Question

Consider the following equilibrium: \(2 \mathrm{H}_{2}(g)+\mathrm{S}_{2}(g) \rightleftharpoons 2 \mathrm{H}_{2} \mathrm{~S}(g) \quad K_{c}=1.08 \times 10^{7}\) at \(700^{\circ} \mathrm{C}\) (a) Calculate \(K_{p}\). (b) Does the equilibrium mixture contain mostly \(\mathrm{H}_{2}\) and \(\mathrm{S}_{2}\) or mostly \(\mathrm{H}_{2} \mathrm{~S}\) ?

Short answer

(a) To calculate Kp, we use the equation \(K_{p} = K_{c}(R T)^{\Delta n}\), where \(\Delta n = -1\), R is the ideal gas law constant, and T = 973.15 K. This gives us \(K_p = 87.03\). (b) Since Kc (1.08 x 10^7) is a large number, the reaction strongly favors the products over the reactants. Therefore, the equilibrium mixture contains mostly H2S.

Step-by-step solution

(a) Calculate Kp.

We can use the equation that relates Kp and Kc for a given reaction at a certain temperature: \[K_{p} = K_{c}(R T)^{\Delta n}\] where \(K_c\) is the equilibrium constant in terms of concentrations, \(K_p\) is the equilibrium constant in terms of pressures, R is the ideal gas law constant, T is the temperature in Kelvin, and \(\Delta n\) is the change in the number of moles of gas in the reaction. In this reaction, \(\Delta n = (\textrm{moles of products}) - (\textrm{moles of reactants}) = 2 - (2+1) = -1\) Since the given temperature is in degrees Celsius, we should convert it to Kelvin: \[T(K) = 700 + 273.15 = 973.15 \, K\] Now we can calculate the value of Kp: \[K_{p} = K_{c}(R T)^{\Delta n} = (1.08 \times 10^7)(8.314 \times 973.15)^{-1}\] \[K_{p} = 1.08 \times 10^7 \times (8.058 \times 10^{-6})\] \[K_p = 87.03\]

(b) Determine the composition of the equilibrium mixture

To determine whether the equilibrium mixture contains mostly reactants or products, we can look at the magnitude of the equilibrium constant, Kc. In this case, Kc is a large number (1.08 x 10^7) which implies that the reaction strongly favors the products over the reactants. Thus, the equilibrium mixture contains mostly H2S.

Key concepts

Equilibrium Constant

The equilibrium constant is a central concept in understanding chemical equilibrium. It provides insight into the composition of the system when it has reached equilibrium. In the given chemical reaction, the equilibrium constant is represented as \(K_c\) for concentrations and \(K_p\) for partial pressures.
The relationship between \(K_c\) and \(K_p\) is vital when dealing with gaseous reactions, allowing conversions based on the ideal gas law.
For this reaction, the large value of \(K_c = 1.08 \times 10^7\) indicates that at equilibrium, the mixture contains a much higher concentration of products, in this case, \(\mathrm{H}_2\mathrm{S}\), compared to the reactants, \(\mathrm{H}_2\) and \(\mathrm{S}_2\).
Understanding the equilibrium constant helps predict the direction of the reaction under set conditions, playing a critical role in chemical manufacturing and laboratory settings.

Le Chatelier's Principle

Le Chatelier's Principle is a fundamental concept in chemistry that helps predict how a change in conditions affects the position of equilibrium. It states that if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium shifts to counteract the change.

• If the concentration of a reactant is increased, the system shifts to the right, favoring the formation of products.
• Conversely, increasing the concentration of a product shifts the equilibrium to the left, favoring the formation of reactants.
• Changes in temperature can also affect equilibrium, depending on whether the reaction is exothermic or endothermic.

In the given reaction at 700°C, a high \(K_c\) value suggests the system naturally favors product formation. However, applying Le Chatelier's Principle helps determine what happens when the parameters change.

Gas Laws

Gas laws, such as the ideal gas law, are crucial for understanding the behavior of gases under varying conditions.
The ideal gas law is expressed as \(PV = nRT\), linking pressure \(P\), volume \(V\), moles \(n\), and temperature \(T\) through the ideal gas constant \(R\). This law is instrumental in converting between concentration-based and pressure-based equilibrium constants.
In the calculation of \(K_p\) from \(K_c\), it's essential to consider the change in the number of moles of gas \((\Delta n)\).
Here, \(\Delta n = -1\), as noted in step 1 of the solution. By manipulating the gas laws, chemists can shift between using concentration and pressure to describe a system at equilibrium, enhancing the flexibility and understanding of reactions involving gases.