Gas Mixtures

6.6 Gas Mixtures

Learning Objective

Learn Dalton’s law of partial pressures.

One of the properties of gases is that they mix with each other. When they do so, they become a solution—a homogeneous mixture. Some of the properties of gas mixtures are easy to determine if we know the composition of the gases in the mix.

In gas mixtures, each component in the gas phase can be treated separately. Each component of the mixture shares the same temperature and volume. (Remember that gases expand to fill the volume of their container; gases in a mixture continue to do that as well.) However, each gas has its own pressure. The partial pressureThe pressure that an individual gas in a mixture has. of a gas, P_i, is the pressure that an individual gas in a mixture has. Partial pressures are expressed in torr, millimeters of mercury, or atmospheres like any other gas pressure; however, we use the term pressure when talking about pure gases and the term partial pressure when we are talking about the individual gas components in a mixture.

Dalton’s law of partial pressuresThe total pressure of a gas mixture, $P_{tot}$ , is equal to the sum of the partial pressures of the components, $P_{i}$ . states that the total pressure of a gas mixture, P_tot, is equal to the sum of the partial pressures of the components, P_i:

P_{tot} = P_{1} + P_{2} + P_{3} + \dots = \sum_{# of gases} P_{i}

Although this may seem to be a trivial law, it reinforces the idea that gases behave independently of each other.

Example 15

A mixture of H₂ at 2.33 atm and N₂ at 0.77 atm is in a container. What is the total pressure in the container?

Solution

Dalton’s law of partial pressures states that the total pressure is equal to the sum of the partial pressures. We simply add the two pressures together:

P_tot = 2.33 atm + 0.77 atm = 3.10 atm

Test Yourself

Air can be thought of as a mixture of N₂ and O₂. In 760 torr of air, the partial pressure of N₂ is 608 torr. What is the partial pressure of O₂?

Answer

152 torr

Example 16

A 2.00 L container with 2.50 atm of H₂ is connected to a 5.00 L container with 1.90 atm of O₂ inside. The containers are opened, and the gases mix. What is the final pressure inside the containers?

Solution

Because gases act independently of each other, we can determine the resulting final pressures using Boyle’s law and then add the two resulting pressures together to get the final pressure. The total final volume is 2.00 L + 5.00 L = 7.00 L. First, we use Boyle’s law to determine the final pressure of H₂:

(2.50 atm)(2.00 L) = P₂(7.00 L)

Solving for P₂, we get

P₂ = 0.714 atm = partial pressure of H₂

Now we do that same thing for the O₂:

(1.90 atm)(5.00 L) = P₂(7.00 L) P₂ = 1.36 atm = partial pressure of O₂

The total pressure is the sum of the two resulting partial pressures:

P_tot = 0.714 atm + 1.36 atm = 2.07 atm

Test Yourself

If 0.75 atm of He in a 2.00 L container is connected to a 3.00 L container with 0.35 atm of Ne and the containers are opened, what is the resulting total pressure?

Answer

0.51 atm

One of the reasons we have to deal with Dalton’s law of partial pressures is because gases are frequently collected by bubbling through water. As we will see in Chapter 10 "Solids and Liquids", liquids are constantly evaporating into a vapor until the vapor achieves a partial pressure characteristic of the substance and the temperature. This partial pressure is called a vapor pressureThe partial pressure exerted by evaporation of a liquid.. Table 6.2 "Vapor Pressure of Water versus Temperature" lists the vapor pressures of H₂O versus temperature. Note that if a substance is normally a gas under a given set of conditions, the term partial pressure is used; the term vapor pressure is reserved for the partial pressure of a vapor when the liquid is the normal phase under a given set of conditions.

Table 6.2 Vapor Pressure of Water versus Temperature

Temperature (°C)	Vapor Pressure (torr)	Temperature (°C)	Vapor Pressure (torr)
5	6.54	30	31.84
10	9.21	35	42.20
15	12.79	40	55.36
20	17.54	50	92.59
21	18.66	60	149.5
22	19.84	70	233.8
23	21.08	80	355.3
24	22.39	90	525.9
25	23.77	100	760.0

Any time a gas is collected over water, the total pressure is equal to the partial pressure of the gas plus the vapor pressure of water. This means that the amount of gas collected will be less than the total pressure suggests.

Example 17

Hydrogen gas is generated by the reaction of nitric acid and elemental iron. The gas is collected in an inverted 2.00 L container immersed in a pool of water at 22°C. At the end of the collection, the partial pressure inside the container is 733 torr. How many moles of H₂ gas were generated?

Solution

We need to take into account that the total pressure includes the vapor pressure of water. According to Table 6.2 "Vapor Pressure of Water versus Temperature", the vapor pressure of water at 22°C is 19.84 torr. According to Dalton’s law of partial pressures, the total pressure equals the sum of the pressures of the individual gases, so

733 torr = P_{H_{2}} + P_{H_{2} O} = P_{H_{2}} + 19 . 84 torr

We solve by subtracting:

P_{H_{2}} = 713 torr

Now we can use the ideal gas law to determine the number of moles (remembering to convert temperature to kelvins, making it 295 K):

(713 torr)(2 .00 L) = n (62.36 \frac{L \cdot atm}{mol \cdot K}) (295 K)

All the units cancel except for mol, which is what we are looking for. So

n = 0.0775 mol H₂ collected

Test Yourself

CO₂, generated by the decomposition of CaCO₃, is collected in a 3.50 L container over water. If the temperature is 50°C and the total pressure inside the container is 833 torr, how many moles of CO₂ were generated?

Answer

0.129 mol

Finally, we introduce a new unit that can be useful, especially for gases. The mole fractionThe ratio of the number of moles of a component in a mixture divided by the total number of moles in the sample., χ_i, is the ratio of the number of moles of component i in a mixture divided by the total number of moles in the sample:

χ_{i} = \frac{moles of component i}{total number of moles}

(χ is the lowercase Greek letter chi.) Note that mole fraction is not a percentage; its values range from 0 to 1. For example, consider the combination of 4.00 g of He and 5.0 g of Ne. Converting both to moles, we get

4 .00 g He \times \frac{1 mol He}{4.00 g He} = 1.00 mol He and 5 .0 g Ne \times \frac{1 mol Ne}{20.0 g Ne} = 0.25 mol Ne

The total number of moles is the sum of the two mole amounts:

total moles = 1.00 mol + 0.025 mol = 1.25 mol

The mole fractions are simply the ratio of each mole amount and the total number of moles, 1.25 mol:

\begin{array}{l} χ_{He} = \frac{1.00 mol}{1.25 mol} = 0.800 \\ χ_{Ne} = \frac{0.25 mol}{1.25 mol} = 0.200 \end{array}

The sum of the mole fractions equals exactly 1.

For gases, there is another way to determine the mole fraction. When gases have the same volume and temperature (as they would in a mixture of gases), the number of moles is proportional to partial pressure, so the mole fractions for a gas mixture can be determined by taking the ratio of partial pressure to total pressure:

χ_{i} = \frac{P_{i}}{P_{tot}}

This expression allows us to determine mole fractions without calculating the moles of each component directly.

Example 18

A container has a mixture of He at 0.80 atm and Ne at 0.60 atm. What are the mole fractions of each component?

Solution

According to Dalton’s law, the total pressure is the sum of the partial pressures:

P_tot = 0.80 atm + 0.60 atm = 1.40 atm

The mole fractions are the ratios of the partial pressure of each component and the total pressure:

\begin{array}{l} χ_{He} = \frac{0.80 atm}{1 .40 atm} = 0.57 \\ χ_{Ne} = \frac{0.60 atm}{1 .40 atm} = 0.43 \end{array}

Again, the sum of the mole fractions is exactly 1.

Test Yourself

What are the mole fractions when 0.65 atm of O₂ and 1.30 atm of N₂ are mixed in a container?

Answer

$χ_{O_{2}} = 0. 33; χ_{N_{2}} = 0. 67$

Food and Drink App: Carbonated Beverages

Carbonated beverages—sodas, beer, sparkling wines—have one thing in common: they have CO₂ gas dissolved in them in such sufficient quantities that it affects the drinking experience. Most people find the drinking experience pleasant—indeed, in the United States alone, over 1.5 × 10⁹ gal of soda are consumed each year, which is almost 50 gal per person! This figure does not include other types of carbonated beverages, so the total consumption is probably significantly higher.

All carbonated beverages are made in one of two ways. First, the flat beverage is subjected to a high pressure of CO₂ gas, which forces the gas into solution. The carbonated beverage is then packaged in a tightly-sealed package (usually a bottle or a can) and sold. When the container is opened, the CO₂ pressure is released, resulting in the well-known hiss of an opening container, and CO₂ bubbles come out of solution. This must be done with care: if the CO₂ comes out too violently, a mess can occur!

The second way a beverage can become carbonated is by the ingestion of sugar by yeast, which then generates CO₂ as a digestion product. This process is called fermentation. The overall reaction is

C₆H₁₂O₆(aq) → 2C₂H₅OH(aq) + 2CO₂(aq)

When this process occurs in a closed container, the CO₂ produced dissolves in the liquid, only to be released from solution when the container is opened. Most fine sparkling wines and champagnes are turned into carbonated beverages this way. Less-expensive sparkling wines are made like sodas and beer, with exposure to high pressures of CO₂ gas.

Key Takeaways

The pressure of a gas in a gas mixture is termed the partial pressure.
Dalton’s law of partial pressure says that the total pressure in a gas mixture is the sum of the individual partial pressures.
Collecting gases over water requires that we take the vapor pressure of water into account.
Mole fraction is another way to express the amounts of components in a mixture.

Exercises

What is the total pressure of a gas mixture containing these partial pressures: $P_{N_{2}} = 0.78 atm$ , $P_{H_{2}} = 0.33 atm$ , and $P_{O_{2}} = 1.59 atm$ ?
What is the total pressure of a gas mixture containing these partial pressures: P_Ne = 312 torr, P_He = 799 torr, and P_Ar = 831 torr?
In a gas mixture of He and Ne, the total pressure is 335 torr and the partial pressure of He is 0.228 atm. What is the partial pressure of Ne?
In a gas mixture of O₂ and N₂, the total pressure is 2.66 atm and the partial pressure of O₂ is 888 torr. What is the partial pressure of N₂?
A 3.55 L container has a mixture of 56.7 g of Ar and 33.9 g of He at 33°C. What are the partial pressures of the gases and the total pressure inside the container?
A 772 mL container has a mixture of 2.99 g of H₂ and 44.2 g of Xe at 388 K. What are the partial pressures of the gases and the total pressure inside the container?
A sample of O₂ is collected over water in a 5.00 L container at 20°C. If the total pressure is 688 torr, how many moles of O₂ are collected?
A sample of H₂ is collected over water in a 3.55 L container at 50°C. If the total pressure is 445 torr, how many moles of H₂ are collected?
A sample of CO is collected over water in a 25.00 L container at 5°C. If the total pressure is 0.112 atm, how many moles of CO are collected?
A sample of NO₂ is collected over water in a 775 mL container at 25°C. If the total pressure is 0.990 atm, how many moles of NO₂ are collected?
A sample of NO is collected over water in a 75.0 mL container at 25°C. If the total pressure is 0.495 atm, how many grams of NO are collected?
A sample of ClO₂ is collected over water in a 0.800 L container at 15°C. If the total pressure is 1.002 atm, how many grams of ClO₂ are collected?
Determine the mole fractions of each component when 44.5 g of He is mixed with 8.83 g of H₂.
Determine the mole fractions of each component when 9.33 g of SO₂ is mixed with 13.29 g of SO₃.
In a container, 4.56 atm of F₂ is combined with 2.66 atm of Cl₂. What are the mole fractions of each component?
In a container, 77.3 atm of SiF₄ are mixed with 33.9 atm of O₂. What are the mole fractions of each component?

Answers

2.70 atm
162 torr, or 0.213 atm
P_Ar = 10.0 atm; P_He = 59.9 atm; P_tot = 69.9 atm
0.183 mol
0.113 mol
0.0440 g
$χ_{He} = 0. 718$ ; $χ_{H_{2}} = 0. 282$
$χ_{F_{2}} = 0. 632$ ; $χ_{{Cl}_{3}} = 0. 368$