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 the correct curve that represents the rate of cooling of a heated metal block, we can follow these steps:

### Step 1: Understand Newton's Law of Cooling
Newton's Law of Cooling states that the rate of change of temperature of an object is proportional to the difference between its temperature and the ambient temperature. Mathematically, this can be expressed as:

\[ \frac{dT}{dt} = -k (T - T_0) \]

where:
- \( T \) is the temperature of the object,
- \( T_0 \) is the ambient temperature,
- \( k \) is a positive constant that depends on the characteristics of the object and the environment.

### Step 2: Set Up the Initial Conditions
Assume the initial temperature of the metal block is \( T_i \) (much higher than \( T_0 \)), and we want to find the temperature \( T_f \) at time \( t \).

### Step 3: Integrate the Equation
We can rearrange the equation and integrate it:

\[ \int \frac{dT}{T - T_0} = -k \int dt \]

This results in:

\[ \ln |T - T_0| = -kt + C \]

where \( C \) is the integration constant.

### Step 4: Solve for the Constant
To find \( C \), we can use the initial condition at \( t = 0 \):

\[ \ln |T_i - T_0| = C \]

### Step 5: Substitute Back
Substituting back into the equation gives:

\[ \ln |T - T_0| = -kt + \ln |T_i - T_0| \]

Exponentiating both sides results in:

\[ |T - T_0| = |T_i - T_0| e^{-kt} \]

### Step 6: Final Temperature Equation
This can be simplified to:

\[ T = T_0 + (T_i - T_0)e^{-kt} \]

This shows that the temperature \( T \) approaches \( T_0 \) as \( t \) increases, and the rate of cooling is exponential.

### Step 7: Analyze the Graphs
From the derived equation, we can see that the temperature decreases exponentially over time. Therefore, we need to identify which of the given curves represents an exponential decay.

### Step 8: Conclusion
After analyzing the options, we find that:

- **Option B** represents an exponential decrease in temperature over time, which matches our derived equation.
- Other options either show linear decrease or do not approach the ambient temperature correctly.

Thus, the correct answer is **Option B**.

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