The activity of a nucleus is inversely proportional to its half of average life. Thus, shorter the half life of an element, greater is its radioactivity, i.e., greater the number of atomsd disintegrating per second. The relation between half life and average life is `t_(1//2) = (0.693)/(lambda) = tau xx 0.693`
or `tau = 1.44 t_(1//2)`
The half-life periods of four isotopes are given 1 = 6.7 years, II = 8000 years, III = 5760 years, `IV = 2.35 xx 10^(5)` years. Which of these is most stable?

Options

A. I
B. II
C. III
D. IV

Explanation

To determine which of the four isotopes is the most stable based on their half-lives, we can follow these steps:

### Step 1: Understand the Relationship Between Half-Life and Stability
The stability of a radioactive isotope is inversely related to its radioactivity, which means that isotopes with longer half-lives are more stable. Therefore, we need to identify the isotope with the longest half-life.

### Step 2: List the Given Half-Lives
The half-lives of the four isotopes are:
- Isotope I: 6.7 years
- Isotope II: 8000 years
- Isotope III: 5760 years
- Isotope IV: \(2.35 \times 10^5\) years (which is 235000 years)

### Step 3: Compare the Half-Lives
Now we will compare the half-lives to determine which one is the longest:
- Isotope I: 6.7 years
- Isotope II: 8000 years
- Isotope III: 5760 years
- Isotope IV: 235000 years

### Step 4: Identify the Longest Half-Life
From the comparison:
- Isotope IV has the longest half-life of 235000 years.

### Step 5: Conclusion
Since the stability of an isotope is directly proportional to its half-life, the isotope with the longest half-life (Isotope IV) is the most stable.

### Final Answer
The most stable isotope is **Isotope IV** with a half-life of \(2.35 \times 10^5\) years.

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