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Chapter 10: Problem 65

A mixture of gases contains \(0.75 \mathrm{~mol} \mathrm{~N}_{2}, 0.30 \mathrm{~mol} \mathrm{O}_{2}\) and \(0.15 \mathrm{~mol} \mathrm{CO}_{2}\). If the total pressure of the mixture is \(1.56 \mathrm{~atm}, \mathrm{what}\) is the partial pressure of each component?

Short Answer

Expert verified
The partial pressures of each gas component in the mixture are approximately: N鈧�: \(0.975 \mathrm{~atm}\), O鈧�: \(0.390 \mathrm{~atm}\), and CO鈧�: \(0.195 \mathrm{~atm}\).

Step by step solution

01

Calculate the sum of moles of all components

Add the moles of each gas component: Nitrogen, Oxygen, and Carbon Dioxide, to get the total moles in the mixture. Total moles = moles of N鈧� + moles of O鈧� + moles of CO鈧� Total moles = 0.75 + 0.30 + 0.15
02

Calculate mole fraction of each component

The mole fraction 'x_i' of a particular gas component 'i' in the mixture is calculated by dividing the moles of the component i by the total moles in the mixture. Mole fraction of N鈧� = Moles of N鈧� / Total moles Mole fraction of O鈧� = Moles of O鈧� / Total moles Mole fraction of CO鈧� = Moles of CO鈧� / Total moles
03

Calculate partial pressure of each component

The partial pressure of each gas in the mixture can be calculated by multiplying the mole fraction of each component by the total pressure of the mixture. This can be calculated using the following formula: Partial pressure of a component i (P_i) = Mole fraction of i (x_i) * Total pressure (P_total) Partial pressure of N鈧� = Mole fraction of N鈧� * Total pressure Partial pressure of O鈧� = Mole fraction of O鈧� * Total pressure Partial pressure of CO鈧� = Mole fraction of CO鈧� * Total pressure Now we have all the information we need to find the partial pressures of each gas component in the mixture.
04

Calculate the partial pressures and provide the final answer

Total moles = 0.75 + 0.30 + 0.15 = 1.20 Mole fraction of N鈧� = Moles of N鈧� / Total moles = 0.75/1.20 Mole fraction of O鈧� = Moles of O鈧� / Total moles = 0.30/1.20 Mole fraction of CO鈧� = Moles of CO鈧� / Total moles = 0.15/1.20 Partial pressure of N鈧� = Mole fraction of N鈧� * Total pressure = (0.75/1.20) * 1.56 atm Partial pressure of O鈧� = Mole fraction of O鈧� * Total pressure = (0.30/1.20) * 1.56 atm Partial pressure of CO鈧� = Mole fraction of CO鈧� * Total pressure = (0.15/1.20) * 1.56 atm Partial pressure of N鈧� 鈮� 0.975 atm Partial pressure of O鈧� 鈮� 0.390 atm Partial pressure of CO鈧� 鈮� 0.195 atm Therefore, the partial pressures of each gas component in the mixture are approximately: N鈧�: 0.975 atm, O鈧�: 0.390 atm, and CO鈧�: 0.195 atm.

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Key Concepts

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

Mole Fraction
The mole fraction is an essential concept when dealing with gas mixtures. It introduces a way to express the concentration of a particular component within a mixture of gases. To calculate the mole fraction, you take the number of moles of the gas of interest and divide it by the total number of moles of all gases in the mixture.

Here is the formula used:
  • Mole fraction of gas i, \( x_i = \frac{n_i}{n_{total}} \)
where \( n_i \) is the number of moles of gas i and \( n_{total} \) is the sum of moles of all the gases in the mixture.

The mole fraction has a few interesting properties:
  • Mole fractions are always less than one, as they represent a part of the whole.
  • They are dimensionless, meaning they have no units.
  • The sum of all mole fractions in the mixture equals one.
Using the mole fraction allows you to determine the proportionate effect that each gas has on the overall behavior of the gas mixture.
Gas Mixtures
Gas mixtures consist of more than one type of gas, each contributing to the properties of the mixture as a whole. Understanding gas mixtures requires considering how the different gases interact and share space. A fundamental property of gas mixtures is that all components distribute evenly among one another, given sufficient time and space for the molecules to mix thoroughly. However, each gas retains its own chemical identity while mixed.

In a mixture, each gas exerts a pressure as if it were the only gas present. This pressure is called partial pressure. The sum of the partial pressures of all gases in the mixture will give you the total pressure of the gas mixture. Therefore, knowing the individual contributions of each gas is crucial when analyzing or predicting the behavior of gas mixtures.

Other than pressure, mole fraction is widely used in describing gas mixtures. It provides information about the relative amount of each gas present in the mixture.
Ideal Gas Law
The Ideal Gas Law is a cornerstone concept in chemistry and physics, describing the relationship between the four critical properties of gases: pressure, volume, temperature, and moles. It is usually written as:\[ PV = nRT \]Where:
  • \( P \) stands for pressure
  • \( V \) is volume
  • \( n \) denotes the number of moles
  • \( R \) is the ideal gas constant
  • \( T \) is temperature in Kelvin
This formula allows you to calculate any one of these variables if the other three are known. While it assumes gases behave ideally, meaning the gas particles do not interact and occupy no volume, this model works well for many gases under a wide range of conditions.

The Ideal Gas Law can be particularly handy when dealing with gas mixtures. Given that partial pressure is linked to the number of moles and the conditions in the container, you can use the law to explore how changes in one variable can affect the others. It's a very powerful tool for predicting the behavior of gases and their interactions in mixtures.

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Most popular questions from this chapter

(a) On Titan, the largest moon of Saturn, the atmospheric pressure is \(1.63105 \mathrm{~Pa}\). What is the atmospheric pressure of Titan in atm? (b) On Venus the surface atmospheric pressure is about 90 Earth atmospheres. What is the Venusian atmospheric pressure in kilopascals?

A \(4.00-\mathrm{g}\) sample of a mixture of \(\mathrm{CaO}\) and \(\mathrm{BaO}\) is placed in a 1.00-L vessel containing \(\mathrm{CO}_{2}\) gas at a pressure of 730 torr and a temperature of \(25^{\circ} \mathrm{C}\). The \(\mathrm{CO}_{2}\) reacts with the \(\mathrm{CaO}\) and \(\mathrm{BaO}\), forming \(\mathrm{CaCO}_{3}\) and \(\mathrm{BaCO}_{3}\). When the reaction is complete, the pressure of the remaining \(\mathrm{CO}_{2}\) is 150 torr. (a) Calculate the number of moles of \(\mathrm{CO}_{2}\) that have reacted. (b) Calculate the mass percentage of \(\mathrm{CaO}\) in the mixture.

A gas forms when elemental sulfur is heated carefully with AgF. The initial product boils at \(15^{\circ} \mathrm{C}\). Experiments on several samples yielded a gas density of \(0.803 \pm 0.010 \mathrm{~g} / \mathrm{L}\) for the gas at \(150 \mathrm{~mm}\) pressure and \(32{ }^{\circ} \mathrm{C}\). When the gas reacts with water, all the fluorine is converted to aqueous HF. Other products are elemental sulfur, \(\mathrm{S}_{8}\), and other sulfur-containing compounds. A 480 -mL sample of the dry gas at \(126 \mathrm{~mm}\) pressure and \(28^{\circ} \mathrm{C}\), when reacted with \(80 \mathrm{~mL}\) of water, yielded a \(0.081 \mathrm{M}\) solution of HF. The initial gaseous product undergoes a transformation over a period of time to a second compound with the same empirical and molecular formula, which boils at \(-10{ }^{\circ} \mathrm{C}\). (a) Determine the empirical and molecular formulas of the first compound formed. (b) Draw at least two reasonable Lewis structures that represent the initial compound and the one into which it is transformed over time. (c) Describe the likely geometries of these compounds, and estimate the single bond distances, given that the \(\mathrm{S}-\mathrm{S}\) bond distance in \(\mathrm{S}_{8}\) is \(2.04 \AA\) and the \(\mathrm{F}-\mathrm{F}\) distance in \(\mathrm{F}_{2}\) is \(1.43 \AA\).

Consider the combustion reaction between \(25.0 \mathrm{~mL}\) of liquid methanol (density \(=0.850 \mathrm{~g} / \mathrm{mL}\) ) and \(12.5 \mathrm{~L}\) of oxygen gas measured at STP. The products of the reaction are \(\mathrm{CO}_{2}(g)\) and \(\mathrm{H}_{2} \mathrm{O}(g)\). Calculate the number of moles of \(\mathrm{H}_{2} \mathrm{O}\) formed if the reaction goes to completion.

A 15.0-L tank is filled with helium gas at a pressure of \(1.00 \times 10^{2}\). How many balloons (each \(2.00 \mathrm{~L}\) ) can be inflated to a pressure of \(1.00 \mathrm{~atm}\), assuming that the temperature remains constant and that the tank cannot be emptied below \(1.00\) atm?
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