Which of the following statement is not true for radioactive decay ? (1) Amount of radioactive substance remained after three half lives is 1/8 th of original amount. (2) Decay constant does not depend upon temperature. (3) Decay constant increases with increase in temperature. (4) Half life is ln 2 times of (1/rate constant)

Key Concepts You Need

1. Radioactive Decay Law
   $N = N_0 e^{-\lambda t}$
   $N$ = amount left after time $t$, $N_0$ = initial amount, $\lambda$ = decay constant.

2. Half-life ($t_{1/2}$)
   Set $N = \frac{N_0}{2}$:
   $\frac{N_0}{2} = N_0 e^{-\lambda t_{1/2}} \;\;\Rightarrow\;\; t_{1/2} = \frac{\ln 2}{\lambda}$

3. Temperature Independence
   Radioactive decay is a nuclear process. Nuclear forces are so strong that ordinary temperature changes (chemical scale, a few hundred Kelvin) do not affect $\lambda$. Hence $\lambda$ is practically constant with temperature.

4. Multiple Half-lives
   After $n$ half-lives the fraction remaining is $\left(\frac12\right)^{n}$. For $n = 3$, that is $\frac18$.

5. Matching Statements to Theory

   • Statement (1) matches point 4.
   • Statement (2) matches point 3.
   • Statement (3) contradicts point 3, so it must be false.
   • Statement (4) rewrites point 2: $t_{1/2} = (\ln 2) \times \frac{1}{\lambda}$.

Thus, a student simply cross-checks each option with these rules to find the incorrect one.

Imagine Super-Fast Melting Ice Cubes

Think of a radioactive atom like an ice cube that melts on its own, but at a perfectly fixed speed no matter how hot or cold the room is.

• Half-life is just the time after which half the original ice cube has melted away.
• After three half-lives, half of a half of a half is left: $\left(\frac12\right)^3 = \frac18$.
• The melting speed (in real science we call it the decay constant $\lambda$) stays the same even if you warm or cool the room.

So if someone says, "Heat it up and it will melt faster," that is wrong for radioactive atoms!

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