Naturally occuring thallium consists of two stable isotopes, Tl-203 and Tl-205 (atomic masses = 203.0 and 205.0, respectively) and has an average atomic massof 204.4. What is the percentage of Tl-205?

Options

A. 0.14
B. 0.7
C. 0.5
D. 0.647

Answer

To find the percentage of the isotope Tl-205 in naturally occurring thallium, we can set up an equation based on the average atomic mass of the isotopes and their respective abundances. Let's denote the percentage of Tl-205 as \( x \) and the percentage of Tl-203 as \( (100 - x) \).

### Step-by-Step Solution:

1. **Define Variables**:
- Let \( x \) = percentage of Tl-205.
- Therefore, the percentage of Tl-203 = \( 100 - x \).

2. **Set Up the Equation**:
The average atomic mass of thallium can be calculated using the formula:
\[ \text{Average Atomic Mass} = \left(\frac{\text{mass of Tl-203} \times \text{percentage of Tl-203}}{100}\right) + \left(\frac{\text{mass of Tl-205} \times \text{percentage of Tl-205}}{100}\right) \]
Plugging in the values we have:
\[ 204.4 = \left(\frac{203 \times (100 - x)}{100}\right) + \left(\frac{205 \times x}{100}\right) \]

3. **Multiply Through by 100**:
To eliminate the fractions, multiply the entire equation by 100:
\[ 20440 = 203 \times (100 - x) + 205 \times x \]

4. **Expand the Equation**:
Expanding the right side gives:
\[ 20440 = 20300 - 203x + 205x \]
Simplifying this, we combine like terms:
\[ 20440 = 20300 + 2x \]

5. **Isolate \( x \)**:
Rearranging the equation to solve for \( x \):
\[ 20440 - 20300 = 2x \]
\[ 1140 = 2x \]
\[ x = \frac{1140}{2} = 570 \]

6. **Calculate the Percentage**:
Since \( x \) represents the percentage of Tl-205, we have:
\[ \text{Percentage of Tl-205} = 57\% \]

7. **Calculate the Percentage of Tl-203**:
To find the percentage of Tl-203:
\[ \text{Percentage of Tl-203} = 100 - x = 100 - 57 = 43\% \]

### Final Answer:
The percentage of Tl-205 is **57%**.