Question

The half-life of Pb-210 is 22 years. If 2 grams of Pb-210 is let to decay, what will be its amount after 11 years?
A: 0.1414 grams.
B: 1.414 grams.
C: 2.828 grams
D: 0.707 grams.

Answer 1

The half-life of the substance is the time taken by it to decay up to half of its initial concentration. The half-life will tell you about the rate constant for the reaction.

Complete step by step answer:
We know that, the reaction of Pb-210 is as follows;
$^{210}Pb \to X$
Hence the kinetics can be shown as given in the table below;

| $^{20}Pb$| $X$
---|---|---
T=0| 2 grams| 0 gram
T=22 years| 1 gram| 1 gram

Now as we know the half-life we can find out the rate constant
Since the decay follows first order kinetics we can say that rate constant is given by;
$\lambda = \dfrac{{\ln (2)}}{T}$
Here, T is the half-life and $\lambda$ is the rate constant.
Thus, we substitute the values we get;
$\lambda = \dfrac{{\ln (2)}}{{22}}year{s^{ - 1}}$
Now as we have to find the amount at a certain time we will use the integrated rate law;
The integrated rate law is given by;
Amount left = $2{e^{\left( {\dfrac{{\ln (2)}}{T}} \right)t}}$.
Now we have to find the amount at t= 11 years and we know that T=22 years.
So by substituting the values we get;
Amount left $= 2{e^{\left( {\dfrac{{\ln 2}}{{22}}} \right)11}}$
$= 2{e^{\ln \sqrt 2 }}$
$= \dfrac{2}{{\sqrt 2 }}$
$= \sqrt 2$
Therefore the amount left after 11 years is 1.141 grams.
So, option B is correct.

Note: No other rate laws should be used. In a decay reaction everything is similar to a first order chemical reaction. The half-life is inversely proportional to the rate constant.