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Chapter 12: Problem 69

Three gases were added to the same 2.0-I. container. The total pressure of the gases was 790 torr at room temperature \(\left(25.0^{\circ} \mathrm{C}\right)\). If the mixture contained \(0.65 \mathrm{~g}\) of oxygen gas, \(0.58 \mathrm{~g}\) of carbon dioxide, and an unknown amount of nitrogen gas, determine the following: (a) the total number of moles of gas in the container (b) the number of grams of nitrogen in the container (c) the partial pressure of each gas in the mixture

Short Answer

Expert verified
Total moles of gas: 0.0856 mol. Mass of N\textsubscript{2}: 1.46 g. Partial pressures: O\textsubscript{2} = 187.51 torr, CO\textsubscript{2} = 121.69 torr, N\textsubscript{2} = 480.80 torr.

Step by step solution

01

Calculate the moles of oxygen gas

Use the molar mass of oxygen (O\textsubscript{2}), which is 32.00 g/mol, to find the moles of oxygen gas: \[ \text{Moles of O\textsubscript{2}} = \frac{0.65 \text{ g}}{32.00 \text{ g/mol}} = 0.0203 \text{ mol} \]
02

Calculate the moles of carbon dioxide gas

Use the molar mass of carbon dioxide (CO\textsubscript{2}), which is 44.01 g/mol, to find the moles of carbon dioxide: \[ \text{Moles of CO\textsubscript{2}} = \frac{0.58 \text{ g}}{44.01 \text{ g/mol}} = 0.0132 \text{ mol} \]
03

Determine total moles of gas from the pressure

Use the ideal gas law: \[ PV = nRT \] Where: - P = 790 torr = 790/760 atm = 1.0395 atm - V = 2.0 L - R = 0.0821 L atm / (K mol) - T = 25.0°C = 298 K Rearrange to find the total moles (n): \[ n = \frac{PV}{RT} = \frac{(1.0395 \text{ atm})(2.0 \text{ L})}{(0.0821 \text{ L atm} / \text{K mol})(298 \text{ K})} = 0.0856 \text{ mol} \]
04

Calculate the moles of nitrogen gas

Subtract the moles of oxygen and carbon dioxide from the total moles: \[ \text{Moles of N\textsubscript{2}} = 0.0856 \text{ mol} - 0.0203 \text{ mol of O\textsubscript{2}} - 0.0132 \text{ mol of CO\textsubscript{2}} = 0.0521 \text{ mol} \]
05

Calculate the mass of nitrogen gas

Use the molar mass of nitrogen (N\textsubscript{2}), which is 28.02 g/mol, to find the grams of nitrogen: \[ \text{Mass of N\textsubscript{2}} = \text{Moles of N\textsubscript{2}} \times \text{Molar mass of N\textsubscript{2}} = 0.0521 \text{ mol} \times 28.02 \text{ g/mol} = 1.46 \text{ g} \]
06

Calculate the partial pressures of each gas

Use Dalton's law of partial pressures: \[ P_i = \frac{n_i}{n_{\text{total}}} P_{\text{total}} \] For O\textsubscript{2}: \[ P_{\text{O\textsubscript{2}}} = \frac{0.0203 \text{ mol}}{0.0856 \text{ mol}} \times 790 \text{ torr} = 187.51 \text{ torr} \] For CO\textsubscript{2}: \[ P_{\text{CO\textsubscript{2}}} = \frac{0.0132 \text{ mol}}{0.0856 \text{ mol}} \times 790 \text{ torr} = 121.69 \text{ torr} \] For N\textsubscript{2}: \[ P_{\text{N\textsubscript{2}}} = \frac{0.0521 \text{ mol}}{0.0856 \text{ mol}} \times 790 \text{ torr} = 480.80 \text{ torr} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Partial Pressure
When we talk about gases in a mixture, each gas exerts a pressure independently. This pressure is known as partial pressure. The partial pressure of a gas is the pressure that gas would exert if it were alone in the container. This concept is crucial for understanding how gases in mixtures behave. To find the partial pressure of each gas in a mixture, you can use Dalton's Law of Partial Pressures. According to Dalton's law, the total pressure of a gas mixture is the sum of the partial pressures of each individual gas.
Moles Calculation
To calculate the amount of a substance, chemists often measure it in 'moles'. A mole is a unit that represents a specific number of particles, usually atoms or molecules. It’s like a dozen, but instead of 12, a mole is approximately 6.02 x 10\textsuperscript{23} particles, also known as Avogadro's number. To find the number of moles of a substance, use the formula: \[ \text{Moles} = \frac{\text{Mass}}{\text{Molar Mass}} \] where mass is the amount of the substance you have (in grams), and molar mass is the mass of one mole of that substance (in grams per mole). This calculation is crucial for determining the quantity of reagents and products in chemical reactions.
Molar Mass
Molar mass is essentially the weight of one mole of a particular substance. It is expressed in grams per mole (g/mol). Each element has a unique molar mass which can be found on the periodic table. For example, \[{\text O\_2}\] (oxygen) has a molar mass of 32.00 g/mol, and \[\text{CO\_2}\] (carbon dioxide) has a molar mass of 44.01 g/mol. The calculation of molar mass helps chemists understand how much of one substance is needed to react completely with another. This allows precise calculations and predictions in experiments and industrial processes.
Dalton's Law of Partial Pressures
Dalton's law of partial pressures is a principle in physics and chemistry that states that for a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases. This can be written as: \[ P\_\text{total} = P\_1 + P\_2 + P\_3 + \ldots \] where \[P\_1, P\_2, P\_3\]... are the partial pressures of the gases. Dalton's law is useful because it lets us understand and predict how gases in a mixture will behave under different conditions. This principle is widely used in fields ranging from meteorology to chemical engineering.

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Most popular questions from this chapter

(a) If you filled up three balloons with equal volumes of hydrogen, argon, and carbon dioxide gas, all at the same temperature and pressure, which balloon would weigh the most? the least? Explain. (b) If you filled up three balloons with equal masses of nitrogen, oxygen, and neon, all to the same volume at the same temperature, which would have the lowest pressure?

A baker is making strawberry cupcakes and wants to ensure they are very light. To do this, he must add enough baking soda to increase the volume of the cupcakes by \(55.0 \%\). All of the carbon dioxide produced by the decomposition of the baking soda is incorporated into the cupcakes. The baking soda or sodium bicarbonate will react with citric acid in the batter according to the reaction \(3 \mathrm{NaHCO}_{3}+\mathrm{H}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7} \longrightarrow \mathrm{Na}_{3} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{7}+3 \mathrm{H}_{2} \mathrm{O}+3 \mathrm{CO}_{2}\) which occurs with a \(63.7 \%\) yield. How many grams of baking soda must be added to \(1.32 \mathrm{~L}\) of cupcake batter at a baking temperature of \(325^{\circ} \mathrm{F}\) and a pressure of 738 torr to achieve the correct consistency?

A glass containing \(345 \mathrm{~mL}\) of a soft drink (a carbonated beverage) was left sitting out on a kitchen counter. If the \(\mathrm{CO}_{2}\) released at room temperature \(\left(25^{\circ} \mathrm{C}\right)\) and pressure \((1 \mathrm{~atm})\) occupies \(1.40 \mathrm{~L}\), at a minimum, what is the concentration (in ppm) of the \(\mathrm{CO}_{2}\) in the original soft drink? (Assume the density of the original soft drink is \(0.965 \mathrm{~g} / \mathrm{mL}\).)

A sample of \(\mathrm{CH}_{4}\) gas was collected over water at \(29^{\circ} \mathrm{C}\) and 749 \(\mathrm{mm} \mathrm{Hg}\). What is the partial pressure of the \(\mathrm{CH}_{4}\) ? (Refer to Appendix 5 for the vapor pressure of water.)

How many grams of methane gas were used to fill a balloon to a volume of \(3.0 \mathrm{~L}\) at STP?
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