A boiler heats water flowing at the rate of 2.0 litres per minute from 27°C to 77°C. If the boiler operates on a gas burner, the rate of consumption of the fuel if its heat of combustion is `4.0xx10^4J//g` , is

Options

A. `31.5g" "mi n^(-1)`
B. `1.05g" "mi n^(-1)`
C. `10.5g" "m in^(-1)`
D. `62.5g" "m in^(-1)`

Answer

To solve the problem of determining the rate of fuel consumption for the boiler, we will follow these steps:

### Step 1: Calculate the amount of water being heated
The rate of water flow is given as 2.0 liters per minute. Since the density of water is approximately 1000 grams per liter, we can convert this to grams:
\[ \text{Mass of water (m)} = \text{Volume} \times \text{Density} = 2.0 \, \text{liters/min} \times 1000 \, \text{g/liter} = 2000 \, \text{g/min} \]

### Step 2: Calculate the temperature change (ΔT)
The water is heated from 27°C to 77°C. Therefore, the change in temperature (ΔT) is:
\[ \Delta T = 77°C - 27°C = 50°C \]

### Step 3: Calculate the heat energy required (Q)
Using the specific heat capacity of water (s = 4.2 J/g°C), we can calculate the total heat energy required to heat the water:
\[ Q = m \cdot s \cdot \Delta T = 2000 \, \text{g/min} \times 4.2 \, \text{J/g°C} \times 50 \, \text{°C} \]
\[ Q = 2000 \times 4.2 \times 50 = 420000 \, \text{J/min} \]

### Step 4: Determine the rate of fuel consumption (R)
The heat of combustion of the fuel is given as \(4.0 \times 10^4 \, \text{J/g}\). To find the rate of fuel consumption, we can use the formula:
\[ R = \frac{Q}{\text{Heat of combustion}} = \frac{420000 \, \text{J/min}}{4.0 \times 10^4 \, \text{J/g}} \]
\[ R = \frac{420000}{40000} = 10.5 \, \text{g/min} \]

### Final Answer
The rate of consumption of the fuel is \(10.5 \, \text{g/min}\).

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