General Chemistry Chapter 9 Section 3: Stoichiometry of Gaseous Substances, Mixtures, and Reactions

When a reaction produces a gas that is collected above water, the trapped gas is a mixture of the gas produced by the reaction and water vapor. If the collection flask is appropriately positioned to equalize the water levels both within and outside the flask, the pressure of the trapped gas mixture will

equal the atmospheric pressure outside the flask *However, there is another factor we must consider when we measure the pressure of the gas by this method. Water evaporates and there is always gaseous water (water vapor) above a sample of liquid water

The vapor pressure of water

is the pressure exerted by water vapor in equilibrium with liquid water in a closed container, depends on the temperature

For gases, molar amount can be derived from convenient experimental measurements of ___________________. Therefore, these measurements are useful in assessing the stoichiometry of pure gases, gas mixtures, and chemical reactions involving gases

pressure, temperature, and volume

Chemical stoichiometry describes the

quantitative relationships between reactants and products in chemical reactions

As a gas is collected over water, it becomes saturated with water vapor and the total pressure of the mixture equals the partial pressure of the gas plus the partial pressure of the water vapor. The pressure of the pure gas is therefore equal to the

total pressure minus the pressure of the water vapor—this is referred to as the "dry" gas pressure, that is, the pressure of the gas only, without water vapor

If we know the volume, pressure, and temperature of a gas,

we can use the ideal gas equation to calculate how many moles of the gas are present

When the identity of a gas is unknown, measurements of the mass, pressure, volume, and temperature of a sample can be used to calculate the molar mass of the gas (a useful property for identification purposes). Combining the ideal gas equation 𝑃𝑉=𝑛𝑅𝑇 and the definition of molarity ℳ = 𝑚/n yields the following equation:

ℳ = 𝑚𝑅𝑇/𝑃𝑉

We can extend Avogadro's law (that the volume of a gas is directly proportional to the number of moles of the gas) to chemical reactions with gases:

Gases combine, or react, in definite and simple proportions by volume, provided that all gas volumes are measured at the same temperature and pressure

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. The pressure exerted by each individual gas in a mixture is called its

Partial pressure

Example: Determining the Molar Mass of a Volatile Liquid The approximate molar mass of a volatile liquid can be determined by: 1.) Heating a sample of the liquid in a flask with a tiny hole at the top, which converts the liquid into gas that may escape through the hole 2.) Removing the flask from heat at the instant when the last bit of liquid becomes gas, at which time the flask will be filled with only gaseous sample at ambient pressure 3.) Sealing the flask and permitting the gaseous sample to condense to liquid, and then weighing the flask to determine the sample's mass Using this procedure, a sample of chloroform gas weighing 0.494 g is collected in a flask with a volume of 129 cm3 at 99.6 °C when the atmospheric pressure is 742.1 mm Hg. What is the approximate molar mass of chloroform?

Since ℳ = 𝑚/𝑛 and 𝑛 =𝑃𝑉/𝑅𝑇, substituting and rearranging gives ℳ = 𝑚𝑅𝑇/𝑃𝑉, then ℳ = 𝑚𝑅𝑇/𝑃𝑉 = [(0.494 g) × 0.08206 L·atm/mol K × 372.8 K] / 0.976 atm × 0.129 L = 120g/mol

Example: Pressure of a Gas Collected Over Water If 0.200 L of argon is collected over water at a temperature of 26 °C and a pressure of 750 torr in a system, what is the partial pressure of argon?

Solution: According to Dalton's law, the total pressure in the bottle (750 torr) is the sum of the partial pressure of argon and the partial pressure of gaseous water: 𝑃T = 𝑃Ar + 𝑃H2O Rearranging this equation to solve for the pressure of argon gives: 𝑃Ar = 𝑃T − 𝑃H2O The pressure of water vapor above a sample of liquid water at 26 °C is 25.2 torr, so: 𝑃Ar = 750torr − 25.2torr = 725torr

Example: Volumes of Reacting Gases Ammonia is an important fertilizer and industrial chemical. Suppose that a volume of 683 billion cubic feet of gaseous ammonia, measured at 25 °C and 1 atm, was manufactured. What volume of H2( g), measured under the same conditions, was required to prepare this amount of ammonia by reaction with N2? N2(𝑔)+3H2(𝑔)⟶2NH3(𝑔)

Solution: Because equal volumes of H2 and NH3 contain equal numbers of molecules and each three molecules of H2 that react produce two molecules of NH3, the ratio of the volumes of H2 and NH3 will be equal to 3:2. Two volumes of NH3, in this case in units of billion ft^3, will be formed from three volumes of H2: 683 billion ft^3 NH3 × 3 billion ft^3 H2 / 2 billion ft^3 NH3 = 1.02×10^3 billion ft^3 H2 The manufacture of 683 billion ft^3 of NH3 required 1020 billion ft^3 of H2. (At 25 °C and 1 atm, this is the volume of a cube with an edge length of approximately 1.9 miles.)

Example: gas unknown Determining the Molecular Formula of a Gas from its Molar Mass and Empirical Formula Cyclopropane, a gas once used with oxygen as a general anesthetic, is composed of 85.7% carbon and 14.3% hydrogen by mass. Find the empirical formula. If 1.56 g of cyclopropane occupies a volume of 1.00 L at 0.984 atm and 50 °C, what is the molecular formula for cyclopropane?

Solution: First determine the empirical formula of the gas. Assume 100 g and convert the percentage of each element into grams. Determine the number of moles of carbon and hydrogen in the 100-g sample of cyclopropane. Divide by the smallest number of moles to relate the number of moles of carbon to the number of moles of hydrogen. In the last step, realize that the smallest whole number ratio is the empirical formula: 85.7 g C × 1 mol C/12.01 g C = 7.136 mol C: 7.136/7.136=1.00 mol C 14.3 g H × 1 mol H/1.01 g H = 14.158 mol H: 14.158/7.136=1.98 mol H Empirical formula is CH2 [empirical mass (EM) of 14.03 g/empirical unit]. Next, use the provided values for mass, pressure, temperature and volume to compute the molar mass of the gas: ℳ = 𝑚𝑅𝑇/𝑃𝑉 = [(1.56g)(0.0821L·atm·mol^−1K^−1)(323K)] / (0.984atm)(1.00L) = 42.0g/mol Comparing the molar mass to the empirical formula mass shows how many empirical formula units make up a molecule: ℳ/𝐸𝑀 = (42.0g/mol)/14.0g/mol = 3 The molecular formula is thus derived from the empirical formula by multiplying each of its subscripts by three: (CH2)3=C3H6

Example: The Pressure of a Mixture of Gases A 10.0-L vessel contains 2.50 × 10^−3 mol of H2, 1.00 × 10^−3 mol of He, and 3.00 × 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 = 𝑛𝑅𝑇/𝑉 𝑃H2 = [(2.50×10^−3mol)(0.08206L atm mol^−1K^−1)(308K)] / 10.0L = 6.32×10^-3 atm 𝑃He= [(1.00×10^−3mol)(0.08206L atm mol^-1K^−1)(308K)] /10.0L = 2.53×10^−3 atm 𝑃Ne = [(3.00×10^−4mol)(0.08206L atm mol^−1K^−1)(308K)] / 10.0L = 7.58×10^-4 atm The total pressure is given by the sum of the partial pressures: 𝑃T=𝑃H2+𝑃He+𝑃Ne = (0.00632+0.00253+0.00076)atm = 9.61×10^-3 atm

Example: 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 =ℳ𝑃/ 𝑅𝑇 = [(28.01g/mol)(1.00atm)] / (0.0821L·atm·mol^−1K^−1)(273K) = 1.25g/L

Example: Reaction of Gases Propane, C3H8( g), is used in gas grills to provide the heat for cooking. What volume of O2( g) measured at 25 °C and 760 torr is required to react with 2.7 L of propane measured under the same conditions of temperature and pressure? *Assume that the propane undergoes complete combustion

Solution: The ratio of the volumes of C3H8 and O2 will be equal to the ratio of their coefficients in the balanced equation for the reaction: C3H8(𝑔)+5O2(𝑔)⟶3CO2(𝑔)+4H2O(𝑙) 1 volume + 5 volume ---> 3 volume + 4 volumes From the equation, we see that one volume of C3H8 will react with five volumes of O2: 2.7L C3H8 × 5L O2/ 1L C3H8 = 13.5 LO2 A volume of 13.5 L of O2 will be required to react with 2.7 L of C3H8

Volume of Gaseous Product What volume of hydrogen at 27 °C and 723 torr may be prepared by the reaction of 8.88 g of gallium with an excess of hydrochloric acid? 2Ga(𝑠)+6HCl(𝑎𝑞)⟶2GaCl3(𝑎𝑞)+3H2(𝑔)

Solution: Convert the provided mass of the limiting reactant, Ga, to moles of hydrogen produced: 8.88g Ga × 1mol Ga / 69.723g Ga × 3 molH2/2molGa = 0.191mol H2 Convert the provided temperature and pressure values to appropriate units (K and atm, respectively), and then use the molar amount of hydrogen gas and the ideal gas equation to calculate the volume of gas: 𝑉 = (𝑛𝑅𝑇/𝑃) = (0.191mol × 0.08206 L atm mol^−1K^−1 × 300 K) / 0.951 atm = 4.94 L

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: 𝑃𝑉 = 𝑛𝑅𝑇 The density d of a gas, on the other hand, is determined by its identity. The density of a substance is a characteristic property that may be used to identify the substance. 𝑑 = 𝑚/𝑉 Rearranging the ideal gas equation to isolate V and substituting into the density equation yields 𝑑 = 𝑚𝑃/ 𝑛𝑅𝑇=(𝑚𝑛)𝑃/𝑅𝑇 The ratio m/ n is the definition of molar mass, ℳ: ℳ = 𝑚/𝑛

The density equation can then be written 𝑑=ℳ𝑃/𝑅𝑇 This relation may be used for calculating the densities of gases of known identities at specified values of pressure and temperature

French nobleman Antoine Lavoisier, widely regarded as the "father of modern chemistry," changed chemistry from a qualitative to a quantitative science through his work with gases. He discovered?

The law of conservation of matter, discovered the role of oxygen in combustion reactions, determined the composition of air, explained respiration in terms of chemical reactions, and more

Here is another example of this concept, but dealing with mole fraction calculations Example: 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?

The mole fraction is given by 𝑋𝐴=𝑛𝐴/𝑛𝑇𝑜𝑡𝑎𝑙 and the partial pressure is PA = XA × PTotal. For O2, 𝑋𝑂2 = 𝑛𝑂2/𝑛𝑇𝑜𝑡𝑎𝑙 = 2.83 mol/ (2.83+8.41)mol = 0.252 and 𝑃𝑂2= 𝑋𝑂2 × 𝑃𝑇𝑜𝑡𝑎𝑙 = 0.252 × 192 kPa = 48.4 kPa For N2O, 𝑋𝑁2 = 𝑛𝑁2/𝑛Total = 8.41 mol/ (2.83+8.41)mol = 0.748 and 𝑃𝑁2 = 𝑋𝑁2 × 𝑃Total= 0.748 × 192 kPa = 143.6 kPa

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: 𝑃𝑇𝑜𝑡𝑎𝑙=𝑃𝐴+𝑃𝐵+𝑃𝐶+...=Σi𝑃i *In the equation PTotal is the total pressure of a mixture of gases, PA is the partial pressure of gas A; PB is the partial pressure of gas B; PC is the partial pressure of gas C; and so on.

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: 𝑃𝐴=𝑋𝐴×𝑃𝑇𝑜𝑡𝑎𝑙 where 𝑋𝐴=𝑛𝐴/𝑛𝑇𝑜𝑡𝑎𝑙 *where PA, XA, and nA are the partial pressure, mole fraction, and number of moles of gas A, respectively, and nTotal is the number of moles of all components in the mixture

use of stoichiometry is the

amount of substance, typically measured in moles ( n)

A simple way to collect gases that do not react with water is to capture them in a bottle that has been filled with water and inverted into a dish filled with water. The pressure of the gas inside the bottle can be made equal to the air pressure outside by raising or lowering the bottle. When the water level is the same both inside and outside the bottle, the pressure of the gas is equal to the atmospheric pressure, which can be measured with a

barometer

If we know how many moles of a gas are involved, we can

calculate the volume of a gas at any temperature and pressure

Avogadro's Law Revisited All gases that show ideal behavior contain the same number of molecules in the same volume (at the same temperature and pressure). Thus, the ratios of volumes of gases involved in a chemical reaction are given by the

coefficients in the equation for the reaction, provided that the gas volumes are measured at the same temperature and pressure

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

d = 0.0847 g/L