Question

Question 5 - Half Lives:

Radioactive cesium (Cs-137) is one of many possible by-products of a nuclear meltdown. The half-life for the decay of Cs-137 is 30.17 yr, what is the rate constant for its decay in yr^-1?

Enter the numerical answer in decimal format, do not include the units.

For a first order reaction how many half-lives (t_1/2) must pass for the concentration of the reactant to reach 1/8 of its original amount?

For a first order reaction having a rate constant of 0.034 sec^-1, what is its half-life in sec?

Answer 1

1)

Given that,

The half-life for the decay of Cs-137 = 30.17 yr

For 1st order reaction , the rate constant = 0.693 / t_1/2

= 0.693 / 30.17

= 0.022 yr^-1

2)

The integrated rate equation is , [A] = [A]₀ e^-kt

Where [A] is concentration of A at any time t

  [A]₀ is initial concentration

k is rate constant

t is time

Now , t_1/2 = ln 2 / k

Substitute these values of k in the above equation

[A] = [A]₀ e^{-ln2 (t) / t_1/2}

Substitute [A]₀ = x [A] = x/8

x/8 = xe ^{-ln2 (t) / t_1/2}

1 / 8 = e ^{-ln2 (t) / t_1/2}

ln2 (t) / t_1/2 = 2.079

t = 3 t_1/2 hours must be passed.

3)

Tha half-life of the first order reaction is

   t_1/2 = 0.693 / k Where k is rate constant , t_1/2 is half life of the reaction

t_1/2 = 0.693 / 0.034

t_1/2 = 20.38 sec