Q. 5.16 - An Introduction to General, Organic, and Biological Chemistry

Question

Write a balanced equation for the beta decay of each of the following radioactive isotopes:

(a) ${}_{19}{}^{44}\mathrm{K}$

(b) iron-$59$

(c) potassium-$42$

(d) ${}_{56}{}^{141}\mathrm{Ba}$

Step-by-Step Solution

Answer

(a) The balanced equation for potassium is${}_{19}{}^{44}\mathrm{K}\to {}_{20}{}^{44}\mathrm{Ca}+{}_{-1}{}^0\mathrm{e}$.

(b) The balanced equation for iron is${}_{26}{}^{59}\mathrm{Fe}\to {}_{27}{}^{59}\mathrm{Co}+{}_{-1}{}^0\mathrm{e}$.

(c) The balanced equation of potassium is${}_{19}{}^{42}\mathrm{K}\to {}_{20}{}^{42}\mathrm{Ca}+{}_{-1}{}^0\mathrm{e}$.

(d) The balanced equation of barium is${}_{56}{}^{141}\mathrm{Ba}\to {}_{57}{}^{141}\mathrm{La}+{}_{-1}{}^0\mathrm{e}$.

Part (a) Step 1: Given Information

We are given the isotope and we have to find out the balanced nuclear reaction for beta decay.

Part (a) Step 2: Explanation

To find out the balanced nuclear equation, we need to find out the mass number and atomic number of the new nucleus atom.

In the beginning, the equation is, ${}_{19}{}^{44}\mathrm{K}\to ?+{}_{-1}{}^0\mathrm{e}$

To complete it, find the mass number,

which is, $44=?+0$

which states that the mass number is$44$.

By equating, find the atomic number,

which is, $19=?-1$

which states that an atomic number is$20$.

So the new nucleus atom is calcium.

Hence, the equation is ${}_{19}{}^{44}\mathrm{K}\to {}_{20}{}^{44}\mathrm{Ca}+{}_{-1}{}^0\mathrm{e}$

Part (b) Step 1: Given Information

We are given the isotope and we have to find the balanced nuclear reaction for beta decay.

Part (b) Step 2: Explanation

To find out the balanced nuclear equation, we need to find out the mass number and atomic number of the new nucleus atom.

In the beginning, the equation is, ${}_{26}{}^{59}\mathrm{Fe}\to ?+{}_{-1}{}^0\mathrm{e}$

To complete it, find the mass number,

which is, $59=?+0$

which states that the mass number is$59$

By equating, find the atomic number,

which is, $26=?-1$

which states that an atomic number is$27$

So the new nucleus atom is cobalt.

Hence, the equation is ${}_{26}{}^{59}\mathrm{Fe}\to {}_{27}{}^{59}\mathrm{Co}+{}_{-1}{}^0\mathrm{e}$

Part (c) Step 1: Given Information

We are given the isotope and we have to find the balanced nuclear reaction for beta decay.

Part (c) Step 2: Explanation

To find out the balanced nuclear equation, we need to find out the mass number and atomic number of the new nucleus atom.

In the beginning, the equation is${}_{19}{}^{42}\mathrm{K}\to ?+{}_{-1}{}^0\mathrm{e}$,

To complete it, find the mass number,

which is$42=?+0$,

which states that the mass number is$42$ .

By equating, find the atomic number,

which is$19=?-1$,

which states that an atomic number is$20$

So the new nucleus atom is calcium.

Hence, the equation is${}_{19}{}^{42}\mathrm{K}\to {}_{20}{}^{42}\mathrm{Ca}+{}_{-1}{}^0\mathrm{e}$

Part (d) Step 1: Given Information

We are given the isotope and we have to find the balanced nuclear reaction for beta decay.

Part (d) Step 2: Explanation

To find out the balanced nuclear equation, we need to find out the mass number and atomic number of the new nucleus atom.

In the beginning, the equation is${}_{56}{}^{141}\mathrm{Ba}\to ?+{}_{-1}{}^0\mathrm{e}$,

To complete it, find the mass number,

which is$141=?+0$,

which states that the mass number is$141$ .

By equating, find the atomic number.

which is$56=?-1$,

which states that an atomic number is$57$

So the new nucleus atom is lanthanum.

Hence, the equation is ${}_{56}{}^{141}\mathrm{Ba}\to {}_{57}{}^{141}\mathrm{La}+{}_{-1}{}^0\mathrm{e}$