A block of metal is heated to a temperature much higher than the room temperature and allowed to cool in a room free from air currents. Which of the following curves correctly represents the rate of cooling

Options

A.
B.
C.
D.

Answer

To solve the problem of determining which curve correctly represents the rate of cooling of a heated metal block in a room free from air currents, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Scenario**:
- A block of metal is heated to a temperature much higher than the room temperature and is allowed to cool in a still environment (no air currents).

2. **Apply Newton's Law of Cooling**:
- According to Newton's Law of Cooling, the rate of cooling of an object is directly proportional to the temperature difference between the object and its surroundings.
- Mathematically, this can be expressed as:
\[ \frac{d\theta}{dt} = -k(\theta - \theta_0) \]
where:
- \(\theta\) = temperature of the object at time \(t\)
- \(\theta_0\) = ambient (room) temperature
- \(k\) = positive constant of proportionality

3. **Rearranging the Equation**:
- Rearranging gives:
\[ d\theta = -k(\theta - \theta_0) dt \]

4. **Integrate the Equation**:
- Integrating both sides, we get:
\[ \int \frac{1}{\theta - \theta_0} d\theta = -k \int dt \]
- This results in:
\[ \ln|\theta - \theta_0| = -kt + C \]
- Exponentiating both sides gives:
\[ \theta - \theta_0 = Ce^{-kt} \]
- Where \(C\) is a constant determined by initial conditions.

5. **Final Temperature Equation**:
- Rearranging gives:
\[ \theta(t) = \theta_0 + Ce^{-kt} \]
- This shows that the temperature of the block decreases exponentially over time.

6. **Identify the Correct Graph**:
- The graph of \(\theta(t)\) as a function of time \(t\) will show an exponential decay, starting from the initial temperature and approaching the ambient temperature \(\theta_0\) asymptotically.
- Among the provided options, we need to identify which curve represents this exponential decay.

### Conclusion:
- The correct curve representing the rate of cooling of the metal block will be the one that shows an exponential decrease in temperature over time.