Question

A reactant decomposes with a half-life of 29.5 s when its initial concentration is 0.229 M. When the initial concentration is 0.639 M, this same reactant decomposes with a half-life of 82.3 s. What is the value and unit of the rate constant for this reaction?

Answer 1

The order of reaction is, 0 (zero order reaction).

The value of rate constant is,
`0.00388Ms^(-1)`

Explanation :

Half life : It is defined as the time in which the concentration of a reactant is reduced to half of its original value.

The general expression of half-life for nth order is:

`t_(1/2)\propto (1)/([A_o]^(n-1))`

or,

`(t_(1/2)_1)/(t_(1/2)_2)=([A_2]^(n-1))/([A_1]^(n-1))`

or,

`n=\left((\log((t_(1/2))_1)/((t_(1/2))_2))/(\log((A)_2)/((A)_1))\right )+1` .............(1)

where,

`t_(1/2)` = half-life of the reaction

n = order of reaction

[A] = concentration

As we are given:

Initial concentration of A = 0.229 M

Final concentration of A = 0.639 M

Initial half-life of the reaction = 29.5 s

Final half-life of the reaction = 82.3 s

Now put all the given values in the above formula 1, we get:

`n=\left ((\log (29.5)/(82.3))/(\log(0.639)/(0.229))\right )+1`

`n=0.000196\approx 0`

Thus, the order of reaction is, 0 (zero order reaction).

Now we have to determine the rate constant.

To calculate the rate constant for zero order the expression will be:

`t_(1/2)=([A_o])/(2k)`

When,

`t_(1/2)` = 29.5 s

`[A_o]` = 0.229 M

`29.5s=(0.229M)/(2k)`

`k=0.00388Ms^(-1)`

Thus, the value of rate constant is,
`0.00388Ms^(-1)`