Question

Two solids dissociates as follows:
${\text{A(s)}}\,\, \to {\text{B(g)}} + {\text{C(g)}}$; ${{\text{K}}_{{\text{p1}}}}\,{\text{ = }}\,{\text{x}}\,\,{\text{at}}{{\text{m}}^{\text{2}}}$
${\text{D(s)}}\,\, \to {\text{C(g) + }}\,{\text{E(g)}}$; ${{\text{K}}_{{\text{p2}}}}\,{\text{ = }}\,{\text{y}}\,\,{\text{at}}{{\text{m}}^{\text{2}}}$
The total pressure when both the solids dissociates simultaneously is:
A. $\sqrt {{\text{x}}\,{\text{ + }}\,{\text{y}}\,} {\text{atm}}$
B.$2(\sqrt {{\text{x}}\,{\text{ + }}\,{\text{y}}\,} {\text{)atm}}$
C. $\,{\text{(x + }}\,{\text{y)}}\,{\text{atm}}$
D. ${{\text{x}}^2}{\text{ + }}\,{{\text{y}}^{2\,}}{\text{atm}}$

Answer 1

We have to determine the pressure in terms of equilibrium constant x and y. So, we have to determine the relation between x and y and pressures. To determine the answer we will write the equilibrium equation with pressure. Then we will determine the relation between pressure and equilibrium constant. Both solid decompose simultaneously, so we will add the pressure of each to determine the total pressure.

Complete solution:
The given reactions are as follows:
${\text{A(s)}}\,\, \to {\text{B(g)}} + {\text{C(g)}}$
${\text{D(s)}}\,\, \to {\text{C(g) + }}\,{\text{E(g)}}$
At equilibrium the reaction will be represented as follows:
${\text{A(s)}}\,\, \rightleftharpoons {\text{B(g)}}\, + \,{\text{C(g)}}$
${\text{D(s)}}\,\, \rightleftharpoons {\text{C(g)}}\, + \,{\text{E(g)}}$
We assume that initial pressure of the A is ${{\text{P}}_{\text{1}}}$ and initial pressure of B and C will be zero. At equilibrium all the products and reactant are present in equal amount so, the pressure of all the species will be same so,
$\mathop {\text{A}}\limits_{{{\text{P}}_1}} {\text{(s)}}\,{\text{ + }}\,\mathop {\text{B}}\limits_{{{\text{P}}_1}} {\text{(g)}}\, \rightleftharpoons \mathop {\text{C}}\limits_{{{\text{P}}_1}} {\text{(g)}}$
Similarly, we assume that initial pressure of the A is ${{\text{P}}_2}$ and initial pressure of C and D will be zero. At equilibrium all the products and reactant are present in equal amount so, the pressure of all the species will be same so,
$\mathop {\text{D}}\limits_{{{\text{P}}_2}} {\text{(s)}}\,\, \rightleftharpoons \mathop {\text{C}}\limits_{{{\text{P}}_2}} {\text{(g) + }}\,\mathop {\text{E}}\limits_{{{\text{P}}_2}} {\text{(g)}}$
When both solids decompose simultaneously, as the product C is common in both reaction so, he pressure of C will be sum of both pressure so, at simultaneously decomposition,
$\mathop {\text{A}}\limits_{{{\text{P}}_1}} {\text{(s)}}\,{\text{ + }}\,\mathop {\text{B}}\limits_{{{\text{P}}_1}} {\text{(g)}}\, \rightleftharpoons \mathop {\text{C}}\limits_{{{\text{P}}_1} + {{\text{P}}_2}} {\text{(g)}}$
$\mathop {\text{D}}\limits_{{{\text{P}}_2}} {\text{(s)}}\,\, \to \mathop {\text{C}}\limits_{{{\text{P}}_1}{\text{ + }}{{\text{P}}_2}} {\text{(g) + }}\,\mathop {\text{E}}\limits_{{{\text{P}}_2}} {\text{(g)}}$
Now, we will write the equilibrium expression for both the reaction as follows:
For $\mathop {\text{A}}\limits_{{{\text{P}}_1}} {\text{(s)}}\,{\text{ + }}\,\mathop {\text{B}}\limits_{{{\text{P}}_1}} {\text{(g)}}\, \rightleftharpoons \mathop {\text{C}}\limits_{{{\text{P}}_1} + {{\text{P}}_2}} {\text{(g)}}$,
${{\text{K}}_{{\text{p1}}}}\,{\text{ = }}\,\left[ {\text{B}} \right]\left[ {\text{C}} \right]$
On substituting the pressure and ${{\text{K}}_{{\text{p1}}}}$value,
${\text{x}}\,{\text{ = }}\,\left[ {{{\text{P}}_{\text{1}}}} \right]\left[ {{{\text{P}}_{\text{1}}}\,{\text{ + }}\,{{\text{P}}_{\text{2}}}} \right]$…..$(1)$
For $\mathop {\text{D}}\limits_{{{\text{P}}_2}} {\text{(s)}}\,\, \to \mathop {\text{C}}\limits_{{{\text{P}}_1}{\text{ + }}{{\text{P}}_2}} {\text{(g) + }}\,\mathop {\text{E}}\limits_{{{\text{P}}_2}} {\text{(g)}}$,
${{\text{K}}_{{\text{p2}}}}\,{\text{ = }}\,\left[ {\text{C}} \right]\left[ {\text{E}} \right]$
${{\text{K}}_{{\text{p2}}}}\,{\text{ = }}\,\left[ {{{\text{P}}_{\text{1}}}\,{\text{ + }}{{\text{P}}_{\text{2}}}} \right]\left[ {{{\text{P}}_{\text{2}}}} \right]$
On substituting the pressure and ${{\text{K}}_{{\text{p2}}}}$value,
${\text{y}}\,{\text{ = }}\,\left[ {{{\text{P}}_{\text{1}}}\,{\text{ + }}\,{{\text{P}}_{\text{2}}}} \right]\left[ {{{\text{P}}_2}} \right]$…..$(2)$
We will determine the total value of equilibrium constant or the relation between x and y and pressure by adding the equation$(1)$ and $(2)$.
${\text{x}}\,\, + \,{\text{y}}\,\,{\text{ = }}\,\left[ {{{\text{P}}_{\text{1}}}} \right]\left[ {{{\text{P}}_{\text{1}}}\,{\text{ + }}\,{{\text{P}}_{\text{2}}}} \right] + \,\,\left[ {{{\text{P}}_{\text{1}}}\,{\text{ + }}\,{{\text{P}}_{\text{2}}}} \right]\left[ {{{\text{P}}_2}} \right]$
${\text{x}}\,\, + \,{\text{y}}\,\,{\text{ = }}\,\left[ {{\text{P}}_1^2} \right]\left[ {{{\text{P}}_{\text{1}}}{{\text{P}}_{\text{2}}}} \right] + \,\,\left[ {{{\text{P}}_{\text{1}}}{{\text{P}}_{\text{2}}}} \right]\left[ {{\text{P}}_2^2} \right]$
${\text{x}}\,\, + \,{\text{y}}\,\,{\text{ = }}\,\left[ {{\text{P}}_1^2} \right] + 2\left[ {{{\text{P}}_{\text{1}}}{{\text{P}}_{\text{2}}}} \right] + \left[ {{\text{P}}_2^2} \right]$
As we know, ${\left( {{\text{a}}\,\,{\text{ + }}\,{\text{b}}} \right)^{\text{2}}}\,\,{\text{ = }}\,\left[ {{{\text{a}}^{\text{2}}}} \right]{\text{ + 2}}\left[ {{\text{ab}}} \right]{\text{ + }}\left[ {{{\text{b}}^{\text{2}}}} \right]$
So, ${\text{x}}\,\, + \,{\text{y}}\,\,{\text{ = }}\,{\left[ {{{\text{P}}_{\text{1}}}\,{\text{ + }}\,{{\text{P}}_{\text{2}}}} \right]^2}$
$\sqrt {{\text{x}}\,\, + \,{\text{y}}} \,\,{\text{ = }}\,\left[ {{{\text{P}}_{\text{1}}}\,{\text{ + }}\,{{\text{P}}_{\text{2}}}} \right]$….$(3)$
Now the total pressure is the sum of pressure of B, C and D. so,
${{\text{P}}_{{\text{total}}}}\,{\text{ = }}\,{{\text{P}}_{\text{B}}}\,{\text{ + }}{{\text{P}}_{\text{C}}}\,{\text{ + }}\,{{\text{P}}_{\text{D}}}$
${{\text{P}}_{{\text{total}}}}\,{\text{ = }}\,{{\text{P}}_1}\,{\text{ + }}\left( {{{\text{P}}_1}\,{\text{ + }}\,{{\text{P}}_2}} \right){\text{ + }}\,{{\text{P}}_2}$
${{\text{P}}_{{\text{total}}}}\,{\text{ = }}\,{\text{2}}\left( {{{\text{P}}_1}\,{\text{ + }}\,{{\text{P}}_2}} \right)$….$(4)$
On substituting the value of $\left[ {{{\text{P}}_{\text{1}}}\,{\text{ + }}\,{{\text{P}}_{\text{2}}}} \right]$from equation $(3)$ in equation$(4)$,
${{\text{P}}_{{\text{total}}}}\,{\text{ = }}\,{\text{2}}\,\sqrt {{\text{x}}\,{\text{ + }}\,{\text{y}}}$
So, the total pressure when both the solids dissociates simultaneously is ${\text{2}}\,\sqrt {{\text{x}}\,{\text{ + }}\,{\text{y}}}$.

Therefore, option (B) is correct.

Note: The pressure of A and D are not included in the equilibrium constant expression. The pressure or concentration of solid is considered as one because solid decomposes very small. As C is a common product, we add the pressure for both C. we can write the pressure of C as ${\text{2P}}$ because C is formed from different reactants and both the reactants have different initial pressures.