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

Hint: 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).