JEE Main · 2019 · Shift-IhardCK-110

A bacterial infection in an internal wound grows as N'(t) = N0(t), where the time t is in hours. A dose of antibiotic,…

Chemical Kinetics · Class 12 · JEE Main Previous Year Question

Question

A bacterial infection in an internal wound grows as N′(t)=N0exp⁡(t)N'(t) = N_0\exp(t), where the time tt is in hours. A dose of antibiotic, taken orally, needs 1 hour to reach the wound. Once it reaches there, the bacterial population goes down as dNdt=−5N2\frac{dN}{dt} = -5N^2. What will be the plot of N0N\frac{N_0}{N} vs tt after 1 hour? image

Options
  1. a

    Option 1

    ✓
  2. b

    Option 2

  3. c

    Option 3

  4. d

    Option 4

Correct Answera

Option 1

Detailed Solution

After 1 hour, the antibiotic reaches the wound. At t=1 ht = 1\,\mathrm{h}, N=N0e1=eN0N = N_0 e^1 = eN_0.

For t>1 ht > 1\,\mathrm{h}: dNdt=−5N2\frac{dN}{dt} = -5N^2 (second order decay)

Integrating: 1N−1N1=5(t−1)\frac{1}{N} - \frac{1}{N_1} = 5(t-1) where N1=eN0N_1 = eN_0

1N=1eN0+5(t−1)\frac{1}{N} = \frac{1}{eN_0} + 5(t-1)

N0N=N0eN0+5N0(t−1)=1e+5N0(t−1)\frac{N_0}{N} = \frac{N_0}{eN_0} + 5N_0(t-1) = \frac{1}{e} + 5N_0(t-1)

This is a linear function of tt (for t>1t > 1), starting at N0/N=1/e<1N_0/N = 1/e < 1 at t=1t=1 and increasing linearly.

The plot of N0/NN_0/N vs tt is a straight line with positive slope starting from a value less than 1 at t=1t=1.

Answer: Option (1)

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