Question

For the reaction ${{CO(g) + }}\dfrac{{{1}}}{{{2}}}{{{O}}_{{2}}}{{(g)}} \rightleftharpoons {{C}}{{{O}}_{{2}}}{{(g)}}$ , what is the value of ${{{K}}_{{c}}}{{/}}{{{K}}_{{P}}}$ ?
(A) ${{RT}}$
(B) ${{{(RT)}}^{ - 1}}$
(C) ${{{(RT)}}^{\dfrac{{{{ - 1}}}}{{{2}}}}}$
(D) ${{{(RT)}}^{\dfrac{{{1}}}{{{2}}}}}$

Answer 1

In the above question, the reaction is given and we have to find out the ratio ${{{K}}_{{c}}}{{/}}{{{K}}_{{P}}}$ . Since, change in moles of gas can be found out from the reaction, we can find the ratio by using the relationship between ${{{K}}_{{c}}}$ and ${{{K}}_{{p}}}$ .

Formula Used
${{{K}}_{{p}}}{{ = }}{{{K}}_{{c}}}{{{(RT)}}^{{{\Delta n}}}}$
Where ${{{K}}_{{p}}}$ = equilibrium constant considering the partial pressure
${{{K}}_{{c}}}$ = equilibrium constant considering the concentration
R = universal gas constant
T= temperature
${{\Delta n}}$ = change in number of moles of gas.

Complete step by step solution:
${{{K}}_{{p}}}$ is the equilibrium constant which is calculated from the partial pressures of the reaction. It is used to express the relationship between product pressures and reactant pressures. It is a unitless number, although it relates the pressures.
${{{K}}_{{c}}}$ is the equilibrium constant which is calculated from the concentration of the reaction. It is used to express the relationship between product concentration and reactant concentration. It is also a unitless number.
In the above question, we have the following reaction:
${{CO(g) + }}\dfrac{{{1}}}{{{2}}}{{{O}}_{{2}}}{{(g)}} \rightleftharpoons {{C}}{{{O}}_{{2}}}{{(g)}}$
Here, ${{\Delta n}}$ can be calculated as:
${{\Delta n}}$ = number of moles of gaseous product – number of moles of gaseous reactant
Hence,
${{\Delta n}}$ = $1 - \left( {1 + \dfrac{1}{2}} \right) = - \dfrac{1}{2}$
Now we can use the relation:
${{{K}}_{{p}}}{{ = }}{{{K}}_{{c}}}{{{(RT)}}^{{{\Delta n}}}}$
Substituting the value, we get:
${{{K}}_{{p}}}{{ = }}{{{K}}_{{c}}}{{{(RT)}}^{\dfrac{{{{ - 1}}}}{{{2}}}}}$
Rearranging the equation, we get:
$\dfrac{{{{{K}}_{{p}}}}}{{{{{K}}_{{c}}}}}{{ = (RT}}{{{)}}^{\dfrac{{{{ - 1}}}}{{{2}}}}}$
Reciprocating both the sides we get:
$\dfrac{{{{{K}}_{{c}}}}}{{{{{K}}_{{p}}}}}{{ = (RT}}{{{)}}^{\dfrac{{{1}}}{{{2}}}}}$
$\therefore$ The value of ${{{K}}_{{c}}}{{/}}{{{K}}_{{p}}}$ is ${{{(RT)}}^{\dfrac{{{1}}}{{{2}}}}}$ .
Hence, the correct option is option D.

Note:
When ${{\Delta n}}$ =0 , the value of ${{{K}}_{{p}}}{{ = }}{{{K}}_{{c}}}$
While calculating the values of ${{{K}}_{{p}}}$ and ${{{K}}_{{c}}}$ using the above formula, we should have the unit of R in $\dfrac{{{{L}}{{.atm}}}}{{{{mol}}{{.K}}}}$ , i.e., the value of R is equal to ${{0}}{{.08206}}\dfrac{{{{L}}{{.atm}}}}{{{{mol}}{{.K}}}}$ and the temperature should be in kelvin(K).