Question

At \(87^{\circ} \mathrm{C}\), the following equilibrium is established. $$ \mathrm{H}_{2}(g)+\mathrm{S}(s) \rightleftharpoons \mathrm{H}_{2} \mathrm{~S}(g) ; K_{c}=0.08 $$ If \(0.3\) mole hydrogen and 2 mole sulphur are heated to \(87^{\circ} \mathrm{C}\) in a \(2 \mathrm{~L}\) vessel, what will be the concentration of \(\mathrm{H}_{2} \mathrm{~S}\) at equilibrium? (a) \(0.011 \mathrm{M}\) (b) \(0.022 \mathrm{M}\) (c) \(0.044 \mathrm{M}\) (d) \(0.08 \mathrm{M}\)

Short answer

The concentration of \( H_2S \) at equilibrium is 0.011 M.

Step-by-step solution

Write the Equlibrium Expression

Write the equilibrium constant expression based on the chemical reaction given: $$ K_{c} = \frac{[H_2S]}{[H_2]} $$Note that solid sulphur (\text{S}) does not appear in the expression as its concentration is constant and thereby included in the value of the equilibrium constant.

Determine the Initial Concentrations

Calculate the initial concentration of hydrogen since the volume of the vessel is 2 L:$$ [H_2]_{initial} = \frac{0.3 \text{ mole}}{2 \text{ L}} = 0.15 \text{ M} $$Sulphur, being a solid, does not have a concentration in the expression.

Set Up the ICE Table

Set up an ICE (Initial, Change, Equilibrium) table to determine the concentrations at equilibrium. Let 'x' be the change in concentration of \(H_2\) that reacts to form \(H_2S\).| | \( [H_2] \) | \( [H_{2}S] \) || Initial (M) | 0.15 | 0 || Change (M) | -x | +x || Equilibrium (M) | 0.15 - x | x |

Use the Equilibrium Constant

Plug the equilibrium concentrations into the equilibrium expression and solve for 'x'.$$ K_{c} = 0.08 = \frac{x}{0.15 - x} $$

Solve the Quadratic Equation

Rearrange and solve the quadratic equation:$$ 0.08(0.15 - x) = x \$$ $$ 0.012 - 0.08x = x \$$ $$ 0.012 = 1.08x \$$ $$ x = \frac{0.012}{1.08} \$$ $$ x = 0.011111... $$

Find the Equilibrium Concentration of H2S

The equilibrium concentration of \(H_2S\) is equal to 'x' which we have found to be 0.011111..., which can be rounded to 0.011 M.

Key concepts

Equilibrium Constant Expression

Understanding equilibrium in a chemical reaction is crucial when predicting the concentrations of reactants and products. The equilibrium constant expression is a mathematical formula that relates the concentrations of the products and reactants at equilibrium. For a general reaction where reactants A and B form products C and D, the equilibrium constant expression, denoted as Kc, is given by:
\[ K_{c} = \frac{[C]^{c}[D]^{d}}{[A]^{a}[B]^{b}} \]
where [A], [B], [C], and [D] represent the molar concentrations of the reactants and products, and a, b, c, and d are their respective stoichiometric coefficients in the balanced chemical equation. It's important to note that the concentrations of pure solids and liquids do not appear in this expression as their concentrations are typically constant.
In our specific problem, the equilibrium constant expression for the formation of hydrogen sulfide gas from hydrogen gas and solid sulfur is simplified to：
\[ K_{c} = \frac{[H_2S]}{[H_2]} \]
This expression provides the essential foundation upon which we can solve equilibrium problems and predict how much product will form from given reactants.

ICE Table Method

The ICE table method is a systematic approach to solving equilibrium problems. ICE stands for Initial, Change, and Equilibrium, representing the different stages of the reaction. The table helps us to organize and calculate the changes in concentration or pressure of the reactants and products from the start of the reaction to the point of equilibrium.
The initial row represents the initial concentrations (or pressures for gases) before the reaction reaches equilibrium. The change row indicates the amount of reactants and products that change as the reaction approaches equilibrium. Typically, the change will be represented by a variable 'x' to show the increase or decrease in concentration or pressure of the reactants and products. The equilibrium row shows the final concentrations or pressures at equilibrium, expressed in terms of 'x'.
By setting up an ICE table, you can apply the equilibrium constant expression to find the value of 'x', which gives you the equilibrium concentrations necessary to solve the problem at hand.

Le Chatelier's Principle

Le Chatelier's Principle is a fundamental concept in chemical equilibrium that predicts the response of a system when subjected to changes in concentration, temperature, volume, or pressure. The principle states that if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium will shift to counteract the change.
For instance, if the concentration of a reactant is increased, the system will shift to form more products, moving the equilibrium position to the right. Conversely, if the concentration of a product is increased, the reaction will shift towards the reactants, moving the equilibrium position to the left. Similarly, changes in temperature and pressure can also cause shifts in the equilibrium position. Understanding how these shifts affect the equilibrium is essential for controlling chemical reactions and optimizing product yields.

Solving Equilibrium Problems

Solving equilibrium problems involves the application of the equilibrium constant expression and the ICE table method, often supplemented by Le Chatelier's Principle for dynamic changes. The process typically includes the following steps:

• Write the balanced chemical equation for the reaction.
• Determine the equilibrium constant expression from the balanced equation.
• Calculate the initial concentrations of reactants and products.
• Set up an ICE table to delineate the initial concentrations, the changes that occur as the system reaches equilibrium, and the final equilibrium concentrations.
• Substitute the equilibrium concentrations into the equilibrium constant expression to solve for the unknown values.
• Apply Le Chatelier’s Principle if there is a change in reaction conditions to predict the direction of the shift in equilibrium.

In the example problem provided, by following these steps, we found the equilibrium concentration of \(H_2S\) to be 0.011 M, matching answer (a). This methodical approach ensures that we thoroughly understand the equilibrium process and accurately predict the outcome of the chemical reaction.