A paramagnetic material has `10^(28) atoms//m^(3)`. Its magnetic susceptibility at temperature `350K` is `2.8xx10^(-4)`. Its susceptibility at `300K` is :

Options

A. `3.267xx10^(-4)`
B. `3.672xx10^(-4)`
C. `3.726xx10^(-4)`
D. `2.672xx10^(-4)`

Explanation

To find the magnetic susceptibility of a paramagnetic material at a different temperature, we can use the relationship that the magnetic susceptibility (χ) is inversely proportional to the temperature (T). This can be expressed mathematically as:

\[ \chi \propto \frac{1}{T} \]

From this relationship, we can derive the following equation for two different temperatures:

\[ \frac{\chi_1}{\chi_2} = \frac{T_2}{T_1} \]

Where:
- \(\chi_1\) is the susceptibility at temperature \(T_1\)
- \(\chi_2\) is the susceptibility at temperature \(T_2\)

Given:
- \(\chi_1 = 2.8 \times 10^{-4}\) at \(T_1 = 350 \, K\)
- We need to find \(\chi_2\) at \(T_2 = 300 \, K\)

Now, we can rearrange the equation to solve for \(\chi_2\):

\[ \chi_2 = \chi_1 \cdot \frac{T_2}{T_1} \]

Substituting the known values:

\[ \chi_2 = 2.8 \times 10^{-4} \cdot \frac{300}{350} \]

Calculating the fraction:

\[ \frac{300}{350} = \frac{6}{7} \]

Now substituting this back into the equation:

\[ \chi_2 = 2.8 \times 10^{-4} \cdot \frac{6}{7} \]

Calculating \(\chi_2\):

\[ \chi_2 = 2.8 \times 10^{-4} \cdot 0.8571 \approx 3.2 \times 10^{-4} \]

To be more precise:

\[ \chi_2 = 2.8 \times 10^{-4} \cdot \frac{6}{7} = 2.8 \times 10^{-4} \cdot 0.8571 \approx 3.2 \times 10^{-4} \]

Thus, the susceptibility at \(300 \, K\) is approximately:

\[ \chi_2 \approx 3.26 \times 10^{-4} \]

Final answer:

\[ \chi_2 = 3.26 \times 10^{-4} \]