Decomposition of X and Y obey first order with half lives 54 and 18 min, respectively. Starting with of [X]/[Y] = 1, calculate the time required for the ratio 4:1.

Answer

To solve the problem, we need to find the time required for the concentration ratio of X to Y to change from 1:1 to 4:1, given that both X and Y decompose according to first-order kinetics with specified half-lives.

### Step-by-Step Solution:

1. **Understand the Half-Lives**:
- The half-life of X (t₁/₂(X)) = 54 minutes.
- The half-life of Y (t₁/₂(Y)) = 18 minutes.

2. **Initial Concentration**:
- Let the initial concentrations of X and Y be [X]₀ = 1 and [Y]₀ = 1. Thus, the initial ratio [X]/[Y] = 1.

3. **First-Order Kinetics**:
- For a first-order reaction, the concentration at time t can be expressed as:
\[ [A] = [A]_0 \cdot \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} \]
- Therefore, the concentrations of X and Y at time t will be:
\[ [X] = 1 \cdot \left(\frac{1}{2}\right)^{\frac{t}{54}} \]
\[ [Y] = 1 \cdot \left(\frac{1}{2}\right)^{\frac{t}{18}} \]

4. **Set Up the Ratio**:
- We want to find the time t when the ratio [X]/[Y] = 4/1.
- This gives us the equation:
\[ \frac{[X]}{[Y]} = \frac{\left(\frac{1}{2}\right)^{\frac{t}{54}}}{\left(\frac{1}{2}\right)^{\frac{t}{18}}} = 4 \]

5. **Simplifying the Equation**:
- This can be rewritten as:
\[ \left(\frac{1}{2}\right)^{\frac{t}{54} - \frac{t}{18}} = 4 \]
- Since \(4 = 2^2\), we can express this as:
\[ \left(\frac{1}{2}\right)^{\frac{t}{54} - \frac{t}{18}} = 2^2 \]
- Taking logarithm base 2 on both sides, we have:
\[ -\left(\frac{t}{54} - \frac{t}{18}\right) = 2 \]

6. **Finding a Common Denominator**:
- The common denominator for the fractions is 54:
\[ -\left(\frac{t}{54} - \frac{3t}{54}\right) = 2 \]
- This simplifies to:
\[ -\left(-\frac{2t}{54}\right) = 2 \]
- Therefore:
\[ \frac{2t}{54} = 2 \]

7. **Solving for t**:
- Cross-multiplying gives:
\[ 2t = 108 \quad \Rightarrow \quad t = 54 \text{ minutes} \]

### Final Answer:
The time required for the ratio of concentrations of X to Y to become 4:1 is **54 minutes**.

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