Two isotopes `P` and `Q` atomic weight `10` and `20`, respectively are mixed in equal amount by weight. After `20` days their weight ratio is found to be `1:4`. Isotope `P` has a half-life of `10` days. The half-life of isotope `Q` is

Options

A. zero
B. `5` days
C. `20` days
D. infinite

Answer

To solve the problem, we need to analyze the decay of the two isotopes P and Q over a period of 20 days.

### Step-by-Step Solution:

1. **Identify Given Data:**
- Isotope P has an atomic weight of 10 and a half-life of 10 days.
- Isotope Q has an atomic weight of 20.
- Both isotopes are mixed in equal weight.
- After 20 days, the weight ratio of P to Q is 1:4.

2. **Determine the Number of Half-Lives for Isotope P:**
- The half-life of P is 10 days.
- In 20 days, the number of half-lives (n) for P is calculated as:
\[ n_P = \frac{20 \text{ days}}{10 \text{ days}} = 2 \]

3. **Calculate the Remaining Mass of Isotope P:**
- If we start with an initial mass \( m_0 \) for P, after 2 half-lives, the remaining mass \( m_P \) can be calculated using the formula:
\[ m_P = m_0 \left(\frac{1}{2}\right)^n = m_0 \left(\frac{1}{2}\right)^2 = \frac{m_0}{4} \]

4. **Set Up the Mass Ratio Equation:**
- Let the initial mass of both isotopes be \( m_0 \) (for P) and \( m_0 \) (for Q).
- After 20 days, the remaining mass of Q is \( m_Q \).
- According to the problem, the mass ratio of P to Q after 20 days is:
\[ \frac{m_P}{m_Q} = \frac{1}{4} \]
- Substituting for \( m_P \):
\[ \frac{\frac{m_0}{4}}{m_Q} = \frac{1}{4} \]
- Cross-multiplying gives:
\[ m_Q = m_0 \]

5. **Determine the Number of Half-Lives for Isotope Q:**
- Since \( m_Q \) is the remaining mass of Q after some number of half-lives \( n_Q \), we can express it as:
\[ m_Q = m_0 \left(\frac{1}{2}\right)^{n_Q} \]
- Setting this equal to the mass we found earlier:
\[ m_0 = m_0 \left(\frac{1}{2}\right)^{n_Q} \]
- Dividing both sides by \( m_0 \) (assuming \( m_0 \neq 0 \)):
\[ 1 = \left(\frac{1}{2}\right)^{n_Q} \]
- This implies:
\[ n_Q = 0 \]

6. **Calculate the Half-Life of Isotope Q:**
- The number of half-lives \( n_Q = 0 \) means that isotope Q has not decayed at all in the 20 days.
- This indicates that the half-life of isotope Q is infinite.

### Final Answer:
The half-life of isotope Q is infinite.