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Chemical Kinetics

Question
CBSEENCH12011375

If 60% of a first order reaction was completed in 60 min, 50% of the same reaction would be completed in approximately:

  • 50 min

  • 45 min

  • 60 min

  • 40 min

Solution

B.

45 min

straight k space equals space fraction numerator 2.303 over denominator straight t end fraction space log subscript 10 space fraction numerator straight a over denominator straight a minus straight x end fraction
straight k subscript 1 space equals space fraction numerator 2.303 over denominator straight t subscript 1 end fraction space log space fraction numerator straight a over denominator straight a subscript 1 minus straight x subscript 1 end fraction
straight k subscript 2 space equals space fraction numerator 2.303 over denominator straight t subscript 2 end fraction space log space fraction numerator straight a over denominator straight a subscript 2 minus straight x subscript 2 end fraction
straight x subscript 1 space space equals 60 over 100 straight a subscript 1 comma space space space straight t subscript 1 space equals space 60
straight x subscript 2 space equals space 50 over 100 space straight a subscript 2 space comma space space space space straight t subscript 2 space equals space ? space

straight k subscript 1 space equals space fraction numerator 2.303 over denominator straight t subscript 1 end fraction space log space fraction numerator straight a over denominator straight a subscript 1 minus straight x subscript 1 end fraction space equals space straight k subscript 2 space equals space fraction numerator 2.303 over denominator straight t subscript 2 end fraction space log space fraction numerator straight a over denominator straight a subscript 2 minus straight x subscript 2 end fraction
fraction numerator 2.303 over denominator 60 end fraction space log space fraction numerator straight a over denominator open parentheses straight a subscript 1 minus begin display style 60 over 100 end style straight a subscript 1 close parentheses end fraction space equals fraction numerator space 2.303 over denominator straight t subscript 1 end fraction space log space fraction numerator straight a subscript 2 over denominator open parentheses straight a subscript 2 minus begin display style 50 over 100 end style straight a subscript 2 close parentheses end fraction
fraction numerator 2.303 over denominator 60 end fraction space log space fraction numerator 100 space straight a subscript 1 over denominator 40 straight a subscript 1 end fraction space equals space fraction numerator 2.303 over denominator straight t subscript 2 end fraction space log space fraction numerator 100 space straight a subscript 2 over denominator 50 space straight a subscript 2 end fraction
1 over 60 space log space 100 over 40 space equals space 1 over straight t subscript 2 space log space 100 over 50
straight t subscript 2 space equals space fraction numerator 60 space log space 100 divided by 50 over denominator log space 100 divided by 40 end fraction
equals space fraction numerator left parenthesis 60 space log space 10 minus log space 5 right parenthesis over denominator left parenthesis log space 10 minus log space 4 right parenthesis end fraction
space equals space fraction numerator 60 space left parenthesis 1 minus 0.06 right parenthesis over denominator left parenthesis 1 minus 0 minus 0.60 right parenthesis end fraction space equals space fraction numerator 60 space straight x space 0.31 over denominator 0.40 end fraction
space equals space 1.5 space straight x space 31 space equals space 46.4 space almost equal to space 45 space min