Problem 190 - A Complete Resource Book in Chemistry for JEE Main

Question

For the reaction \(\mathrm{CO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{~g}) \rightleftharpoons-\mathrm{COCl}_{2}(\mathrm{~g})\), the value of \(\mathrm{Kp} / \mathrm{K}_{c}\) is equal to [2004] (a) RT (b) \(1 / \mathrm{RT}\) (c) I (d) \(\sqrt{\mathrm{RT}}\)

Step-by-Step Solution

Answer

The answer is (b) \(1 / \mathrm{RT}\).

Step 1: Understanding the relation between Kp and Kc

For the given reaction \(\mathrm{CO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{g}) \rightleftharpoons \mathrm{COCl}_{2}(\mathrm{g})\), we need to find the relationship between \(K_p\) and \(K_c\). The relation is given by the equation: \[ K_p = K_c (RT)^{\Delta n} \] where \(\Delta n\) is the change in the number of moles of gas from reactants to products.

Step 2: Calculate Δn

To find \(\Delta n\), we calculate the difference in moles of gaseous products and reactants: - Products: 1 mole of \(\mathrm{COCl}_{2}(\mathrm{g})\)- Reactants: 1 mole of \(\mathrm{CO}(\mathrm{g})\) and 1 mole of \(\mathrm{Cl}_{2}(\mathrm{g})\) making \(1+1=2\) moles total.\[ \Delta n = (\text{moles of products}) - (\text{moles of reactants}) = 1 - 2 = -1 \]

Step 3: Applying Δn to Kp and Kc relationship

Now apply \(\Delta n = -1\) to the equation relating \(K_p\) and \(K_c\): \[ K_p = K_c (RT)^{-1} \] This simplifies to: \[ \frac{K_p}{K_c} = (RT)^{-1} \]

Step 4: Determine the Correct Answer Option

From the equation \(\frac{K_p}{K_c} = \frac{1}{RT}\), we identify the correct answer as option (b) \(1 / \mathrm{RT}\).

Key Concepts

Equilibrium Constant

The equilibrium constant is a crucial concept in understanding chemical reactions that reach a state of equilibrium. In such reactions, the rate of the forward reaction equals the rate of the reverse reaction, resulting in constant concentrations of reactants and products over time. The equilibrium constant, denoted as \( K_c \) when referring to concentrations and \( K_p \) when referring to partial pressures, provides valuable information about the reaction at equilibrium.

For a general reaction where \((aA + bB \rightleftharpoons cC + dD)\), the equilibrium constant expression is based on the law of mass action:

• \( K_c = \frac{[C]^c [D]^d}{[A]^a [B]^b} \)
• \( K_p = \frac{(P_C)^c (P_D)^d}{(P_A)^a (P_B)^b} \)

Where \([C]\) and \((P_C)\) represent the concentration and partial pressure of species \(C\), respectively.

These constants provide insights into the proportions of products and reactants at equilibrium, indicating whether products or reactants are favored. A high \( K_c \) or \( K_p \) value means the reaction favors product formation, while a low value indicates it favors the reactants. Understanding these relationships allows chemists to predict the direction of reactions and optimize conditions for desired outcomes.

Relation between Kp and Kc

The relationship between \( K_p \) and \( K_c \) offers insight into how changes in gas volume and temperature affect chemical reactions involving gases. The two constants are related through the equation:\[ K_p = K_c (RT)^\Delta n \]Here, \( R \) represents the universal gas constant, and \( T \) is the temperature in Kelvin. The exponent \( \Delta n \) is the change in moles of gases, calculated as the difference between the moles of gaseous products and reactants.

In the example reaction \( \mathrm{CO} + \mathrm{Cl}_2 \rightleftharpoons \mathrm{COCl}_2 \), the change in moles is \( \Delta n = 1 - 2 = -1 \). Substituting this into our equation reveals: \[ \frac{K_p}{K_c} = (RT)^{-1} \]This relationship implies that \( K_p \) and \( K_c \) will be affected differently based on the conditions.

If \( \Delta n = 0 \), meaning no change in moles of gas, then \( K_p = K_c \). If \( \Delta n > 0 \), \( K_p \) increases with increasing pressure, and if \( \Delta n < 0 \) as in our example, \( K_p \) decreases with increasing pressure. Understanding this relation helps in predicting how equilibrium shifts with pressure and temperature changes.

Stoichiometry in Chemical Reactions

Stoichiometry involves calculating the quantitative relationships of reactants and products in chemical reactions, serving as the mathematical foundation to describe reaction dynamics. It is crucial for determining how much of each substance is needed or produced in a reaction.

In any balanced chemical equation, the coefficients of reactants and products reveal the precise ratio in which chemicals react and form. For the reaction \( \mathrm{CO} + \mathrm{Cl}_2 \rightleftharpoons \mathrm{COCl}_2 \), the stoichiometry is simple: one mole of carbon monoxide reacts with one mole of chlorine to produce one mole of phosgene gas.
Understanding stoichiometry enables the calculation of \( \Delta n \), the change in moles of gases, which is essential in determining the relationship between \( K_p \) and \( K_c \). Correct stoichiometric coefficients ensure accurate computation of equilibrium constants.

Moreover, stoichiometry is integral for scaling reactions up or down, allowing chemists to predict the outcomes in industrial applications and lab settings alike.