To solve the problem of how many years are required for the decomposition of 5 micrograms of X into 2.5 micrograms, we follow these steps:
### Step-by-Step Solution:
1. **Identify the Reaction Order**:
- We are given the rate constant \( k = 0.5 \, \mu g/year \).
- The units of the rate constant suggest that this is a zero-order reaction because the unit of \( k \) is mass per time (i.e., \( \mu g/year \)).
2. **Write the Zero-Order Reaction Equation**:
- For a zero-order reaction, the relationship between the initial concentration \( A_0 \), final concentration \( A \), rate constant \( k \), and time \( t \) is given by:
\[
A = A_0 - kt
\]
- Here, \( A_0 = 5 \, \mu g \) (initial amount), \( A = 2.5 \, \mu g \) (final amount), and \( k = 0.5 \, \mu g/year \).
3. **Substitute the Known Values into the Equation**:
- Substitute the values into the equation:
\[
2.5 \, \mu g = 5 \, \mu g - (0.5 \, \mu g/year) \cdot t
\]
4. **Rearrange the Equation to Solve for Time \( t \)**:
- Rearranging gives:
\[
0.5 \, \mu g/year \cdot t = 5 \, \mu g - 2.5 \, \mu g
\]
\[
0.5 \, \mu g/year \cdot t = 2.5 \, \mu g
\]
5. **Solve for \( t \)**:
- Divide both sides by \( 0.5 \, \mu g/year \):
\[
t = \frac{2.5 \, \mu g}{0.5 \, \mu g/year} = 5 \, years
\]
### Final Answer:
The time required for the decomposition of 5 micrograms of X into 2.5 micrograms is **5 years**.
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