Most substances contract on freezing . However, water does not belong to this category. We know that water expands on freezing. Further , coefficient of volume expansion of water in the temperature range from `0^(@)C` to `4^(@)C` is negative and above `4^(@)C` it is positive . This behaviour of water shapes the freezing of lakes as the atmospheric temperature goes down and it is still above `4^(@)C`.
If the atmospheric temperature is below `0^(@)C` and ice begins to form at t = 0 , thickness of ice sheet formed up to a time 't' will be directly proprotional to

Options

A. `t^(4)`
B. `t^(2)`
C. `t`
D. `t^(1//2)`

Explanation

To solve the problem regarding the thickness of the ice sheet formed on a lake as the atmospheric temperature drops below 0°C, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- Water expands when it freezes, which is contrary to most substances that contract. The coefficient of volume expansion for water is negative between 0°C and 4°C, indicating expansion, and positive above 4°C.
- When the atmospheric temperature falls below 0°C, ice begins to form on the surface of the water.

2. **Heat Flow and Ice Formation**:
- The rate of heat flow (heat current) can be expressed as:
\[ \frac{dq}{dt} = \frac{\Delta T}{x} \]
where \( \Delta T \) is the temperature difference and \( x \) is the thickness of the ice.

3. **Latent Heat of Fusion**:
- The heat required to form ice can be expressed as:
\[ dq = dm \cdot L \]
where \( dm \) is the mass of ice formed and \( L \) is the latent heat of fusion.

4. **Differentiating Heat Equation**:
- Differentiating both sides gives:
\[ \frac{dq}{dt} = \frac{dm}{dt} \cdot L \]

5. **Relating Mass to Volume**:
- The mass of ice can be related to its volume:
\[ dm = \rho \cdot dV \]
where \( \rho \) is the density of ice and \( dV = A \cdot dx \) (area times thickness).

6. **Substituting into the Heat Equation**:
- Substitute \( dm \) into the heat equation:
\[ \frac{dq}{dt} = \frac{dm}{dt} \cdot L = \frac{d(\rho \cdot A \cdot dx)}{dt} \cdot L \]

7. **Equating Heat Flow and Mass Formation**:
- Equate the expressions for heat flow:
\[ \frac{\Delta T}{x} = \rho A \frac{dx}{dt} \cdot L \]

8. **Rearranging the Equation**:
- Rearranging gives:
\[ \rho A L \frac{dx}{dt} = \frac{\Delta T}{x} \]

9. **Integrating**:
- Rearranging and integrating gives:
\[ x \, dx = k \, dt \]
where \( k \) is a constant that includes \( \rho, A, L, \Delta T \).

10. **Final Integration**:
- Integrating both sides from 0 to \( x \) and 0 to \( t \):
\[ \int_0^x x \, dx = \int_0^t k \, dt \]
leads to:
\[ \frac{x^2}{2} = kt \]
which implies:
\[ x^2 = 2kt \]

11. **Finding Proportionality**:
- From the equation \( x^2 = 2kt \), we can conclude that:
\[ x \propto t^{1/2} \]
Thus, the thickness of the ice sheet formed is directly proportional to the square root of time.

### Conclusion:
The thickness of the ice sheet formed up to time \( t \) is directly proportional to \( t^{1/2} \).