Question

Relationship of \(K_{c}\) and \(K_{\mathrm{p}}\) (a) \(K_{\mathrm{p}}\) for the following reaction is 0.16 at \(25^{\circ} \mathrm{C}\) What is the value of \(K_{c} ?\) $$ 2 \mathrm{NOBr}(\mathrm{g}) \rightleftharpoons 2 \mathrm{NO}(\mathrm{g})+\mathrm{Br}_{2}(\mathrm{g}) $$ (b) The equilibrium constant, \(K_{c}\) for the following reaction is 1.05 at 350 K. What is the value of \(K_{\mathrm{p}} ?\) $$ 2 \mathrm{CH}_{2} \mathrm{Cl}_{2}(\mathrm{g}) \rightleftharpoons \mathrm{CH}_{4}(\mathrm{g})+\mathrm{CCl}_{4}(\mathrm{g}) $$

Short answer

a) \( K_c = 0.0067 \) b) \( K_p = 1.05 \)

Step-by-step solution

Understand the Relationship Between Kc and Kp

The equilibrium constants Kc and Kp are related to each other by the equation: \[ K_{p} = K_{c} (RT)^{\Delta n} \] where \( R \) is the ideal gas constant (0.0821 L atm / K mol), \( T \) is the temperature in Kelvin, and \( \Delta n \) is the change in moles of gas between products and reactants.

Calculate Δn for Reaction (a)

For the reaction \( 2\, \text{NOBr}(\text{g}) \rightleftharpoons 2\, \text{NO}(\text{g}) + \text{Br}_2(\text{g}) \), the number of moles of gas in products is 2 + 1 = 3, and in reactants, it is 2. Therefore, \( \Delta n = 3 - 2 = 1 \).

Calculate Kc for Reaction (a)

We have \( K_{p} = 0.16 \), \( \Delta n = 1 \), and \( T = 25^{\circ}C = 298\,K \). Substitute into the formula: \[ K_{c} = \frac{K_{p}}{(RT)^{\Delta n}} = \frac{0.16}{(0.0821 \times 298)^{1}} \] Calculate the denominator and solve for \( K_{c} \).

Calculate Δn for Reaction (b)

For the reaction \( 2\, \text{CH}_2\text{Cl}_2(\text{g}) \rightleftharpoons \text{CH}_4(\text{g}) + \text{CCl}_4(\text{g}) \), the number of moles of gas in products is 1 + 1 = 2, and in reactants, it is 2. Therefore, \( \Delta n = 2 - 2 = 0 \).

Calculate Kp for Reaction (b)

Since \( \Delta n = 0 \), the equation simplifies to \( K_{p} = K_{c} \), which means \( K_{p} = 1.05 \) because the exponential term \((RT)^{0}\) equals 1.

Key concepts

relationship_between_Kc_and_Kp

The relationship between the equilibrium constants, \( K_c \) and \( K_p \), is fundamental in understanding chemical equilibrium in gaseous reactions. These constants describe the ratio of concentrations (\( K_c \)) and pressures (\( K_p \)) of the products and reactants at equilibrium. The equation that connects \( K_c \) and \( K_p \) is:\[ K_{p} = K_{c} (RT)^{\Delta n} \]Where:

• \( R \) is the ideal gas constant, specifically 0.0821 L atm / K mol when dealing with pressure in atm.
• \( T \) is the temperature in Kelvin.
• \( \Delta n \) represents the change in moles of gaseous reactants and products.

This equation highlights the influence of temperature and the change in the number of moles on the equilibrium state. If \( \Delta n \) is zero, as in reaction (b), \( K_p \) equals \( K_c \) because the term \((RT)^{0}\) becomes 1.

ideal_gas_constant

The ideal gas constant is a key component in many equations in chemistry, particularly when dealing with gases. Known as \( R \), the ideal gas constant provides a link between volume, pressure, temperature, and the number of moles. Its value is typically given as 0.0821 L atm / K mol in contexts involving pressure in atmospheres and volume in liters.
When using the equation connecting \( K_c \) and \( K_p \), \( R \) helps quantify the effect of temperature and changes in volume and pressure according to the ideal gas law. It's important to ensure \( R \) is consistent with the units of pressure being used, to maintain accuracy in calculations.

change_in_moles

The change in moles, denoted as \( \Delta n \), plays a crucial role in the relationship between \( K_c \) and \( K_p \). It represents the difference in the number of moles of gaseous products and reactants. To determine \( \Delta n \), simply subtract the moles of gaseous reactants from the moles of gaseous products.
For example, in reaction (a), \( \Delta n = 3 - 2 = 1 \) since there are 3 moles of gaseous products and 2 moles of gaseous reactants. This information affects how we adjust \( K_c \) to find \( K_p \) and vice versa.
Additionally, the value of \( \Delta n \) determines whether the change in equilibrium favors the products or reactants when pressures change. If products exceed reactants, more pressure can shift equilibrium towards reactants, unless the reaction is exothermic and temperature is increasing.