The ratio $\frac{K_P}{K_C}$ for the reaction:
- $(RT)^{1/2}$
- RT
- 1
- $\frac{1}{\sqrt{RT}}$
The Correct Option is D
Approach Solution - 1
The problem asks to determine the ratio of the equilibrium constant in terms of partial pressure (\(K_p\)) to the equilibrium constant in terms of molar concentration (\(K_c\)) for the reaction: \(CO_{(g)} + \frac{1}{2}O_{2(g)} \rightleftharpoons CO_{2(g)}\).
Concept Used:
The relationship between \(K_p\) and \(K_c\) for a reversible gaseous reaction is given by the equation:
\[ K_p = K_c (RT)^{\Delta n_g} \]where:
\(R\) is the ideal gas constant.
\(T\) is the absolute temperature in Kelvin.
\(\Delta n_g\) is the change in the number of moles of gaseous components, calculated as the difference between the sum of the stoichiometric coefficients of the gaseous products and the sum of the stoichiometric coefficients of the gaseous reactants.
\[ \Delta n_g = (\text{Total moles of gaseous products}) - (\text{Total moles of gaseous reactants}) \]Step-by-Step Solution:
Step 1: Identify the stoichiometric coefficients of the gaseous reactants and products from the given balanced equation.
The reaction is:
\[ CO_{(g)} + \frac{1}{2}O_{2(g)} \rightleftharpoons CO_{2(g)} \]The stoichiometric coefficient of the gaseous product (\(CO_2\)) is 1.
The sum of the stoichiometric coefficients of the gaseous reactants (\(CO\) and \(O_2\)) is \(1 + \frac{1}{2} = \frac{3}{2}\).
Step 2: Calculate the value of \(\Delta n_g\).
Using the formula for \(\Delta n_g\):
\[ \Delta n_g = (1) - \left(1 + \frac{1}{2}\right) \] \[ \Delta n_g = 1 - \frac{3}{2} \] \[ \Delta n_g = -\frac{1}{2} \]Step 3: Substitute the value of \(\Delta n_g\) into the relationship \(K_p = K_c(RT)^{\Delta n_g}\).
\[ K_p = K_c (RT)^{-1/2} \]Step 4: Rearrange the equation to find the required ratio \(\frac{K_p}{K_c}\).
\[ \frac{K_p}{K_c} = (RT)^{-1/2} \]Final Computation & Result:
The expression can be simplified as follows:
\[ \frac{K_p}{K_c} = \frac{1}{(RT)^{1/2}} = \frac{1}{\sqrt{RT}} \]Therefore, the ratio \(\frac{K_p}{K_c}\) for the given reaction is \(\frac{1}{\sqrt{RT}}\). This corresponds to option (4).
Approach Solution -2
\(\text{CO}_{(g)}\) + \(\frac{1}{2}\)\(\text{O}_2{(g)}\) \(\rightarrow\) \(\text{CO}_2{(g)}\)
\( \Delta n_g = 1 - \left( 1 + \frac{1}{2} \right) = -\frac{1}{2} \)
\( \frac{K_P}{K_C} = (RT)^{\Delta n_g} = \frac{1}{\sqrt{RT}} \)
Thus the correct answer is option 4.
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