Question

Which of the following statement is not true for radioactive decay?

• A. Amount of radioactive substance remained after three half lives is \( \frac{1}{8} \)th of original amount.
• B. Decay constant does not depend upon temperature.
• C. Decay constant increases with increase in temperature.
• D. Half life is in 2 times of \( \frac{1}{\text{rate constant}} \).

Hint:

The decay constant is constant for a given radioactive substance, and it does not depend on temperature.

Answer 1

The Correct Option is C

Let's evaluate the statements one by one to determine which is not true for radioactive decay.

Statement 1: The amount of radioactive substance remaining after three half-lives is \( \frac{1}{8} \)th of the original amount.

This statement is true. Radioactive substances decay by half over each half-life period. After one half-life, \( \frac{1}{2} \) remains; after two half-lives, \( \frac{1}{4} \) remains; and after three half-lives, \( \frac{1}{8} \) remains. Mathematically, this is expressed as \( \frac{1}{2^n} \) for \( n \) half-lives.

Statement 2: Decay constant does not depend upon temperature.

This statement is true. The decay constant (\( \lambda \)) is a characteristic of each radioactive isotope and does not change with temperature or pressure.

Statement 3: Decay constant increases with increase in temperature.

This statement is false. As established, the decay constant is independent of temperature changes, making this statement incorrect in the context of radioactive decay.

Statement 4: Half-life is \( \frac{\ln(2)}{\text{rate constant}} \).

This is a true statement. The half-life \( t_{\frac{1}{2}} \) is related to the decay constant by the formula \( t_{\frac{1}{2}} = \frac{\ln(2)}{\lambda} \).

Therefore, the incorrect statement regarding radioactive decay is: "Decay constant increases with increase in temperature."

Answer 2

Radioactive decay is a nuclear process and is independent of temperature, pressure, or chemical state of the substance.

The decay constant (λ) and half-life (t_1/2) are related as: \[ t_{1/2} = \frac{\ln 2}{\lambda} \]

Hence:

• Option 1 → True (After 3 half-lives, remaining = \( \frac{1}{8} \))
• Option 2 → True (λ does not depend on temperature)
• Option 3 → False (λ does not increase with temperature)
• Option 4 → True (Half-life is inversely proportional to λ)

 

Final Answer:

✅ Option 3: Decay constant increases with increase in temperature. (This statement is not true.)