Question

Suppose that you place exactly 100 bacteria into a flask containing nutrients for the bacteria and that you find the following data at 37 C:
t(min) Number of bacteria
0 100
16.0 200
32.0 400
48.0 800
64.0 1600

a. What is the order of the rate of production of the bacteria?
b. How many bacteria will be present after 112 min?
c. What is the rate constant for the process?

Answer 1

Answer:

The order of the production of bacteria is 1.

12,787 bacteria will be present after 112 min.

$0.04331 min^{-1}$ is the rate constant for the process.

Explanation:

Rate of formation of bacteria when there 100 bacteria: R

Rate of formation of bacteria when there 200 bacteria: R' =2R

Rate of formation of bacteria is directly proportional to the number of bacteria:

$\frac{R}{R'}=\frac{k[200]^x}{k[400]^x}$

$\frac{R}{2R}=(\frac{200}{400})^x$

x = 1

The order of the production of bacteria is 1.

The half life of the process = $t_{1/2}=16 minutes$

Rate constant of the process =  k

For first order process k is only dependent on half life:

$k=\frac{0.693}{t_{1/2}}$

$k=\frac{0.693}{16 min}=0.04331 min^{-1}$

For the first order process the increment numbers is given ;

$N=N_o\times e^{kt}$

Where:

$N_o$= Initial population

N = Population after time t.

We have ;

$N_o=100$

t = 112 minutes

$N=100\times e^{0.04331 min^{-1}\times 112 min}$

$N = 12,786.82\approx 12,787$

12,787 bacteria will be present after 112 min.

$0.04331 min^{-1}$ is the rate constant for the process.