Question

A body takes 4 minutes to cool from 100°C to 70°C. If the room temperature is 15°C, the time taken to cool from 70°C to 40°C will be:
1. 7 minutes
2. 6 minutes
3. 5 minutes
4. 2 minutes

Answer 1

Correct Answer - Option 1 : 7 minutes

CONCEPT:

• Heat always flows from a hot to a cold body.
• A body at a temperature higher than its surrounding cools down by giving away the heat to the surrounding and a body at a temperature lower than its surrounding warms up by taking heat from the surrounding.
• The rate at which a body loses heat to the surrounding is given by Newton's law of cooling.

Newton’s law of cooling

• According to Newton’s law of cooling, the rate of loss of heat from a body is directly proportional to the difference in the temperature of the body, and its surroundings.
• Mathematically it is given as,

\(\Rightarrow \frac{T_1-T_2}{t} = k\left [ \frac{T_1+T_2}{2} - T_s \right ]\)

Where T₁ = temperature before cooling, T₂ = temperature after cooling, T_s = temperature of surrounding, t = time required and 

k = Positive constant that depends on the area and nature of the surface of the body under consideration

CALCULATION:

For case 1: (T₁ = 100 ^oC, T₂ = 70 ^oC, t = 4 minutes, T_s = 15 ^oC)

\(\Rightarrow \frac{100-70}{4}=k[\frac{100+70}{2} - 15] \\ \Rightarrow\frac{30}{4}=k[85 - 15]\\\Rightarrow k=\frac{3}{28}\)

For case 2: (T₁ = 70 oC, T₂ = 40 oC, T_s = 15 oC, \(k= \frac{3}{28}\))

• The time taken to cool from 70°C to 40°C is

\(\Rightarrow\frac{70-40}{t}=\frac{3}{28}[\frac{70+40}{2} - 15] \\\Rightarrow \frac{30}{t}=\frac{3}{28}[55 - 15]\\ \Rightarrow t = 7\ minutes\)

• Hence, option 1 is correct.