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Equilibrium

Question
CBSEENCH11006424

At 700K, equilibrium constant for the reaction:
   straight H subscript 2 left parenthesis straight g right parenthesis space plus space straight I subscript 2 left parenthesis straight g right parenthesis rightwards harpoon over leftwards harpoon space 2 HI left parenthesis straight g right parenthesis 
is 54.8. If 0.5 mole/litre of HI(g) is present at equilibrium at 700K, what are the concentration of H2(g) and I2(g) assuming that we initially started with HI(g) and allowed it to reach equilibrium at 700 K?

Solution
Here, equilibrium constant for the reaction,
       straight H subscript 2 left parenthesis straight g right parenthesis space plus space straight I subscript 2 left parenthesis straight g right parenthesis space space space rightwards harpoon over leftwards harpoon space space space 2 HI left parenthesis straight g right parenthesis  
is 54.8 . Thus the equilibrium constant for the reverse reaction
         2 HI left parenthesis straight g right parenthesis space space rightwards harpoon over leftwards harpoon space space straight H subscript 2 left parenthesis straight g right parenthesis space plus space straight I subscript 2 left parenthesis straight g right parenthesis space would space be space fraction numerator 1 over denominator 54.8 end fraction
Concentration of HI at equilibrium = 0.5 mole/litre[given]
The concentration of H2 and I2 at equilibrium will be same.
Suppose each of them = x mole /litre
Applying law of chemical equilibrium to the above reaction,
           straight K subscript straight c space equals space fraction numerator open square brackets straight H subscript 2 left parenthesis straight g right parenthesis close square brackets space open square brackets straight I subscript 2 left parenthesis straight g right parenthesis close square brackets over denominator space open square brackets HI left parenthesis straight g right parenthesis close square brackets end fraction
Substituting the values, we have,
        fraction numerator 1 over denominator 54.8 end fraction space equals space fraction numerator straight x space cross times space straight x over denominator left parenthesis 0.5 right parenthesis squared end fraction space equals space fraction numerator straight x squared over denominator 0.25 end fraction
or space space space straight x squared space equals space fraction numerator 0.25 over denominator 54.8 end fraction space equals space 0.00456
or space space space space straight x space equals space 0.068 space mol space straight L to the power of negative 1 end exponent