Q. 11 - An Introduction to General, Organic, and Biological Chemistry

Question

$K^{+}$ is an electrolyte required by the human body and found in many foods as well as salt substitutes. One of the isotopes of potassium is potassium-$40$, which has a natural abundance of $0.012\%$ and a half-life of $1.30\times {10}^{9 }\mathrm{yr}$. The isotope potassium-$40$ decays to calcium$‐40$ or to argon-$40$. A typical activity for potassium-$40$ is $7.0 \mathrm{\mu Ci}$per gram.$(5.2 ,5.3 , 5.4)$

Potassium chloride is used as a salt substitute.

a. Write a balanced nuclear equation for each type of decay and identify the particle emitted.

b. A shaker of salt substitute contains $1.6 \mathrm{oz}$of $K$. What is the activity, in millicuries and becquerels, of the potassium in the shaker?

Step-by-Step Solution

Answer

a. A  balanced nuclear equation is${}_{19}{}^{40}\mathrm{K}\to {}_{18}{}^{40}\mathrm{Ar}+{}_{+1}{}^0\mathrm{e}$.

b. The  activity is$1.67\times {10}^{12}\mathrm{Bq}$.

Part (a) step 1: Given Information

We need to write a balanced nuclear equation for each type of decay and identify the particle emitted. 

Part (a) step 2: Simplify

Beta-decay: 

${}_{19}{}^{40}K\to {}_{20}{}^{40}Ca+{}_{-1}{}^{0}e$

Positron emission: 

${}_{19}{}^{40}\mathrm{K}\to {}_{18}{}^{40}\mathrm{Ar}+{}_{+1}{}^0\mathrm{e}$

Part (b) step 1: Given Information

We need to find the activity, in millicuries and becquerels, of the potassium in the shaker. 

Part (b) step 2: Simplify

Consider:

Activity$=7 \mathrm{m}\raisebox{1ex}{$\mathrm{Ci}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{g}$}\right.$ mass$=1.6 oz K$

$$\begin{gathered} 1 \mathrm{oz}=28.34 \mathrm{g} \\ 1.6 \mathrm{oz} \mathrm{K}=1.6\times 28.34 \mathrm{g}=45.344 \mathrm{g} \end{gathered}$$

$$\begin{gathered} 1 \mathrm{Ci}=3.7\times {10}^{10} \\ \mathrm{A}=45.344\times 3.7\times {10}^{10} \\ =1.67\times {10}^{12} \mathrm{Bq} \end{gathered}$$