Answer
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Hint :We know that $ {K_c} $ is equal to the ratio of concentration of products to the concentration of reactants. Finding the value of $ {K_c} $ for the considered reaction and then substituting it in the ideal gas equation provides us the relation between $ {K_p} $ and $ {K_c} $ . Also, remember that $ {K_p} $ is equal to the ratio of partial pressure of products to the partial pressure of reactants.
Complete Step By Step Answer:
This question belongs to the concept of chemical equilibrium. Let us see the basic terminology used in this question.
Here we have to find the relation between $ {K_p} $ and $ {K_c} $ . But let us first get an idea of what $ {K_p} $ and $ {K_c} $ are,
$ {K_p} $ is the equilibrium constant of an ideal gas mixture. Specifically it is the equilibrium constant which is used when the concentration of a given mixture or gas is expressed in terms of pressure. Whereas $ {K_c} $ is also the equilibrium constant for an ideal gas mixture but it is used when the concentrations of the ideal gas mixture are expressed in terms of molarity .
So, in order to find the relation between $ {K_p} $ and $ {K_c} $ we will take a gaseous reaction at equilibrium.
Let the gaseous reaction which is in a state of equilibrium is,
$ aA(g) + bB(g) \rightleftharpoons cC(g) + dD(g) $
Let us consider pA, pB, pC and pD as the partial pressure of A,B,C and D respectively.
Therefore,
$ {K_c} = \dfrac{{{{[C]}^c}{{[D]}^d}}}{{{{[A]}^a}{{[B]}^b}}} - - - - (1) $ and
$ {K_p} = \dfrac{{p{C^c}p{D^d}}}{{p{A^a}p{B^b}}} - - - - (2) $
We know that ideal gas equation is,
$ PV = nRT $
$ \Rightarrow P = \dfrac{{nRT}}{V} = CRT $ ,where C is the concentration ( $ C = n/V $ where n is number of moles and V is volume)
Now let us write some relations,
$ pA = [A]RT $
$ pB = [B]RT $
$ pC = [C]RT $
$ pD = [D]RT $
Now let us substitute these values in equation (2), therefore we will get,
$ {K_p} = \dfrac{{{{[C]}^c}{{(RT)}^c}{{[D]}^d}{{(RT)}^d}}}{{{{[A]}^a}{{(RT)}^a}{{[B]}^b}{{(RT)}^b}}} $
$ \Rightarrow {K_p} = \dfrac{{{{[C]}^c}{{[D]}^d}{{(RT)}^{(c + d) - (a + b)}}}}{{{{[A]}^a}{{[B]}^b}}} $
But we know that $ {K_c} = \dfrac{{{{[C]}^c}{{[D]}^d}}}{{{{[A]}^a}{{[B]}^b}}} $ from equation (1), so substituting this in this above equation we get,
$ \Rightarrow {K_p} = {K_c}{(RT)^{(c + d) - (a + b)}} $
Or we can write it as,
$ \Rightarrow {K_p} = {K_c}^{\Delta ng} $
Here, $ \Delta ng = $ Total number of moles of gaseous product $ - $ Total number of moles of gaseous reactant
Hence, we can conclude that the relation between $ {K_p} $ and $ {K_c} $ is $ {K_p} = {K_c}^{\Delta ng} $ .
Note :
$ {K_p} $ and $ {K_c} $ both are equilibrium constant but expressed in different quantities. $ {K_p} $ and $ {K_c} $ are dimensionless because they are ratios of concentrations only. $ {K_p} $ and $ {K_c} $ are equal to each other in a reaction where the number of moles gaseous reactants is equal to the number of moles gaseous products.
Complete Step By Step Answer:
This question belongs to the concept of chemical equilibrium. Let us see the basic terminology used in this question.
Here we have to find the relation between $ {K_p} $ and $ {K_c} $ . But let us first get an idea of what $ {K_p} $ and $ {K_c} $ are,
$ {K_p} $ is the equilibrium constant of an ideal gas mixture. Specifically it is the equilibrium constant which is used when the concentration of a given mixture or gas is expressed in terms of pressure. Whereas $ {K_c} $ is also the equilibrium constant for an ideal gas mixture but it is used when the concentrations of the ideal gas mixture are expressed in terms of molarity .
So, in order to find the relation between $ {K_p} $ and $ {K_c} $ we will take a gaseous reaction at equilibrium.
Let the gaseous reaction which is in a state of equilibrium is,
$ aA(g) + bB(g) \rightleftharpoons cC(g) + dD(g) $
Let us consider pA, pB, pC and pD as the partial pressure of A,B,C and D respectively.
Therefore,
$ {K_c} = \dfrac{{{{[C]}^c}{{[D]}^d}}}{{{{[A]}^a}{{[B]}^b}}} - - - - (1) $ and
$ {K_p} = \dfrac{{p{C^c}p{D^d}}}{{p{A^a}p{B^b}}} - - - - (2) $
We know that ideal gas equation is,
$ PV = nRT $
$ \Rightarrow P = \dfrac{{nRT}}{V} = CRT $ ,where C is the concentration ( $ C = n/V $ where n is number of moles and V is volume)
Now let us write some relations,
$ pA = [A]RT $
$ pB = [B]RT $
$ pC = [C]RT $
$ pD = [D]RT $
Now let us substitute these values in equation (2), therefore we will get,
$ {K_p} = \dfrac{{{{[C]}^c}{{(RT)}^c}{{[D]}^d}{{(RT)}^d}}}{{{{[A]}^a}{{(RT)}^a}{{[B]}^b}{{(RT)}^b}}} $
$ \Rightarrow {K_p} = \dfrac{{{{[C]}^c}{{[D]}^d}{{(RT)}^{(c + d) - (a + b)}}}}{{{{[A]}^a}{{[B]}^b}}} $
But we know that $ {K_c} = \dfrac{{{{[C]}^c}{{[D]}^d}}}{{{{[A]}^a}{{[B]}^b}}} $ from equation (1), so substituting this in this above equation we get,
$ \Rightarrow {K_p} = {K_c}{(RT)^{(c + d) - (a + b)}} $
Or we can write it as,
$ \Rightarrow {K_p} = {K_c}^{\Delta ng} $
Here, $ \Delta ng = $ Total number of moles of gaseous product $ - $ Total number of moles of gaseous reactant
Hence, we can conclude that the relation between $ {K_p} $ and $ {K_c} $ is $ {K_p} = {K_c}^{\Delta ng} $ .
Note :
$ {K_p} $ and $ {K_c} $ both are equilibrium constant but expressed in different quantities. $ {K_p} $ and $ {K_c} $ are dimensionless because they are ratios of concentrations only. $ {K_p} $ and $ {K_c} $ are equal to each other in a reaction where the number of moles gaseous reactants is equal to the number of moles gaseous products.
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