Two elements `P` and `Q` have half-line of `10` and `15` minutes repectively. Freshly preapared sample of mixuture containing equal number of atoms is allowed to decay for `30` minutes. The ratio of number of atoms of `P` and `Q` in left in mixture is:

Options

A. 0.5
B. `2.0`
C. `1.0`
D. `3.0`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understanding the half-lives
We have two elements, P and Q, with half-lives of 10 minutes and 15 minutes, respectively. This means that after each half-life, the number of radioactive atoms will reduce to half of its previous value.

### Step 2: Initial conditions
Let's assume that the initial number of atoms of both elements P and Q is \(N_0\). Since the mixture contains equal numbers of atoms, we can denote the initial number of atoms of P as \(N_0\) and the initial number of atoms of Q as \(N_0\).

### Step 3: Decay of element P
For element P, with a half-life of 10 minutes:
- After the first 10 minutes, the number of atoms left = \( \frac{N_0}{2} \)
- After the next 10 minutes (20 minutes total), the number of atoms left = \( \frac{N_0}{4} \)
- After the next 10 minutes (30 minutes total), the number of atoms left = \( \frac{N_0}{8} \)

So, after 30 minutes, the remaining number of atoms of P is \( N_P = \frac{N_0}{8} \).

### Step 4: Decay of element Q
For element Q, with a half-life of 15 minutes:
- After the first 15 minutes, the number of atoms left = \( \frac{N_0}{2} \)
- After the next 15 minutes (30 minutes total), the number of atoms left = \( \frac{N_0}{4} \)

So, after 30 minutes, the remaining number of atoms of Q is \( N_Q = \frac{N_0}{4} \).

### Step 5: Finding the ratio of remaining atoms
Now, we need to find the ratio of the remaining atoms of P and Q:
\[ \text{Ratio} = \frac{N_P}{N_Q} = \frac{\frac{N_0}{8}}{\frac{N_0}{4}} = \frac{N_0}{8} \times \frac{4}{N_0} = \frac{4}{8} = \frac{1}{2} \]

### Final Answer
The ratio of the number of atoms of P and Q left in the mixture after 30 minutes is \(1:2\).

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