Question

A gas based geyser heats water flowing at the rate of 5.0 litres per minute from 27 °C to 87 °C. The rate of consumption of the gas is_________ g/s. (Take heat of combustion of gas = \(5.0 \times 10^4\) J/g, specific heat capacity of water = 4200 J/kg.°C).

• A. 0.21
• B. 2.1
• C. 0.42
• D. 4.2

Hint:

In calorimetry problems, always ensure your units are consistent. Here, the specific heat is in J/kg, but the heat of combustion is in J/g. It's crucial to either convert everything to kg or everything to g. The question asks for the answer in g/s, so working with grams for the gas part is convenient.

Answer 1

The Correct Option is C

To determine the rate of consumption of the gas, we need to calculate the total energy required to heat the water and then relate it to the energy provided by the combustion of the gas.

First, calculate the energy required to heat the water:

1. Volume of water heated per minute = 5.0 litres. Since the density of water is 1 kg/litre, this corresponds to 5.0 kg of water per minute.
2. The specific heat capacity of water is given as 4200 \, \text{J/kg} \cdot \degree C.
3. The temperature change \Delta T is from 27°C to 87°C:
4. \Delta T = 87 - 27 = 60 \, \degree C.
5. Using the formula for heat transfer, Q = m \cdot c \cdot \Delta T, where m is mass, c is specific heat capacity, and \Delta T is temperature change:
6. Q = 5.0 \, \text{kg} \cdot 4200 \, \text{J/kg}\cdot\degree C \cdot 60 \, \degree C = 1,260,000 \, \text{J}.

Next, relate this energy requirement to the gas consumption:

1. The heat of combustion of the gas is given as 5.0 \times 10^4 \, \text{J/g}.
2. Let x be the mass of gas consumed in grams per minute. The total energy provided by x grams of gas is:
3. x \cdot 5.0 \times 10^4 \, \text{J/g} = 1,260,000 \, \text{J}.
4. Solve for x:
5. x = \frac{1,260,000}{5.0 \times 10^4} = 25.2 \, \text{g/minute}

To convert this rate to grams per second, divide by 60:

1. x = \frac{25.2}{60} \approx 0.42 \, \text{g/s}

Thus, the rate of consumption of the gas is 0.42 g/s.