Question

An experiment takes $10\, min$ to raise the temperature of water in a container from $0^{\circ} C$ to $100^{\circ} C$ and another $55\, min$ to convert it totally into steam by a heater supplying heat at a uniform rate. Neglecting the specific heat of the container and taking specific heat of water to be $1\, cal / g { }^{\circ} C$, the heat of vaporisation according to this experiment will come out to be

• A. 530 cal/ g
• B. 540 cal/g
• C. 550 cal/g
• D. 560 cal/g

Answer 1

The Correct Option is C

To solve this problem, we need to determine the heat of vaporization of water based on the experiment described. The process involves two stages:

1. Heating the water from 0°C to 100°C.
2. Converting the water at 100°C into steam.

The experiment tells us that it takes 10 minutes to raise the temperature of water from 0°C to 100°C. Since the heater supplies heat at a uniform rate, this same rate applies during the phase change to steam. Let's denote this rate as \( R \) in calories per minute.

The specific heat capacity of water is given as \( 1 \, \text{cal/g}^\circ C \). The mass \( m \) of water is unknown but it is consistent between both heating stages.

Stage 1: Heating the water

In the first stage, the heat required to heat water from 0°C to 100°C can be calculated using the formula:

Q_1 = m \cdot c \cdot \Delta T = m \cdot 1 \, \text{cal/g}^\circ C \cdot (100^\circ C - 0^\circ C) = 100m \, \text{cal}

Since this process takes 10 minutes, we relate the heat absorbed to the power (rate of heat supply):

Q_1 = R \cdot 10 \, \text{min} \Rightarrow 100m = 10R \Rightarrow R = 10m

Stage 2: Vaporizing the water

In the second stage, the heat required to convert water at 100°C to steam is based on the heat of vaporization \( L \). This process takes 55 minutes:

Q_2 = m \cdot L = 55R

Substitute R = 10m from the first stage:

m \cdot L = 55 \cdot 10m \Rightarrow L = 550 \, \text{cal/g}

This calculation shows that the heat of vaporization of water based on this experiment is 550 cal/g, matching the correct answer.