Question

Gas pipelines are used to deliver natural gas (methane, \(\mathrm{CH}_{4}\) ) to the various regions of the United States. The total volume of natural gas that is delivered is on the order of \(2.7 \times 10^{12} \mathrm{~L}\) per day, measured at STP. Calculate the total enthalpy change for combustion of this quantity of methane.

Short answer

The total enthalpy change for the combustion of \(2.7 \times 10^{12}\) L of methane gas at STP is approximately \(-1.09 \times 10^{14}\) kJ.

Step-by-step solution

Find the number of moles of methane

First, we need to find the number of moles of methane gas. The equation to determine the amount of gas in moles is given by: n = PV/(RT) Where: - n is the number of moles - P is the pressure - V is the volume - R is the universal gas constant - T is the temperature Since our volume is given under STP (standard temperature and pressure) conditions, P = 1 atm, T = 273.15K, and R = 0.0821 L atm/(mol K). We can now plug in the values: n = (1 atm × 2.7 × 10^12 L) / (0.0821 L atm/(mol K) × 273.15 K)

Calculate the number of moles of methane

Now, we'll use a calculator to find the number of moles of methane: n ≈ 1.22 × 10^11 moles

Determine enthalpy change per mole

Next, we need to find the enthalpy change of combustion per mole of methane. This value can be found in literature: ΔH_comb = -890.4 kJ/mol This indicates that 890.4 kJ of energy is released per mole of methane during combustion.

Calculate the total enthalpy change

Finally, we can calculate the total enthalpy change by multiplying the number of moles by the enthalpy change per mole: ΔH_total = n × ΔH_comb ΔH_total ≈ (1.22 × 10^11 moles) × (-890.4 kJ/mol) ΔH_total ≈ -1.09 × 10^14 kJ So, the total enthalpy change for the combustion of 2.7 × 10^12 L of methane gas at STP is approximately -1.09 × 10^14 kJ.

Key concepts

Standard Temperature and Pressure (STP)

In chemistry, standard temperature and pressure, commonly referred to as STP, are conditions often used as a reference point for experiments and calculations. At STP, the temperature is defined as 273.15 Kelvin (0°C) and the pressure is set at 1 atmosphere (atm). These conditions are adopted to provide a standard reference where measurements and calculations have been simplified.

This is particularly important for gases like methane used in the exercise. Under STP, one mole of an ideal gas occupies a volume of 22.414 liters. This allows for straightforward calculations in exercises assessing the volume and moles of gases, as adjustments for differing temperatures or pressures are unnecessary. It simplifies converting volume into other units important in chemical reactions, such as moles, without additional complications.

Combustion Reaction

Combustion reactions, such as the burning of methane, are important in both chemical processes and real-world applications. A combustion reaction involves the reaction of a substance with oxygen, often producing heat and light. In the case of methane, it reacts with oxygen to form carbon dioxide and water, along with a release of energy:

CH₄ + 2O₂ → CO₂ + 2H₂O + energy.

Combustion reactions are a type of exothermic reaction where energy is released into the surroundings. This energy release can be harnessed for various practical uses, like generating power or heating homes. Moreover, understanding the energy release helps in calculating the enthalpy change for the reaction as shown in the exercise.

Molecular Calculations

Molecular calculations are essential for understanding chemical reactions by determining quantities like moles, mass, or volume in a reaction's context. In the given exercise, molecular calculations begin with using the ideal gas law to convert the volume of methane at STP into moles.

The ideal gas law is defined by the equation: \[ PV = nRT \]
where 'n' is the number of moles of the gas. Given the conditions at STP, calculations become more straightforward, as standard values for pressure, temperature, and the gas constant are used. Understanding these molecular relationships is key to predicting how much product is formed or how much reactant is used in a combustion reaction.

It lays the foundation for further calculations involving energies or transformations in chemical reactions like measuring the enthalpy change.

Gas Laws

Gas laws are fundamental principles in chemistry that describe how gases behave under various conditions of pressure, temperature, and volume. They include laws such as Boyle's Law, Charles's Law, and Avogadro's Law, which individually describe the relationship between these variables.

The ideal gas law combines all these relationships in a single equation: \[ PV = nRT \]
This equation aids in calculating unknown properties of gas if other parameters are known. In the natural gas example, this law allowed us to determine the number of moles of methane from its volume at STP—a crucial step in calculating the total enthalpy change for the combustion reaction.

Understanding and applying these gas laws is critical to handling many real-world and theoretical chemical problems, ensuring precise and effective results.