The half life of a radioactive substance is 20s. Calculate (i) the decay constant, and (ii) time take by the sample to decay by `7//8th` of its inital value

Explanation

To solve the problem, we will follow these steps:

### Step 1: Calculate the Decay Constant (λ)
The decay constant (λ) is related to the half-life (T) of a radioactive substance by the formula:

\[ \lambda = \frac{0.693}{T} \]

Given that the half-life (T) is 20 seconds, we can substitute this value into the equation:

\[ \lambda = \frac{0.693}{20} \]

Calculating this gives:

\[ \lambda = 0.03465 \, \text{s}^{-1} \, (\text{approximately } 0.0346 \, \text{s}^{-1}) \]

### Step 2: Calculate the Time to Decay by 7/8 of its Initial Value
To find the time taken for the sample to decay by 7/8 of its initial value, we first determine the fraction of the substance remaining after this decay. If the sample decays by 7/8, then:

\[ \text{Remaining fraction} = 1 - \frac{7}{8} = \frac{1}{8} \]

Using the relationship between the remaining quantity and the number of half-lives (n), we have:

\[ \frac{N}{N_0} = \left(\frac{1}{2}\right)^n \]

Setting \(\frac{N}{N_0} = \frac{1}{8}\), we can write:

\[ \left(\frac{1}{2}\right)^n = \frac{1}{8} \]

Recognizing that \(\frac{1}{8} = \left(\frac{1}{2}\right)^3\), we find:

\[ n = 3 \]

Now, to find the time (t) taken for this decay, we use the formula:

\[ t = n \cdot T \]

Substituting the values we have:

\[ t = 3 \cdot 20 \, \text{s} = 60 \, \text{s} \]

### Final Answers
(i) The decay constant (λ) is approximately **0.0346 s⁻¹**.
(ii) The time taken for the sample to decay by 7/8 of its initial value is **60 seconds**.

---