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The equilibrium constant for${{N}_{2}}\left( g \right)+{{O}_{2}}\left( g \right)\rightleftharpoons 2NO\left( g \right)$is K, the equilibrium constant for$\dfrac{1}{2}{{N}_{2}}\left( g \right)+\dfrac{1}{2}{{O}_{2}}\left( g \right)\rightleftharpoons NO\left( g \right)$ will be:
A. K
B. ${{K}^{2}}$
C. ${{K}^{\dfrac{1}{2}}}$
D. $\dfrac{1}{2}K$

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Last updated date: 20th Jun 2024
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Answer
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Hint: Equilibrium constant constant is basically a proportionality constant, which indicates the relation between the molar concentration of reactants and the rate of reaction. And the expression for equilibrium constant is given by the equation:
\[K=\dfrac{concentration\text{ }of\text{ }product}{concentration\text{ }of\text{ }reac\tan t}\]

Complete step by step answer:
- The first reaction given is: ${{N}_{2}}\left( g \right)+{{O}_{2}}\left( g \right)\rightleftharpoons 2NO\left( g \right)$
- Here, the value of equilibrium constant is K. We can also write as:
\[K=\dfrac{{{\left[ NO \right]}^{2}}}{\left[ {{N}_{2}} \right]\left[ {{O}_{2}} \right]}\]
- For the second reaction the value of equilibrium constant will be:
$\dfrac{1}{2}{{N}_{2}}\left( g \right)+\dfrac{1}{2}{{O}_{2}}\left( g \right)\rightleftharpoons NO\left( g \right)$
 \[\begin{align}& K=\dfrac{\left[ NO \right]}{\sqrt{\left[ {{N}_{2}} \right]\left[ {{O}_{2}} \right]}} \\ & =\sqrt{\dfrac{{{\left[ NO \right]}^{2}}}{\left[ {{N}_{2}} \right]\left[ {{O}_{2}} \right]}} \\ & =\sqrt{K} \\ & ={{K}^{\dfrac{1}{2}}} \\ \end{align}\]
- Here, the value of equilibrium constant is ${{K}^{\dfrac{1}{2}}}$.
Hence, we can conclude that the correct option is (C), that is the value of equilibrium constant ${{K}^{\dfrac{1}{2}}}$.
Additional Information: It is found that the equilibrium constant depends upon temperature. In case of exothermic reactions increasing the temperature will reduce equilibrium constant, and in endothermic reactions increasing the temperature will increase equilibrium constant.And change in concentration, catalyst, pressure etc. have no effect on equilibrium constant.
Equilibrium constant is used to predict the extent of a reaction or we can say to predict the degree of disappearance of the reactants. The equilibrium constant magnitude gives an idea of the amount of reactants and the products.
Note: - We should not get confused in terms ${{k}_{C}}$and${{Q}_{C}}$ . As ${{k}_{C}}$is the equilibrium constant, that is the ratio of concentrations of products and reactants, when reaction is at equilibrium. While ${{Q}_{C}}$ is the reaction quotient, which is used to determine in which direction a reaction will proceed.