A) \[\frac{{{x}^{3}}}{(2+x){{(1-x)}^{2}}}\times P\]
B) \[x=\sqrt[3]{\frac{P}{2{{K}_{P}}}}\]
C) \[x=\sqrt{\frac{2{{K}_{p}}}{P}}\]
D) \[\frac{{{x}^{2}}}{(2+x){{(1-x)}^{2}}}\times P\]
Correct Answer: A
Solution :
\[2A{{B}_{2}}\rightleftharpoons 2AB+{{B}_{2}}\] \[\begin{matrix} \text{Initial} & 2 & 0 & 0 \\ \text{Final} & 2-2x & 2x & x \\ \end{matrix}\] Total no. of moles\[=2-2x+2x+x=2+x\] \[{{P}_{A{{B}_{2}}}}=\frac{2-2x}{2+x}\times P,\,{{p}_{AB}}=\frac{2x\times P}{2+x},\,{{p}_{{{B}_{2}}}}=\frac{x}{2+x}\times P\] \[{{K}_{p}}=\frac{{{({{p}_{AB}})}^{2}}({{p}_{{{B}_{2}}}})}{{{({{P}_{A{{B}_{2}}}})}^{2}}}=\frac{{{\left( \frac{2x}{2+x} \right)}^{2}}{{P}^{2}}\times \left( \frac{x}{2+x} \right)\times P}{{{\left( \frac{2-2x}{2+x} \right)}^{2}}\times {{p}^{2}}}\] \[=\frac{{{x}^{3}}\times P}{(2+x)\,{{(1-x)}^{2}}}\]
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