The degree of dissociation of water in a `0.1 M` aqueous solution of `HCl` at a certain temperature `t^(@)C` is `3.6 xx 10^(-15)`. The temperature `t` must be : [density of water at `t^(@)C = 1 gm//mL`.]

Options

A. `lt 25^(@)C`
B. `= 25^(@)C`
C. `gt 25^(@)C`
D. insufficient data to predict

Explanation

To solve the problem, we need to understand the concept of the degree of dissociation of water and how it relates to the temperature and the concentration of HCl in the solution.

### Step-by-Step Solution:

1. **Understanding Degree of Dissociation**:
The degree of dissociation (α) of water is defined as the fraction of water molecules that dissociate into ions (H⁺ and OH⁻). In pure water, this dissociation is very small, and at room temperature (25°C), it is about \(1 \times 10^{-14}\) for the ion product of water (Kw).

2. **Given Values**:
- Degree of dissociation of water, α = \(3.6 \times 10^{-15}\)
- Concentration of HCl, C = 0.1 M

3. **Relation to Ion Product of Water**:
The ion product of water (Kw) at temperature T can be expressed as:
\[ K_w = [H^+][OH^-] \]
In pure water, at equilibrium:
\[ [H^+] = [OH^-] = \alpha \cdot C \]
Therefore, we can write:
\[ K_w = (\alpha \cdot C)^2 \]

4. **Substituting Values**:
Now substituting the values into the equation:
\[ K_w = (3.6 \times 10^{-15} \cdot 0.1)^2 \]
\[ K_w = (3.6 \times 10^{-16})^2 = 1.296 \times 10^{-31} \]

5. **Finding Temperature**:
The value of Kw changes with temperature. At 25°C, Kw is \(1 \times 10^{-14}\). As temperature increases, Kw increases. We need to find a temperature where Kw is approximately \(1.296 \times 10^{-31}\).

6. **Estimating Temperature**:
Since the value of Kw at 25°C is \(1 \times 10^{-14}\) and the given Kw at the temperature T is much lower, we can conclude that the temperature T must be significantly lower than 25°C. However, since the density of water is given as 1 gm/mL, we can estimate that the temperature T must be around 0°C.

7. **Conclusion**:
Therefore, the temperature T must be less than 25°C, and since the density of water is 1 gm/mL, we can conclude that T is likely around 0°C.

### Final Answer:
The temperature \( t \) must be less than 25°C.