The susceptibility of magnesium at `300 K` is `1.2 xx 10^(-5)`. At what temperature will the susceptibility increase to `1.8 xx 10^(-5)`?

Options

A. `200 K`
B. `250 K`
C. `400K`
D. `150 K`

Explanation

To solve the problem, we will use Curie's law, which states that the susceptibility (χ) of a paramagnetic material is inversely proportional to its absolute temperature (T). This can be expressed mathematically as:

\[ \chi = \frac{C}{T} \]

where \(C\) is a constant.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Initial susceptibility, \( \chi_1 = 1.2 \times 10^{-5} \) at \( T_1 = 300 \, K \)
- Final susceptibility, \( \chi_2 = 1.8 \times 10^{-5} \)

2. **Use Curie's Law:**
According to Curie's law, we can write the relationship between the two states as:
\[ \frac{\chi_1}{\chi_2} = \frac{T_2}{T_1} \]

3. **Substitute the Known Values:**
Substitute \( \chi_1 \), \( \chi_2 \), and \( T_1 \) into the equation:
\[ \frac{1.2 \times 10^{-5}}{1.8 \times 10^{-5}} = \frac{T_2}{300} \]

4. **Simplify the Left Side:**
Calculate the left side:
\[ \frac{1.2}{1.8} = \frac{2}{3} \]
So, we have:
\[ \frac{2}{3} = \frac{T_2}{300} \]

5. **Cross-Multiply to Solve for \(T_2\):**
Cross-multiplying gives:
\[ 2 \cdot 300 = 3 \cdot T_2 \]
\[ 600 = 3 \cdot T_2 \]

6. **Divide to Find \(T_2\):**
Now, divide both sides by 3:
\[ T_2 = \frac{600}{3} = 200 \, K \]

### Final Answer:
The temperature at which the susceptibility increases to \(1.8 \times 10^{-5}\) is \(T_2 = 200 \, K\).

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