Problem 100 - Chemistry: The Central Science

Question

How many grams of methane \(\left[\mathrm{CH}_{4}(g)\right]\) must be combusted to heat \(1.00 \mathrm{~kg}\) of water from \(25.0^{\circ} \mathrm{C}\) to \(90.0^{\circ} \mathrm{C}\), assuming \(\mathrm{H}_{2} \mathrm{O}(l)\) as a product and \(100 \%\) efficiency in heat transfer?

Step-by-Step Solution

Answer

4.9 grams of methane must be combusted to heat 1.00 kg of water from 25.0°C to 90.0°C, assuming 100% efficiency in heat transfer.

Step 1: Balanced chemical equation for the combustion of methane

Write down the balanced chemical equation for the combustion of methane. \[ \mathrm{CH}_{4}(g) + 2\mathrm{O}_{2}(g) \rightarrow \mathrm{CO}_{2}(g) + 2\mathrm{H}_{2}\mathrm{O}(l) \]

Step 2: Calculate the heat needed to heat the water

Use the specific heat formula to calculate the amount of heat needed to raise the temperature of 1 kg of water from 25.0°C to 90.0°C. q = mcΔT, where q is the heat, m is the mass of water, c is the specific heat capacity, and ΔT is the change in temperature. \[ q = (1000\ \mathrm{g}) \times (4.18 \frac{\mathrm{J}}{\mathrm{g°C}}) \times (90.0 \mathrm{°C} - 25.0 \mathrm{°C}) \] \[ q = 272,200 \mathrm{J} \]

Step 3: Convert heat needed to moles of methane

Use the heat of combustion of methane to determine how many moles of methane are required to produce the necessary heat. The heat of combustion of methane is -890 kJ/mol, and 1 mol of methane can release 890 kJ of heat. \[ \frac{272,200 \mathrm{J}}{890,000 \frac{\mathrm{J}}{\mathrm{mol}}} = 0.306 \mathrm{mol} \]

Step 4: Convert moles of methane to grams of methane

Now, we will convert the moles of methane required into grams using the molecular weight of methane (16 g/mol). \[ 0.306\ \mathrm{mol} \times \frac{16\ \mathrm{g}}{1\ \mathrm{mol}} = 4.9\ \mathrm{g} \] Hence, 4.9 grams of methane must be combusted to heat 1.00 kg of water from 25.0°C to 90.0°C, assuming 100% efficiency in heat transfer.

Key Concepts

Chemical Equation Balancing

Understanding the principles of chemical equation balancing is crucial when studying reactions, like the combustion of methane. This process involves ensuring that the number of atoms for each element is equal on both the reactant and product sides of the equation. For the combustion of methane (\text{CH}_4), oxygen (\text{O}_2) is needed for complete combustion to carbon dioxide (\text{CO}_2) and water (\text{H}_2\text{O}), thus the balanced equation is:\[\mathrm{CH}_{4}(g) + 2\mathrm{O}_{2}(g) \rightarrow \mathrm{CO}_{2}(g) + 2\mathrm{H}_{2}\mathrm{O}(l)\] Each carbon (C), hydrogen (H), and oxygen (O) atom on the left is accounted for on the right, fulfilling the Law of Conservation of Mass.

Balancing chemical equations is a key skill in stoichiometry, which allows us to calculate how much reactants are needed or products are formed in a given chemical reaction.

Specific Heat Capacity

Specific heat capacity is a measure of how much energy is needed to raise the temperature of a substance by a certain amount. In the exercise, we calculate the energy required to heat up water, which has a high specific heat capacity of \(4.18 \frac{\text{J}}{\text{g°C}}\). This means for each gram of water, 4.18 joules of energy are needed to raise its temperature by one degree Celsius.

Role in Heat Transfer

Specific heat capacity plays a pivotal role in heat transfer in chemical reactions. It allows us to calculate the exact amount of energy exchanged during the processes, which is especially important for endothermic and exothermic reactions like the combustion of methane. By using the specific heat capacity of water, we can determine that \(272,200 \text{J}\) of energy is required to heat 1 kg of water from \(25.0^\circ \text{C}\) to \(90.0^\circ \text{C}\).

Heat Transfer in Chemical Reactions

In a chemical reaction, heat transfer occurs when energy is released or absorbed, changing the temperature of the substances involved. The combustion of methane is an exothermic reaction, meaning it releases heat. With 100% efficiency assumed, all the energy released during the combustion process is used to heat the water.

Calculating Energy Transfer

To determine the actual amount of methane needed, we must convert the energy required to heat the water into moles of methane using its heat of combustion value, which is \(-890 \text{kJ/mol}\). By understanding the heat transfer, we can correlate the amount of energy released by the burning methane to the exact amount of energy needed to achieve the desired water temperature increase.

Stoichiometry

Stoichiometry is about the quantitative relationships between the amounts of reactants and products in a chemical reaction. Once we've balanced the chemical equation and understood the amount of heat transfer, we can use stoichiometry to calculate how much methane is required to heat the water.

From Energy to Mass

Knowing that the combustion of methane releases \(890 \text{kJ}\) per mole, we can calculate that \(0.306 \text{mol}\) of methane is required to provide the necessary energy. Then, using the molar mass of methane (\text{16 g/mol}), we find that \(4.9 \text{g}\) of methane must be combusted. This exemplifies stoichiometry's role in relating the molar amounts of reacting substances to the physical mass that must be used to achieve a specific chemical process.