An element A exist in two isotopic forms `A^15` and `A^16` . If the average atomic mass of A was found to be 15.24, then the % relative abundance of `A^15` will be

Options

A. 0.1
B. 0.2
C. 0.4
D. 0.76

Answer

To find the percentage relative abundance of the isotopes \( A^{15} \) and \( A^{16} \) given the average atomic mass of element A is 15.24, we can follow these steps:

### Step 1: Define Variables
Let:
- \( X \) = relative abundance of \( A^{15} \) (in percentage)
- \( 100 - X \) = relative abundance of \( A^{16} \) (in percentage)

### Step 2: Set Up the Equation for Average Atomic Mass
The average atomic mass can be calculated using the formula:
\[ \text{Average Atomic Mass} = \frac{( \text{mass of } A^{15} \times \text{relative abundance of } A^{15} ) + ( \text{mass of } A^{16} \times \text{relative abundance of } A^{16} )}{100} \]
Substituting the known values:
\[ 15.24 = \frac{(15 \times X) + (16 \times (100 - X))}{100} \]

### Step 3: Multiply Both Sides by 100
To eliminate the fraction, multiply both sides by 100:
\[ 15.24 \times 100 = 15X + 16(100 - X) \]
This simplifies to:
\[ 1524 = 15X + 1600 - 16X \]

### Step 4: Simplify the Equation
Combine like terms:
\[ 1524 = 1600 - X \]

### Step 5: Solve for X
Rearranging gives:
\[ X = 1600 - 1524 \]
Calculating this gives:
\[ X = 76 \]

### Step 6: Calculate Percentage
Since \( X \) is the relative abundance of \( A^{15} \):
\[ \text{Percentage relative abundance of } A^{15} = 76\% \]

### Step 7: Convert to Decimal Form
To express this as a decimal for the options provided:
\[ \text{Decimal form} = \frac{76}{100} = 0.76 \]

### Final Answer
The percentage relative abundance of \( A^{15} \) is \( 76\% \) or \( 0.76 \).

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