JEE MAIN - Chemistry (2008 - No. 7)
The equilibrium constants KP1 and KP2 for the reactions X $$\leftrightharpoons$$ 2Y and Z $$\leftrightharpoons$$ P + Q, respectively are in
the ratio of 1 : 9. If the degree of dissociation of X and Z be equal then the ratio of total pressure at
these equilibria is :
1 : 36
1 : 1
1 : 3
1 : 9
Explanation
Let the initial moles of $$X$$ be $$'a'$$
and that of $$Z$$ be $$'b'$$ the for the given reactions,
we have $$X\,\,\,\,\,\,\,\,\,\rightleftharpoons\,\,\,\,\,\,\,\,\,2Y$$
$$\eqalign{ & Initial\,\,\,\,\,\,\,\,\,\,\,\,a\,\,moles\,\,\,\,\,\,\,0 \cr & At\,\,equi.\,\,\,\,\,\,\,a\left( {1 - \alpha } \right)\,\,\,\,\,\,\,2a\alpha \cr & (moles) \cr} $$
Total no. of moles $$ = a\left( {1 - \alpha } \right) + 2a\alpha $$
$$ = a - a\alpha + 2a\alpha $$ $$ = a\left( {1 + \alpha } \right)$$
Now, $$\,\,\,{K_{{p_1}}} = {{{{\left( {{n_y}} \right)}^2}} \over {{n_x}}} \times {\left( {{{{P_{{T_1}}}} \over {\sum n }}} \right)^{\Delta n}}$$
or, $$\,\,\,{K_{{p_1}}} = {{{{\left( {2a\alpha } \right)}^2}.{P_{{T_1}}}} \over {\left[ {a\left( {1 - \alpha } \right)} \right]\left[ {a\left( {1 + \alpha } \right)} \right]}}$$
$$\,\,\,\,\,\,\,\,\,\,\,\,$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$Z\,\,\,\,\,\,\,\,\,\rightleftharpoons\,\,\,\,\,\,\,\,\,P+Q$$
$$\eqalign{ & Initial\,\,\,\,\,\,\,\,\,\,\,b\,\,moles\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,0 \cr & Ar\,\,equi.\,\,\,\,\,\,b\left( {1 - \alpha } \right)\,\,\,\,\,\,\,\,b\alpha \,\,\,\,\,\,\,b\alpha \cr & (moles) \cr} $$
Total no. of moles
$$ = b\left( {1 - \alpha } \right) + b\alpha + b\alpha $$
$$ = b - b\alpha + b\alpha + b\alpha $$
$$ = b\left( {1 + \alpha } \right)$$
Now $$\,\,\,{K_{{P_2}}} = {{{n_Q} \times {n_P}} \over {{n_z}}} \times {\left[ {{{{P_{{T_2}}}} \over {\sum\nolimits_n \, }}} \right]^{\Delta n}}$$
or$$\,\,\,{K_{{P_2}}} = {{\left( {b\alpha } \right)\left( {b\alpha } \right).{P_{{T_2}}}} \over {\left[ {b\left( {1 - \alpha } \right)} \right]\left[ {b\left( {1 + \alpha } \right)} \right]}}$$
or$$\,\,\,$$ $$\,\,\,{{{K_{{P_1}}}} \over {{K_{P2}}}} = {{4{\alpha ^2}.{P_{{T_1}}}} \over {\left( {1 - {\alpha ^2}} \right)}} \times {{{{\left( {1 - \alpha } \right)}^2}} \over {{P_{{T_2}}}.{\alpha ^2}}}$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$ = {{4{P_{{T_1}}}} \over {{P_{{T_2}}}}}$$
or$$\,\,\,{{{P_{{T_1}}}} \over {{P_{T2}}}} = {1 \over 9}\,\,\,\,\,$$ [ as $$\,\,\,{{{K_{{P_1}}}} \over {{K_{{P_1}}}}} = {1 \over 9}\,\,$$ given ]
or$$\,\,\,{{{P_{{T_1}}}} \over {{P_{T2}}}} = {1 \over {36}}$$
or$$\,\,\,1:36$$
and that of $$Z$$ be $$'b'$$ the for the given reactions,
we have $$X\,\,\,\,\,\,\,\,\,\rightleftharpoons\,\,\,\,\,\,\,\,\,2Y$$
$$\eqalign{ & Initial\,\,\,\,\,\,\,\,\,\,\,\,a\,\,moles\,\,\,\,\,\,\,0 \cr & At\,\,equi.\,\,\,\,\,\,\,a\left( {1 - \alpha } \right)\,\,\,\,\,\,\,2a\alpha \cr & (moles) \cr} $$
Total no. of moles $$ = a\left( {1 - \alpha } \right) + 2a\alpha $$
$$ = a - a\alpha + 2a\alpha $$ $$ = a\left( {1 + \alpha } \right)$$
Now, $$\,\,\,{K_{{p_1}}} = {{{{\left( {{n_y}} \right)}^2}} \over {{n_x}}} \times {\left( {{{{P_{{T_1}}}} \over {\sum n }}} \right)^{\Delta n}}$$
or, $$\,\,\,{K_{{p_1}}} = {{{{\left( {2a\alpha } \right)}^2}.{P_{{T_1}}}} \over {\left[ {a\left( {1 - \alpha } \right)} \right]\left[ {a\left( {1 + \alpha } \right)} \right]}}$$
$$\,\,\,\,\,\,\,\,\,\,\,\,$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$Z\,\,\,\,\,\,\,\,\,\rightleftharpoons\,\,\,\,\,\,\,\,\,P+Q$$
$$\eqalign{ & Initial\,\,\,\,\,\,\,\,\,\,\,b\,\,moles\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,0 \cr & Ar\,\,equi.\,\,\,\,\,\,b\left( {1 - \alpha } \right)\,\,\,\,\,\,\,\,b\alpha \,\,\,\,\,\,\,b\alpha \cr & (moles) \cr} $$
Total no. of moles
$$ = b\left( {1 - \alpha } \right) + b\alpha + b\alpha $$
$$ = b - b\alpha + b\alpha + b\alpha $$
$$ = b\left( {1 + \alpha } \right)$$
Now $$\,\,\,{K_{{P_2}}} = {{{n_Q} \times {n_P}} \over {{n_z}}} \times {\left[ {{{{P_{{T_2}}}} \over {\sum\nolimits_n \, }}} \right]^{\Delta n}}$$
or$$\,\,\,{K_{{P_2}}} = {{\left( {b\alpha } \right)\left( {b\alpha } \right).{P_{{T_2}}}} \over {\left[ {b\left( {1 - \alpha } \right)} \right]\left[ {b\left( {1 + \alpha } \right)} \right]}}$$
or$$\,\,\,$$ $$\,\,\,{{{K_{{P_1}}}} \over {{K_{P2}}}} = {{4{\alpha ^2}.{P_{{T_1}}}} \over {\left( {1 - {\alpha ^2}} \right)}} \times {{{{\left( {1 - \alpha } \right)}^2}} \over {{P_{{T_2}}}.{\alpha ^2}}}$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$ = {{4{P_{{T_1}}}} \over {{P_{{T_2}}}}}$$
or$$\,\,\,{{{P_{{T_1}}}} \over {{P_{T2}}}} = {1 \over 9}\,\,\,\,\,$$ [ as $$\,\,\,{{{K_{{P_1}}}} \over {{K_{{P_1}}}}} = {1 \over 9}\,\,$$ given ]
or$$\,\,\,{{{P_{{T_1}}}} \over {{P_{T2}}}} = {1 \over {36}}$$
or$$\,\,\,1:36$$
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