Decomposition of X exhibits a rate constant of `0.5 mu g //` year. How many years are required for the decomposition of `5mu g` of X into `2.5 mu g` ?

Options

A. 40
B. 20
C. 50
D. 25

Explanation

To solve the problem of how many years are required for the decomposition of 5 µg of X into 2.5 µg, we will follow these steps:

### Step 1: Understand the Reaction Order
The decomposition of X is given to be a zero-order reaction. In a zero-order reaction, the rate of reaction is constant and does not depend on the concentration of the reactant.

### Step 2: Write the Rate Law for Zero-Order Reaction
For a zero-order reaction, the rate law can be expressed as:
\[ R = k \]
where \( R \) is the rate of reaction and \( k \) is the rate constant.

### Step 3: Identify the Given Values
From the problem, we have:
- Initial amount of X, \( A_0 = 5 \, \mu g \)
- Final amount of X, \( A = 2.5 \, \mu g \)
- Rate constant, \( k = 0.5 \, \mu g/year \)

### Step 4: Calculate the Change in Concentration
The change in concentration (or amount) of X can be calculated as:
\[ \Delta A = A_0 - A = 5 \, \mu g - 2.5 \, \mu g = 2.5 \, \mu g \]

### Step 5: Use the Zero-Order Reaction Formula
For a zero-order reaction, the relationship between the amount of reactant, the rate constant, and time is given by:
\[ \Delta A = k \cdot t \]
Rearranging this formula to solve for time \( t \):
\[ t = \frac{\Delta A}{k} \]

### Step 6: Substitute the Values
Now, substituting the values we have:
\[ t = \frac{2.5 \, \mu g}{0.5 \, \mu g/year} \]

### Step 7: Calculate the Time
Calculating the above expression:
\[ t = \frac{2.5}{0.5} = 5 \, years \]

### Conclusion
Thus, the time required for the decomposition of 5 µg of X into 2.5 µg is **5 years**.

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