3. The civilization on the planet Ecton is endangered by the continual illegal d

Question

3. The civilization on the planet Ecton is endangered by the continual illegal disposal of Klingon hairspray, which is causing the atmosphere to decompose rst-order rate law. Science Spock has that atmosphere is decomposing with a half life of 12.50 min, Dr. McCoy, the medical officer, has determined that the Ectonians need a minimum of 6.25% of the original atmosphere to survive. Meanwhile, Scottie, the chief engineer is desperately trying to fix the so the entire population can be transported to safe planet. You are the acting Commander, replacing Captain Kirk, who is on vacation. How long does Scottie have to repair the transporter?

Explanation / Answer

Given:

First order decomposition

-r = k C or - dC/dt = k C

-dC/C = k dt

integrating from Co at t = 0 to C at t

- (ln C - ln Co) = kt

- ln ( C / Co ) = Kt ------> (1)

where k is rate constant of reaction

C is the Concentration of atmosphere at time t

Given:

half life t1/2 = 12.5 min

for first order reaction

t1/2 = 0.693 / k because

at t = t1/2 C = 1/2 Co

so (1) ==> - ln ( 1/2 Co / Co ) = k t1/2

- ln (1/2) = k t1/2

0.693 = k t1/2

k = 0.693 / t1/2

k = 0.693 / 12.5 min = 0.055 min-1

Maximum low concentration upto which atmosphere can go is 6.25 % i.e C = 0.0625 Co

sub in (1)

- ln ( 0.0625 Co / Co ) = 0.055 * t

2.7726 = 0.055 /min * t

t = 50 min

so scottie has 50 min to repair the transporter

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