Question
Explain with example, whether the neutron-proton ratio in a nucleus increases or decreases due to $\beta-$decay.
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Answer
In $\beta-$decay a neutron is converted into a proton, so the neutron-proton ratio decreases. Equation of $\beta-$decay is:
$\text{zX}^\text{A}\rightarrow\text{Z}+_1\text{Y}^\text{A}+_{-1}\beta^0+\bar{\text{v}}$
$_{90}\text{Th}^{234}\ \rightarrow\ _{91}\text{Pa}^{234}+_{-1}\beta^0+\bar{\text{v}}$
Neutron to proton ratio before $\beta-$decay $=\frac{234-91}{90}=\frac{144}{90}=1.60$
Neutron to proton ratio aftere $\beta-$decay $=\frac{234-91}{91}=\frac{143}{91}=1.57$
$\frac{143}{91}<\frac{144}{90},$ so neutron to proton ratio in $\beta-$decay decreases.
$\text{zX}^\text{A}\rightarrow\text{Z}+_1\text{Y}^\text{A}+_{-1}\beta^0+\bar{\text{v}}$
$_{90}\text{Th}^{234}\ \rightarrow\ _{91}\text{Pa}^{234}+_{-1}\beta^0+\bar{\text{v}}$
Neutron to proton ratio before $\beta-$decay $=\frac{234-91}{90}=\frac{144}{90}=1.60$
Neutron to proton ratio aftere $\beta-$decay $=\frac{234-91}{91}=\frac{143}{91}=1.57$
$\frac{143}{91}<\frac{144}{90},$ so neutron to proton ratio in $\beta-$decay decreases.
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