Library Guides: Chemistry Textbook: Gas Density, Mixtures and Reactions

Gas Density, Mixtures and Reactions

By the end of this section, you will be able to:

Use the ideal gas law to compute gas densities and molar masses
State Dalton’s law of partial pressures and use it in calculations involving gaseous mixtures

Gas Density and Molar Mass

The ideal gas law described previously in this chapter relates the properties of pressure P, volume V, temperature T, and molar amount n. This law is universal, relating these properties in identical fashion regardless of the chemical identity of the gas:

P V = n R T

The density d of a gas, on the other hand, is determined by its identity. As described in another chapter of this text, the density of a substance is a characteristic property that may be used to identify the substance.

d = \frac{m}{V}

Rearranging the ideal gas equation to isolate V and substituting into the density equation yields

d = \frac{m P}{n R T} = (\frac{m}{n}) \frac{P}{R T}

The ratio m/n is the definition of molar mass, ℳ:

ℳ = \frac{m}{n}

The density equation can then be written

d = \frac{ℳ P}{R T}

This relation may be used for calculating the densities of gases of known identities at specified values of pressure and temperature as demonstrated in example 6.9.

EXAMPLE 6.9

Measuring Gas Density

What is the density of molecular nitrogen gas at STP?

Solution:

The molar mass of molecular nitrogen, N₂, is 28.01 g/mol. Substituting this value along with standard temperature and pressure into the gas density equation yields

d = \frac{ℳ P}{R T} = \frac{(28.01 g/mol) (1.00 atm)}{(0.0821 {L·atm·mol}^{−1} K^{− 1}) (273 K)} = 1.25 g/L

Check Your Learning:

What is the density of molecular hydrogen gas at 17.0 °C and a pressure of 760 torr?

Answer: d = 0.0847 g/L

The Pressure of a Mixture of Gases: Dalton’s Law

Unless they chemically react with each other, the individual gases in a mixture of gases do not affect each other’s pressure. Each individual gas in a mixture exerts the same pressure that it would exert if it were present alone in the container (Figure 6.8). The pressure exerted by each individual gas in a mixture is called its partial pressure. This observation is summarized by Dalton’s law of partial pressures: The total pressure of a mixture of ideal gases is equal to the sum of the partial pressures of the component gases:

P_{T o t a l} = P_{A} + P_{B} + P_{C} + ... = Σ_{i} P_{i}

In the equation P_Total is the total pressure of a mixture of gases, P_A is the partial pressure of gas A; P_B is the partial pressure of gas B; P_C is the partial pressure of gas C; and so on.

This figure includes images of four gas-filled cylinders or tanks. Each has a valve at the top. The interior of the first cylinder is shaded blue. This region contains 5 small blue circles that are evenly distributed. The label “300 k P a” is on the cylinder. The second cylinder is shaded lavender. This region contains 8 small purple circles that are evenly distributed. The label “600 k P a” is on the cylinder. To the right of these cylinders is a third cylinder. Its interior is shaded pale yellow. This region contains 12 small yellow circles that are evenly distributed. The label “450 k P a” is on this region of the cylinder. An arrow labeled “Total pressure combined” appears to the right of these three cylinders. This arrow points to a fourth cylinder. The interior of this cylinder is shaded a pale green. It contains evenly distributed small circles in the following quantities and colors; 5 blue, 8 purple, and 12 yellow. This cylinder is labeled “1350 k P a.” — Figure 6.8 If equal-volume cylinders containing gas A at a pressure of 300 kPa, gas B at a pressure of 600 kPa, and gas C at a pressure of 450 kPa are all combined in the same-size cylinder, the total pressure of the mixture is 1350 kPa.

The partial pressure of gas A is related to the total pressure of the gas mixture via its mole fraction (X), a unit of concentration defined as the number of moles of a component of a solution divided by the total number of moles of all components:

P_{A} = X_{A} \times P_{T o t a l} where X_{A} = \frac{n_{A}}{n_{T o t a l}}

where P_A, X_A, and n_A are the partial pressure, mole fraction, and number of moles of gas A, respectively, and n_Total is the number of moles of all components in the mixture.

EXAMPLE 6.10

The Pressure of a Mixture of Gases

A 10.0-L vessel contains 2.50 $\times$ 10⁻³ mol of H₂, 1.00 $\times$ 10⁻³ mol of He, and 3.00 $\times$ 10⁻⁴ mol of Ne at 35 °C.

(a) What are the partial pressures of each of the gases?

(b) What is the total pressure in atmospheres?

Solution:

The gases behave independently, so the partial pressure of each gas can be determined from the ideal gas equation, using $P = \frac{n R T}{V}$ :

P_{H_{2}} = \frac{(2.50 \times 10^{−3} mol) (0.08206 L atm {mol}^{−1} K^{−1}) (308 K)}{10.0 L} = 6.32 \times 10^{−3} atm

P_{He} = \frac{(1.00 \times 10^{−3} mol) (0.08206 L atm {mol}^{−1} K^{−1}) (308 K)}{10.0 L} = 2.53 \times 10^{−3} atm

P_{Ne} = \frac{(3.00 \times 10^{−4} mol) (0.08206 L atm {mol}^{−1} K^{−1}) (308 K)}{10.0 L} = 7.58 \times 10^{−4} atm

The total pressure is given by the sum of the partial pressures:

P_{T} = P_{H_{2}} + P_{He} + P_{Ne} = (0.00632 + 0.00253 + 0.00076) atm = 9.61 \times 10^{−3} atm

Check Your Learning:

A 5.73-L flask at 25 °C contains 0.0388 mol of N₂, 0.147 mol of CO, and 0.0803 mol of H₂. What is the total pressure in the flask in atmospheres?

Answer: 1.137 atm

EXAMPLE 6.11

The Pressure of a Mixture of Gases

A gas mixture used for anesthesia contains 2.83 mol oxygen, O₂, and 8.41 mol nitrous oxide, N₂O. The total pressure of the mixture is 192 kPa.

(a) What are the mole fractions of O₂ and N₂O?

(b) What are the partial pressures of O₂ and N₂O?

Solution:

The mole fraction is given by $X_{A} = \frac{n_{A}}{n_{T o t a l}}$ and the partial pressure is P_A = X_A $\times$ P_Total.

For O₂,

X_{O_{2}} = \frac{n_{O_{2}}}{n_{T o t a l}} = \frac{2.83 mol}{(2.83 + 8.41) mol} = 0.252

and $P_{O_{2}} = X_{O_{2}} \times P_{T o t a l} = 0.252 \times 192 kPa = 48.4 kPa$

For N₂O,

X_{N_{2}} = \frac{n_{N_{2}}}{n_{Total}} = \frac{8.41 mol}{(2.83 + 8.41) mol} = 0.748

and

$P_{N_{2}} = X_{N_{2}} \times P_{Total} = 0.748 \times 192 kPa = 143.6 kPa$

Check Your Learning:

What is the pressure of a mixture of 0.200 g of H₂, 1.00 g of N₂, and 0.820 g of Ar in a container with a volume of 2.00 L at 20 °C?

Answer: 1.87 atm

Key Equations:

P_Total = P_A + P_B + P_C + … = Ʃ_iP_i
P_A = X_A P_Total
$X_{A} = \frac{n_{A}}{n_{T o t a l}}$

Footnotes

1 “Quotations by Joseph-Louis Lagrange,” last modified February 2006, accessed February 10, 2015, http://www-history.mcs.st-andrews.ac.uk/Quotations/Lagrange.html