Question

A liquid cools from a temperature of 368 K to 358 K in 22 minutes. In the same room, the same liquid takes 12.5 minutes to cool from 358 K to 353 K. The room temperature is:

• A. \( 27.5^\circ C \)
• B. \( 27.5 K \)
• C. \( 30.5^\circ C \)
• D. \( 30.5 K \)

Hint:

Newton’s Law of Cooling states that the rate of cooling is proportional to the difference in temperature between the object and the surroundings. Use logarithmic ratios to solve for room temperature.

Answer 1

The Correct Option is A

Step 1: Apply Newton’s Law of Cooling According to Newton’s Law of Cooling: \[ \frac{dT}{dt} = -k (T - T_r) \] where: - \( T \) is the temperature of the body, - \( T_r \) is the room temperature, - \( k \) is a constant. Rearranging the equation: \[ T - T_r = (T_i - T_r) e^{-kt} \] Taking two temperature conditions and solving for \( T_r \): \[ \frac{(T_1 - T_r)}{(T_2 - T_r)} = \left( \frac{t_2}{t_1} \right) \] where: - \( T_1 = 368 K \), - \( T_2 = 358 K \), - \( T_3 = 353 K \), - \( t_1 = 22 \) minutes, - \( t_2 = 12.5 \) minutes. Using the equation: \[ \frac{(368 - T_r)}{(358 - T_r)} = \frac{22}{12.5} \] Solving for \( T_r \): \[ T_r = 27.5^\circ C \]