Question

In a large building, oil is used in a steam boiler heating system. The combustion of \(1.0 \mathrm{lb}\) of oil provides \(2.4 \times 10^{7} \mathrm{~J}\). a. How many kilograms of oil are needed to heat \(150 \mathrm{~kg}\) of water from \(22^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C}\) ?

Short answer

Approximately 0.93 kg of oil is needed.

Step-by-step solution

Calculate the energy needed to heat the water

Use the formula for heat energy: \( Q = mc\Delta T \)where\( m \) is the mass of the water (150 kg),\( c \) is the specific heat capacity of water \( (4.186 \, \text{J/g}^\text{°C}) \),\( \Delta T \) is the temperature change (100°C - 22°C).First, convert the specific heat capacity to \( \, \text{J/kg}^\text{°C} \):\( c = 4.186 \, \text{J/g}^\text{°C} = 4186 \, \text{J/kg}^\text{°C} \).Calculate the energy:\( \Delta T = 100^\text{°C} - 22^\text{°C} = 78^\text{°C} \)\[ Q = 150 \, \text{kg} \times 4186 \, \text{J/kg}^\text{°C} \times 78^\text{°C} = 4.9 \times 10^7 \, \text{J} \]

Find energy provided by 1 kg of oil

First, convert 1 lb of oil to kilograms:\( 1 \, \text{lb} = 0.4536 \, \text{kg} \).Given that 1 lb of oil provides \( 2.4 \times 10^7 \, \text{J} \), we need to calculate how much energy is provided by 1 kg of oil:\[ \text{Energy per kg of oil} = \frac{2.4 \times 10^7 \, \text{J}}{0.4536 \, \text{kg}} \approx 5.29 \times 10^7 \, \text{J/kg} \]

Calculate the amount of oil needed

Use the energy needed to heat the water and the energy provided by oil per kg to find the mass of oil needed:\[ \text{Mass of oil needed} = \frac{4.9 \times 10^7 \, \text{J}}{5.29 \times 10^7 \, \text{J/kg}} \approx 0.93 \, \text{kg} \]

Key concepts

specific heat capacity

The specific heat capacity of a substance tells us how much heat energy is needed to raise one kilogram of the substance by one degree Celsius. It is denoted by the symbol c and is expressed in units of J/kg°C.

In this exercise, we used the specific heat capacity of water, which is 4186 J/kg°C. This means that 4186 joules of energy are required to raise the temperature of 1 kg of water by 1°C. Understanding this concept helps in calculating how much energy we need to heat a particular mass of a substance by a certain temperature.

energy conversion

Energy conversion is the process of changing one form of energy into another. Here, we are converting the energy provided by burning oil into thermal energy used to heat water.

In the exercise, we used the fact that burning 1 lb of oil provides 2.4 × 10^7 J of energy. Converting this to the energy per kilogram, we find that each kilogram of oil provides approximately 5.29 × 10^7 J. This is crucial for determining how much oil is needed to generate the required heat energy.

temperature change

Temperature change, denoted as ΔT, is the difference between the final temperature and the initial temperature of a substance. It is a key factor in calculating heat energy.

In our problem, we needed to heat water from 22°C to 100°C. The temperature change is calculated as:
ΔT = 100°C - 22°C = 78°C.

This value is then used in the formula for heat energy, Q = mcΔT, to find out the total energy required for this temperature increase.

mass of oil

The mass of oil needed to achieve a certain amount of heat energy can be found by using the energy values and oil combustion energy per kg.

Given the energy required to heat the water, 4.9 × 10^7 J, and the energy provided by 1 kg of oil, approximately 5.29 × 10^7 J, we calculate the mass of oil needed as:
Mass of oil needed = Energy required / Energy per kg of oil = 4.9 × 10^7 J / 5.29 × 10^7 J/kg ≈ 0.93 kg.

So, to heat 150 kg of water from 22°C to 100°C, we need approximately 0.93 kg of oil.