Question

The enthalpy of combustion of methane is \(-890 \mathrm{~kJ}\). The volume of methane at \(0{ }^{\circ} \mathrm{C}\) and 1 atm to be burnt to produce \(2670 \mathrm{~kJ}\) heat is (a) \(33.61\) (b) \(67.21\) (c) \(7.471\) (d) \(11.21\)

Short answer

The volume of methane at 0 degrees Celsius and 1 atm to be burnt to produce 2670 kJ of heat is 67.2 liters (b) 67.21.

Step-by-step solution

- Understanding the enthalpy of combustion

Enthalpy of combustion is the heat released when one mole of a substance is completely burnt in oxygen under standard conditions. Here, it is given that the enthalpy of combustion of methane is -890 kJ. This means that for every mole of methane burnt, 890 kJ of heat is released.

- Calculate moles of methane to produce the required heat

To find the number of moles of methane needed to produce 2670 kJ of heat, we can set up the proportion: (-890 kJ) / (1 mole) = (2670 kJ) / (x moles). From this proportion, we calculate x = (2670 kJ) / (-890 kJ/mole).

- Solve for the number of moles

Solve for x (the number of moles of methane) from the previous step: x = (2670 kJ) / (-890 kJ/mole) = -3 moles. Since the heat of combustion is negative, denoting exothermic reaction, we take the absolute value to find the moles of methane: 3 moles.

- Convert moles of methane to volume

At standard temperature and pressure (0 degrees Celsius and 1 atm), one mole of any gas occupies 22.4 liters. Therefore, the volume of methane required can be calculated as follows: Volume = Number of moles \(\times\) Molar volume = 3 moles \(\times\) 22.4 L/mole.

- Calculate the final volume

Multiply the number of moles of methane by the molar volume to get the final volume: Volume = 3 moles \(\times\) 22.4 L/mole = 67.2 liters.

Key concepts

Stoichiometry

Stoichiometry is like a recipe for chemistry; it's the branch of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. Think of it as the calculations of a cook who needs to know exactly how much of each ingredient to use to get the desired amount of food.

In the context of our methane combustion problem, stoichiometry allows us to determine the exact amount of methane needed to produce a particular quantity of heat energy. To do this, we start with a balanced chemical equation and use the coefficients to establish molar ratios. For methane combustion, the balanced equation is:\[ \text{CH}_4 + 2\text{O}_2 \rightarrow \text{CO}_2 + 2\text{H}_2\text{O} \]
The molar ratio between methane and the heat can be derived from the enthalpy of combustion. For every 1 mole of methane burned, \( -890 \text{kJ} \) of heat is released. To get \( 2670 \text{kJ} \), we use a simple ratio and cross-multiplication to find the number of moles needed.

Chemical Thermodynamics

Chemical Thermodynamics focuses on the study of energy changes in chemical reactions, particularly the concept of enthalpy which is a measure of heat content in a system. The enthalpy of combustion is a key point and tells us about the energy change when a substance burns in oxygen.

The enthalpy of combustion is typically negative for exothermic reactions, as they release heat. In our exercise, the enthalpy of combustion for methane is \( -890 \text{kJ} \), meaning that for each mole of methane burned, 890 kJ of heat energy is released. To find out how much methane is needed to produce \( 2670 \text{kJ} \), we need to understand this concept of energy change well enough to do the correct conversions. We're essentially 'unpacking' the amount of methane required to pack a powerful energy punch equivalent to 2670 kJ!

Mole Concept

The mole concept is a cornerstone of chemistry—a mole is a unit similar to a dozen, but instead of 12, it represents approximately \(6.022 \times 10^{23} \) entities (Avogadro's number). In chemistry, it allows us to count particles by weighing, because one mole of any substance contains the same number of entities, whether they're molecules, atoms, or ions.

In relation to our methane problem, we calculate that 3 moles are needed to produce 2670 kJ of heat. But how can we picture this amount? One mole occupies a volume of 22.4 liters at 0°C and 1 atmosphere according to the molar gas volume concept. So to find the total volume of methane at these conditions, we multiply the number of moles by the molar volume:\[ \text{Volume} = 3 \text{moles} \times 22.4 \text{L/mole} = 67.2 \text{liters} \]
Just like measuring out flour for a cake, we are measuring out methane for a reaction. The mole concept ensures our 'baking'—or rather, our chemistry—is precise!