Question

The combustion of methane gas, the principal constituent of natural gas, is represented by the equation $$\begin{aligned} \mathrm{CH}_{4}(\mathrm{g})+2 \mathrm{O}_{2}(\mathrm{g}) \longrightarrow \mathrm{CO}_{2}(\mathrm{g})+& 2 \mathrm{H}_{2} \mathrm{O}(1) \\ \Delta H^{\circ} &=-890.3 \mathrm{kJ} \end{aligned}$$ (a) What mass of methane, in kilograms, must be burned to liberate \(2.80 \times 10^{7} \mathrm{kJ}\) of heat? (b) What quantity of heat, in kilojoules, is liberated in the complete combustion of \(1.65 \times 10^{4} \mathrm{L}\) of \(\mathrm{CH}_{4}(\mathrm{g})\) measured at \(18.6^{\circ} \mathrm{C}\) and \(768 \mathrm{mmHg} ?\) (c) If the quantity of heat calculated in part (b) could be transferred with \(100 \%\) efficiency to water, what volume of water, in liters, could be heated from 8.8 to \(60.0^{\circ} \mathrm{C}\) as a result?

Short answer

In part (a), the approximate mass of the methane is 50268 kg. For part (b), the heat liberated is roughly \(1.372 \times 10^{9}\) kJ. Finally, for part (c), the volume of the water that can be heated is about \(3.279 \times 10^{7}\) L.

Step-by-step solution

Calculate the mass of methane for Part (a)

The heat of combustion of methane (\(CH_{4}\)) is -890.3 kJ. This means that 890.3 kJ of heat is produced when 1 mole of \(CH_{4}\) (which is approximately 16 g) is burned. Use the relationship to find mass of \(CH_{4}\) that produces \(2.80 \times 10^{7}\) kJ of heat. Set up a ratio: \[ \frac{890.3 \mathrm{ kJ}}{16 \mathrm{g}} = \frac{2.80 \times 10^{7}\mathrm{ kJ}}{ x} \]Solve for x (mass of methane).

Calculate quantity of heat for Part (b)

The volume of methane \(1.65 \times 10^{4} \mathrm{L}\) is given under conditions of \(18.6^{\circ} \mathrm{C}\) and \(768 \mathrm{mmHg}\). Convert this to number of moles (\(n\)) using the ideal gas law, \(PV=nRT\), where R is the ideal gas constant in appropriate units. Next, use the heat of combustion to calculate the total heat produced.

Calculate the volume of water for Part (c)

Assume that all the heat generated heats the water with 100% efficiency. Use the formula for heat transfer, \(q=mc\Delta T\), where \(m\) is the mass of water, \(c\) is the specific heat capacity of water (\(4.184 \mathrm{ kJ/kg \cdot K}\)), and \(\Delta T\) is the change in temperature (final - initial). Rearrange the formula to solve for mass (m) and then convert mass to volume using the density of water (1 g/mL or 1 kg/L).

Key concepts

Methane Combustion

The combustion of methane, represented by the chemical equation \( \mathrm{CH}_{4} + 2 \mathrm{O}_{2} \rightarrow \mathrm{CO}_{2} + 2 \mathrm{H}_{2} \mathrm{O} \), is an exothermic reaction. This means that it releases energy in the form of heat. The enthalpy change, \( \Delta H \), for this reaction is \(-890.3 \mathrm{kJ/mol} \), showing that burning one mole of methane releases 890.3 kJ of heat. Methane is the simplest alkane and is a major constituent of natural gas.
This efficient energy release makes methane a valuable resource for heating and power generation. The process involves breaking chemical bonds in methane and oxygen and forming new bonds in carbon dioxide and water, which is where the energy is released. Understanding the combustion of methane helps in calculating the mass of methane needed to produce a certain amount of energy, as seen in exercises and practical applications.

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and number of moles of a gas. It's often written as \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is temperature. When dealing with gases, conditions such as temperature and pressure have significant effects on the volume and behavior of the gas.
In the context of methane combustion, the ideal gas law is essential to determine how much methane (in moles) is involved under specific conditions. For example, to find out how much heat is produced when a certain volume of methane combusts, you first convert the volume into moles using the ideal gas law. This helps in understanding real-life applications, like calculating the energy output from natural gas storage tanks.

Heat Transfer

Heat transfer refers to the movement of heat from one substance or material to another. In chemical reactions like methane combustion, heat is transferred to the surroundings. This concept is key when you calculate how much heat is produced from a reaction and how it's used to change the temperature of other substances.
When methane combusts, the heat can be transferred to water, as in the exercise, where heat raises the water's temperature. The formula \( q = mc\Delta T \) describes this process, where \( q \) is the heat transferred, \( m \) is the mass of the substance being heated, \( c \) is the specific heat capacity, and \( \Delta T \) is the temperature change. This understanding aids in imagining how efficiently energy from methane is utilized in practical situations, like heating water in boilers.

Specific Heat Capacity

Specific heat capacity is the amount of heat needed to raise the temperature of one gram of a substance by one degree Celsius. It is a property that varies between substances and plays a crucial role in thermal calculations. Methane combustion can transfer heat to different materials, for example, water, which has a specific heat capacity of \( 4.184 \mathrm{kJ/kg \, \cdot \, K} \).
Having a higher specific heat capacity like water means more energy is required to change its temperature compared to substances with lower specific heat capacities. This property allows large volumes of water to absorb significant amounts of energy, making it effective in heat absorption and temperature regulation applications. In the exercise, understanding specific heat capacity is vital to calculating how much water can be heated by the combustion of a given amount of methane.