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Chapter 18: Problem 72

Natural gas consists primarily of methane, \(\mathrm{CH}_{4}(g) .\) (a) Write a balanced chemical equation for the complete combustion of methane to produce \(\mathrm{CO}_{2}(g)\) as the only carbon-containing product. (b) Write a balanced chemical equation for the incomplete combustion of methane to produce \(\mathrm{CO}(g)\) as the only carbon-containing product. (c) At \(25^{\circ} \mathrm{C}\) and 1.0 atm pressure, what is the minimum quantity of dry air needed to combust \(1.0 \mathrm{~L}\) of \(\mathrm{CH}_{4}(g)\) completely to \(\mathrm{CO}_{2}(g) ?\)

Short Answer

Expert verified
(a) The balanced chemical equation for the complete combustion of methane is: \[\mathrm{CH_4(g) + 2O_2(g) \rightarrow CO_2(g) + 2H_2O(g)}\] (b) The balanced chemical equation for the incomplete combustion of methane is: \[\mathrm{CH_4(g) + 3/2O_2(g) \rightarrow CO(g) + 2H_2O(g)}\] (c) At 25°C and 1.0 atm pressure, the minimum quantity of dry air needed to combust 1.0 L of methane gas completely to carbon dioxide gas is 0.3887 L.

Step by step solution

01

Complete Combustion Equation

The complete combustion of methane (\(\mathrm{CH_4}\)) involves the reaction between methane and oxygen (\(\mathrm{O_2}\)), producing carbon dioxide (\(\mathrm{CO_2}\)) and water (\(\mathrm{H_2O}\)). The balanced chemical equation for this reaction is: \[\mathrm{CH_4(g) + 2O_2(g) \rightarrow CO_2(g) + 2H_2O(g)}\] (b) Incomplete Combustion of Methane
02

Incomplete Combustion Equation

The incomplete combustion of methane (\(\mathrm{CH_4}\)) occurs when there is not enough oxygen to create CO2 as a product. In such cases, the products include carbon monoxide (\(\mathrm{CO}\)) and water (\(\mathrm{H_2O}\)). The balanced chemical equation for this reaction is: \[\mathrm{CH_4(g) + 3/2O_2(g) \rightarrow CO(g) + 2H_2O(g)}\] (c) Calculation of Minimum Dry Air
03

Molar Volume of an Ideal Gas

At 25°C and 1.0 atm pressure, we can use the molar volume of an ideal gas, which is approximately 24.5 L/mol. In other words, one mole of gas occupies a volume of 24.5 L under these conditions.
04

Moles of Methane and Oxygen Needed

We have 1.0 L of methane gas, so we can calculate the number of moles of methane in this volume: \[\frac{1.0\,\text{L}\,\mathrm{CH_4}}{24.5\,\text{L}\,/\,\text{mol}} = 0.04082\,\text{mol}\,\mathrm{CH_4}\] Looking at the balanced equation for complete combustion, we see that one mole of methane reacts with two moles of oxygen to produce one mole of carbon dioxide. Therefore, to completely combust 0.04082 moles of methane, we will need: \[(0.04082\,\text{mol}\,\mathrm{CH_4})\left(\frac{2\,\text{mol}\,\mathrm{O_2}}{1\,\text{mol}\,\mathrm{CH_4}}\right) = 0.08163\,\text{mol}\,\mathrm{O_2}\]
05

Composition of Dry Air and Volume of Air Needed

Dry air is composed of approximately 21% oxygen by volume. To calculate the minimum quantity of dry air needed to provide 0.08163 moles of oxygen, we can use the following proportion: \[\frac{0.08163\,\text{mol}\,\mathrm{O_2}}{x\,\text{L}\,\text{air}} = \frac{0.21\,\text{mol}\,\mathrm{O_2}}{1\,\text{L}\,\text{air}}\] Now we solve for x: \[x = \frac{0.08163\,\text{mol}\,\mathrm{O_2}}{0.21\,\text{mol}\,\mathrm{O_2}\,/\,\text{L}\,\text{air}} = 0.3887\,\text{L}\,\text{air}\]
06

Conclusion

At 25°C and 1.0 atm pressure, the minimum quantity of dry air needed to combust 1.0 L of methane gas completely to carbon dioxide gas is 0.3887 L.

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Most popular questions from this chapter

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