Question

In a closed system: \[A(s) \rightleftharpoons 2B(g) + 3C(g)\] if the partial pressure of C is doubled then partial pressure of B will be:
A. Twice the original pressure
B. Half of its original pressure
C. $\dfrac{1}{{\sqrt 2 }}$ times the original pressure
D. $\dfrac{1}{{2 \times \sqrt 2 }}$ times the original pressure

Answer 1

Hint: In a mixture of gases, each constituent gas has a partial pressure which is the notional pressure of that constituent gas if it alone occupied the entire volume of the original mixture at the same temperature. The total pressure of an ideal gas mixture is the sum of the partial pressures of the gases in the mixture.

Complete step by step answer:
 According to the question, the given reaction is as follows:
\[A(s) \rightleftharpoons 2B(g) + 3C(g)\]
The equilibrium constant at a constant pressure can be written as follows:
${K_p} = {({P_B})^2} \times {({P_C})^3}$ ….(i)
The partial vapor pressure of A is zero because it exists in a solid phase.
Thus, equation (i) can also be written as:
$\dfrac{{{K_p}}}{{{{({P_C})}^3}}} = {({P_B})^2}$…(ii)
As we have been given that the partial vapor pressure of C becomes double, then the new vapor pressure of C can be represented by $P_C'$ and it is related to the initial vapor pressure as follows:
$P_C' = 2{P_C}$ … (iii)
As the vapor pressure of C changes, the vapor pressure of B is also affected and it changes to $P_B'$ . Thus, the new equation of equilibrium constant will be designated as:
 $\dfrac{{{K_p}}}{{{{(P_C')}^3}}} = {(P_B')^2}$….(iv)
Substituting the value of equation (iii) in equation (iv), we have:
 $ \Rightarrow \dfrac{{{K_p}}}{{{{(2{P_C})}^3}}} = {(P_B')^2}$… (v)
Dividing equation (iv) and (v) and on solving, we have:
$ \Rightarrow {(P_B')^2} = \sqrt {\dfrac{{{{({P_B})}^2}}}{8}} $
$ \Rightarrow P_B' = \dfrac{1}{{2 \times \sqrt 2 }}{P_B}$
Thus, the correct option is D. $\dfrac{1}{{2 \times \sqrt 2 }}$ times the original pressure.

Note:
The partial pressure of a gas is a measure of thermodynamic activity of the gas's molecules. Gases dissolve, diffuse, and react according to their partial pressures, and not according to their concentrations in gas mixtures or liquids. This general property of gases is also true in chemical reactions of gases in biology. For example, the necessary amount of oxygen for human respiration, and the amount that is toxic, is set by the partial pressure of oxygen alone. This is true across a very wide range of different concentrations of oxygen present in various inhaled breathing gases or dissolved in blood.