Chapter 30, Problem 26P

To determine

Part (a):

Using the result of a calculation whether the following decays are possible.

${}_{92}{}^{233}\text{U}\to {}_{92}{}^{232}\text{U}+\text{n}$

Answer to Problem 26P

Solution:

No.

Explanation of Solution

Formula used:

Binding energy is $\Delta m{c}^2$.

Solution:

Calculation:

Mass of ${}_{92}{}^{233}\text{U}$ is $233.039636\text{u}$. Mass of neutron ${}_0{}^1\text{n}$ is $1.008665\text{u}$. Mass of proton ${}_{92}{}^{232}\text{U}$ is $232.037156\text{u}$. Mass of daughter nucleus and neutron is $232.037156\text{u}+1.008665\text{u}=\text{233}\text{.045821u}$. Since $\text{m}\left({}_{92}{}^{233}\text{U}\right)$ is less than $\text{233}\text{.045821u}$, the decay ${}_{92}{}^{233}\text{U}\to {}_{92}{}^{232}\text{U}+\text{n}$ is not possible.

Conclusion:

Mass is equivalent to energy. Mass in a decay reaction cannot increase.

To determine

Part (b):

Using the result of a calculation whether the following decays are possible.

${}_7{}^{14}\text{N}\to {}_7{}^{13}\text{N}+\text{n}$

Answer to Problem 26P

Solution:

No.

Explanation of Solution

Formula used:

Binding energy is $\Delta m{c}^2$.

Solution:

Calculation:

Mass of ${}_7{}^{14}\text{N}$ is $14.003074\text{u}$. Mass of neutron ${}_0{}^1\text{n}$ is $1.008665\text{u}$. Mass of proton ${}_7{}^{13}\text{N}$ is $13.005739\text{u}$. Mass of daughter nucleus and neutron is $13.005739\text{u}+1.008665\text{u}=\text{14}\text{.014404}\text{u}$. Since $\text{m}\left({}_7{}^{14}\text{N}\right)$ is less than $\text{14}\text{.014404}\text{u}$, the decay ${}_7{}^{14}\text{N}\to {}_7{}^{13}\text{N}+\text{n}$ is not possible.

Conclusion:

Mass is equivalent to energy. Mass in a decay reaction cannot increase.

To determine

Part (c):

Using the result of a calculation whether the following decays are possible.

${}_{19}{}^{40}\text{K}\to {}_{19}{}^{39}\text{K}+\text{n}$

Answer to Problem 26P

Solution:

No.

Explanation of Solution

Formula used:

Binding energy is $\Delta m{c}^2$.

Solution:

Calculation:

Mass of ${}_{19}{}^{40}\text{K}$ is $39.963998\text{u}$. Mass of neutron ${}_0{}^1\text{n}$ is $1.008665\text{u}$. Mass of proton ${}_{19}{}^{39}\text{K}$ is $38.963706\text{u}$. Mass of daughter nucleus and neutron is $38.963706\text{u}+1.008665\text{u}=\text{39}\text{.972371}\text{u}$. Since $\text{m}\left({}_{19}{}^{40}\text{K}\right)$ is less than $\text{39}\text{.972371}\text{u}$, the decay ${}_{19}{}^{40}\text{K}\to {}_{19}{}^{39}\text{K}+\text{n}$ is not possible.

Conclusion:

Mass is equivalent to energy. Mass in a decay reaction cannot increase.