The half-life of a radioactive nuclide is 100 hours. The fraction of original activity that will remain after 150 hours would be :

The correct option is 1/2√2.

Step-by-step Derivation and Explanation:

Radioactive decay follows first-order kinetics. The relationship between the remaining activity of a radioactive sample and its initial activity after a certain time is given by the formula:

$\frac{A}{A_0}={\left(\frac{1}{2}\right)}^n$

where:
- $A$ is the activity remaining after time $t$.
- $A_0$ is the initial activity at $t=0$.
- $\frac{A}{A_0}$ is the fraction of original activity that remains.
- $n$ is the number of half-lives that have elapsed.

Step 1: Calculate the number of half-lives ($n$)
The number of half-lives is the total elapsed time divided by the half-life duration:

$n=\frac{t}{T_{1/2}}$

Given:
- Half-life, $T_{1/2}=100\text{hours}$
- Total time, $t=150\text{hours}$

Substituting these values:

$n=\frac{150}{100}=\frac{3}{2}=1.5$

Step 2: Calculate the remaining fraction
Substitute the value of $n$ back into the fraction formula:

$\frac{A}{A_0}={\left(\frac{1}{2}\right)}^{\frac{3}{2}}$

This expression can be simplified as follows:

${\left(\frac{1}{2}\right)}^{\frac{3}{2}}=\frac{1^{3/2}}{2^{3/2}}=\frac{1}{\sqrt{2^3}}$

Since $2^3=8$, we have:

$\frac{1}{\sqrt{8}}=\frac{1}{\sqrt{4\cdot 2}}=\frac{1}{2\sqrt{2}}$

Thus, the fraction of the original activity remaining after 150 hours is $\frac{1}{2\sqrt{2}}$.