$ into more $C$ until a new equilibrium was established while maintaining the equilibrium constant $K$, showcasing the dynamic adaptability of equilibrium systems to concentration changes.

Equilibrium Calculations

To comprehensively understand chemical equilibrium, one must be adept at performing equilibrium calculations. These calculations involve applying the equilibrium constant expression to determine the concentrations of reactants and products at equilibrium.

The process typically requires identifying the initial concentrations, applying any changes due to external conditions, such as addition or removal of substances, and finally calculating the new equilibrium concentrations using the equilibrium constant as a guiding reference.

In our example, the initial equilibrium featured concentrations of $2\text{ M}$, $1\text{ M}$, and $4\text{ M}$ for $A$, $B$, and $C$ respectively. After the addition of $B$, the new equilibrium concentrations were shown to be $1\text{ M}$, $3\text{ M}$, and $6\text{ M}$, for $A$, $B$, and $C$, proving that the equilibrium shifted correctly, and upholding the constant nature of $K$ under the given conditions.