Question

You want to heat the air in your house with natural gas \(\left.\left(\mathrm{CH}_{4}\right) . \text { Assume your house has } 275 \mathrm{m}^{2} \text { (about } 2800 \mathrm{ft}^{2}\right)\) of floor area and that the ceilings are 2.50 m from the floors. The air in the house has a molar heat capacity of \(29.1 \mathrm{J} / \mathrm{mol} \cdot \mathrm{K} .\) (The number of moles of air in the house can be found by assuming that the average molar mass of air is \(28.9 \mathrm{g} / \mathrm{mol}\) and that the density of air at these temperatures is \(1.22 \mathrm{g} / \mathrm{L} .\) ) What mass of methane do you have to burn to heat the air from \(15.0^{\circ} \mathrm{C}\) to \(22.0^{\circ} \mathrm{C} ?\)

Short answer

Burn 118.4 g of methane.

Step-by-step solution

Calculate the volume of the house

First, we determine the volume of the house using the floor area and the height of the ceiling. The volume \( V \) can be calculated using the formula: \( V = \text{floor area} \times \text{ceiling height} \). Thus,\[ V = 275 \, \text{m}^2 \times 2.5 \, \text{m} = 687.5 \, \text{m}^3 \] Convert cubic meters to liters (since 1 cubic meter is 1000 liters):\[ V = 687.5 \, \text{m}^3 \times 1000 \, \text{L/m}^3 = 687500 \, \text{L} \]

Determine the mass of air in the house

Using the density of air, we calculate the mass \( m \) of air in the house:\[ \text{Density} = 1.22 \text{ g/L} \] Thus,\[ \text{mass of air} = 1.22 \, \text{g/L} \times 687500 \, \text{L} = 839750 \, \text{g} = 839.75 \, \text{kg} \]

Calculate the number of moles of air

Using the molar mass of air, calculate the number of moles \( n \) in the house:\[ \text{Molar mass of air} = 28.9 \, \text{g/mol} \] Thus,\[ n = \frac{839750 \, \text{g}}{28.9 \, \text{g/mol}} = 29065.4 \, \text{mol} \]

Calculate the energy required to heat the air

Calculate the energy \( Q \) required to heat the air using the molar heat capacity \( C_m = 29.1 \, \text{J/mol} \cdot \text{K} \) and temperature change:\[ \Delta T = 22.0^{\circ} \text{C} - 15.0^{\circ} \text{C} = 7.0 \, \text{K} \] \[ Q = n \times C_m \times \Delta T = 29065.4 \, \text{mol} \times 29.1 \, \text{J/mol}\cdot \text{K} \times 7.0 \, \text{K} = 5935520.8 \, \text{J} \]

Calculate the mass of methane required

The combustion of methane releases energy; the heat of combustion of methane is approximately \( 802 \text{ kJ/mol} \) or \( 802000 \text{ J/mol} \). Let \( m_{\text{CH}_4} \) be the mass of methane needed:First, calculate the moles of methane required to produce the energy:\[ n_{\text{CH}_4} = \frac{Q}{\Delta H_f} = \frac{5935520.8 \, \text{J}}{802000 \, \text{J/mol}} = 7.4 \, \text{mol} \]Then, convert moles of methane to mass using its molar mass \( = 16.0 \, \text{g/mol} \):\[ m_{\text{CH}_4} = 7.4 \, \text{mol} \times 16.0 \, \text{g/mol} = 118.4 \, \text{g} \]

Conclusion

To heat the air in the house from 15.0°C to 22.0°C, you need to burn approximately 118.4 g of methane.

Key concepts

Molar Heat Capacity

The molar heat capacity is an essential concept in thermochemistry. It refers to the amount of heat needed to raise the temperature of one mole of a substance by one Kelvin.
Molar heat capacity is typically expressed in units of J/mol·K. For example, in this exercise, the molar heat capacity of air is given as 29.1 J/mol·K.
This indicates that for each mole of air, 29.1 Joules of energy are required to increase its temperature by 1 Kelvin. This property helps us understand how much energy is involved in heating substances and is crucial in energy calculations.
This is often used in various practical applications, such as heating homes, cooling systems, and even in industrial processes.
Understanding the concept of molar heat capacity is vital when calculating energy transfers in any system, especially those involving gases like air in a given space.

Energy Calculation

Energy calculation involves finding out how much energy is required or produced during a chemical process. In this exercise, we are focused on the energy needed to heat the air in a house.

To determine this, we use the formula: \[ Q = n \times C_m \times \Delta T \]where:

• \( Q \) is the amount of heat energy (in Joules),
• \( n \) is the number of moles of the substance (air in our case),
• \( C_m \) is the molar heat capacity,
• \( \Delta T \) is the change in temperature.

In this problem, we calculated the number of moles of air in the house to be 29,065.4 mol and found the energy required to heat the air from 15°C to 22°C to be approximately 5,935,520.8 J.

This calculated energy is then used to figure out how much methane needs to be burned to produce that amount of energy. Such calculations are fundamental in ensuring efficient energy use in heating systems.

Combustion of Methane

Combustion is a chemical process where a substance reacts quickly with oxygen, releasing heat. Methane combustion is a common reaction used to generate energy.
The combustion of methane can be represented by the equation:\[\text{CH}_4 + 2\text{O}_2 \rightarrow \text{CO}_2 + 2\text{H}_2\text{O} + \, \text{Energy generated} \]
In our exercise, the heat of combustion for methane is given as 802 kJ/mol. This value indicates the energy produced when one mole of methane is burned.
Using the energy calculated for heating the house, we determine how many moles of methane are needed. Here, about 7.4 moles or 118.4 grams of methane must be combusted to produce the required energy.
Understanding methane combustion is vital in efficiently converting chemical energy to heat energy, which is essential in industrial applications and residential heating. Such efficiency ensures lower fuel use and cost-effectiveness.