Question

A hot metal piece kept in air gets cool from 90°C to 80°C in t\(_1\) seconds, from 80 to 70°C in t\(_2\) seconds and from 70 to 60°C in t\(_3\) seconds. Under these circumstances, which of the following describes the relationship between t\(_1\), t\(_2\) and t\(_3\) ?

• A. t\(_1\) < t\(_2\) < t\(_3\)
• B. t\(_1\) > t\(_2\) > t\(_3\)
• C. t\(_1\) = t\(_2\) = t\(_3\)
• D. t\(_1\) = t\(_2\) + t\(_3\)

Hint:

As a hot object cools and approaches the ambient temperature, its rate of cooling slows down.
Therefore, it always takes progressively more time to cool through equal temperature intervals.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
According to Newton's law of cooling, the rate of cooling (rate of temperature change) of a hot body is directly proportional to the temperature difference between the body and its surroundings.
\[ \frac{dT}{dt} = -C (T - T_{\text{surr}}) \]

Step 2: Detailed Explanation:
Let us analyze the cooling rate at different temperature intervals:
- In the first interval (\( 90^\circ\text{C} \) to \( 80^\circ\text{C} \)), the average temperature of the metal is \( 85^\circ\text{C} \).
The temperature difference between the metal and the air is at its largest, resulting in the highest rate of heat transfer and the fastest cooling rate.
Thus, the time \( t_1 \) taken to cool by \( 10^\circ\text{C} \) will be the shortest.
- In the second interval (\( 80^\circ\text{C} \) to \( 70^\circ\text{C} \)), the average temperature is \( 75^\circ\text{C} \).
The temperature difference has decreased, meaning the rate of cooling is slower, and the time \( t_2 \) taken is longer than \( t_1 \).
- In the third interval (\( 70^\circ\text{C} \) to \( 60^\circ\text{C} \)), the average temperature is \( 65^\circ\text{C} \).
The temperature difference is at its smallest, resulting in the slowest cooling rate and the longest time \( t_3 \).
Therefore, the relationship between the time intervals is \( t_1 < t_2 < t_3 \).

Step 3: Final Answer:
The correct relationship is \( t_1 < t_2 < t_3 \).