The susceptibility of a paramagnetic material is Kat `27^(@)C.` At what temperature will its susceptibility be K/2?

Options

A. `600^(@)`C
B. `287^(@)C`
C. `54^(@)`C
D. `327^(@)`C

Explanation

To solve the problem, we need to use the relationship between the susceptibility (χ) of a paramagnetic material and its temperature (T). The susceptibility of a paramagnetic material is inversely proportional to its temperature. This can be mathematically expressed as:

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

This means:

\[ \chi_1 T_1 = \chi_2 T_2 \]

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

Given:
- \(\chi_1 = K\) at \(T_1 = 27^\circ C\)
- We need to find \(T_2\) when \(\chi_2 = \frac{K}{2}\)

### Step 1: Convert the temperature from Celsius to Kelvin
First, we need to convert \(T_1\) from degrees Celsius to Kelvin:

\[ T_1 = 27^\circ C + 273 = 300 \, K \]

### Step 2: Set up the equation using the proportionality
Now we can set up the equation using the relationship:

\[ \chi_1 T_1 = \chi_2 T_2 \]

Substituting the known values:

\[ K \cdot 300 = \frac{K}{2} \cdot T_2 \]

### Step 3: Solve for \(T_2\)
Now, we can simplify and solve for \(T_2\):

\[ 300K = \frac{K}{2} T_2 \]

Dividing both sides by \(K\) (assuming \(K \neq 0\)):

\[ 300 = \frac{1}{2} T_2 \]

Multiplying both sides by 2:

\[ T_2 = 600 \, K \]

### Step 4: Convert \(T_2\) back to Celsius
Now, we convert \(T_2\) back to degrees Celsius:

\[ T_2 = 600 \, K - 273 = 327^\circ C \]

### Final Answer
The temperature at which the susceptibility will be \(\frac{K}{2}\) is:

\[ \boxed{327^\circ C} \]