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Gas Laws Master Reference Sheet Formula: Clue: 1. Graham's Law _________________________________________________________________________________________ 2. Dalton's Law _________________________________________________________________________________________ 3. Boyle's Law _________________________________________________________________________________________ 4. Charles' Law _________________________________________________________________________________________ 5. Gay Lussac's Law _________________________________________________________________________________________ 6. Combined Gas Law _________________________________________________________________________________________ 7. Ideal Gas Law _________________________________________________________________________________________

Variables in Gas Law Equations n = # moles V = in L or cm3 T = in K (K = 273 + oC) P = in 1 atm = 101.3 kPa (kilopascals) = 760. mm Hg (millimeters of mercury) = 760. Torr R = 0.0821 Latm mol.K = 62.4 Lmm Hg = 62.4 LTorr mol.K mol.K = 8.31 LkPa mol.K = "a constant"

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Kinetic (Molecular) Theory (for gases) “Particles of matter are always in motion and this motion has consequences.”

1. Gases consist of large numbers of tiny particles, which have mass.

The distance between particles is great. Gas particles are neither attracted to nor repelled by each other.

2. a) Gas particles are in constant, rapid, straight-line, random motion. They possess kinetic energy

b) Gas particles have elastic collisions (with each other and container walls)

(no net loss of KE) e.g. elastic : pool balls (do not lose KE) inelastic: car crash (lose lots of KE)

3. The average KE of the gas particles is directly proportional to the Kelvin temperature of the gas.

(Reminder: KE = ½ mv2)

Properties of Gases

• very low density (V(g) x 1000 V(l or s)) • compressible and expandable: they have an indefinite volume • fluid • diffuse through each other (small) • have mass • exert pressure • 1 mol at STP = 22.4 L (STP = 0oC or 273 K; 1 atm)

Graham's Law The speed or rate of gas diffusion is related to the molar mass of the gas. e.g. At the same T, He diffuses faster than K. KE = 1/2 mv2 m = molar mass in g/mol v = velocity of particles in m/s 1/2 mava

2 = 1/2 mbvb2

va = mb vb ma

√

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Gas Pressure Pressure = Force

Area Units: Force measured in Newtons (N) Pressure measured in: Pascals (1 N/m2) – 1 kPa = 103 Pa torr mm Hg atmospheres

Standard Pressure (sea level) = 101.325 kPa = 760. mm Hg

= 760. torr = 1 atm = 14.7 psi

Instruments for Measuring Gas Pressure (diagrams on p. 389 in textbook)

Barometer – instrument used to measure atmospheric pressure, using a column of Hg • invented by Evangelista Torricelli in 1643 • atmospheric pressure presses down on a bowl of mercury, which causes a column of

mercury equal to that pressure to rise into the vacuum column.

Manometer – measures P of an enclosed gas relative to atmospheric P (open end)

Gas pressure = atmospheric pressure ± pressure of liquid in U-tube Ask: Is the gas pressure higher or lower than atmospheric pressure? If higher, add the pressure of the liquid to atmospheric P. If lower, subtract the pressure of the liquid from atmospheric P.

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Gases Chemistry

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Dalton’s Law of Partial Pressures

Ptotal = Pa + Pb + …..

Practice Problem: A 1 L sample contains 78% N2, 21% O2 and 1.0% Ar. The sample is at a pressure of 1 atm. a) What is the partial pressure of each gas in mm Hg?

b) What is the partial volume of each gas in mL?

Collecting Gas by Water Displacement

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The gas bubbles through the water in the jar and collects at the top due to its lower density. The gas has water vapor mixed with it. Ptotal = Pgas + PH2O Calculate the pressure of the dry gas: Pgas = Ptotal – PH2O Ptotal is what is measured (= atmospheric pressure), PH2O can be found in standard tables of vapor pressure of water at different temperatures

Vapor Pressure of H2O at Various Temperatures oC kPa oC kPa 0 0.61 26 3.36 5 0.87 27 3.56 10 1.23 28 3.77 15 1.71 29 4.00 16 1.81 30 4.24 17 1.93 40 7.37 18 2.07 50 12.33 19 2.20 60 19.92 20 2.33 70 31.15 21 2.49 80 47.33 22 2.64 90 70.01 23 2.81 100 101.3 24 2.99 105 120.8 25 3.17 110 143.2

Note: At 100oC, the normal boiling point, vapor pressure = atmospheric pressure = 101.3 kPa e.g. Hydrogen gas is collected over water at a total pressure of 95.0 kPa and temperature of 25oC. What is the partial pressure of hydrogen gas? (A: 91.8 kPa)

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Mole Fraction mole fraction of gas A = moles gas A = PgasA total # moles of gas Ptotal Since partial pressures of gases reflect the quantity of a particular gas, comparing partial pressure with total pressure will give you mole fraction. 1. The partial pressure of oxygen was observed to be 156 torr in air with a total

atmospheric pressure of 743 torr. Calculate the mole fraction of O2 present.

PO2 = 156 torr = 0.210 Ptotal 743 torr

2. The partial pressure of nitrogen was observed to be

590 mm Hg in air with a total atmospheric pressure of 760. mm Hg. Calculate the mole fraction of N2 present. PN2 = 590 mm Hg = 0.78 Ptotal 760. mm Hg

Note: Mole fraction has NO units.

Dalton’s Law of Partial Pressures and Mole Fraction Practice Problems

1. Determine the partial pressure of oxygen (O2) collected over water if the temperature is 20.0oC and the total (atmospheric) gas pressure is 98.0 kPa. (A: (95.7 kPa)

2. The barometer at an indoor pool reads 105.00 kPa. If the temperature in the room is

30.0oC, what is the partial pressure of the “dry” air? (A: 100.76 kPa)

3. What is the mole fraction of hydrogen (H2) in a gas mixture that has a PH2 of 5.26 kPa? The other gases in the mixture are oxygen (O2), with a PO2 of 35.2 kPa and carbon dioxide with a PCO2 of 16.1 kPa. (A: 0.0929)

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Describing Gases To describe a gas, you need: • Volume • Pressure • Temperature (K) • # particles (moles)

“Gas Laws”

Constant Temperature What happens to pressure when volume decreases?

Constant Pressure What happens to volume when temperature increases?

Constant Volume What happens to pressure when temperature increases?

Constant Volume and Temperature What happens to pressure when the # of particles is increased?

Constant Temperature and Pressure What happens to volume when the # of particles is increased?

The Combined Gas Law What happens to a gas when various conditions are changed? P1V1 = P2V2 T1 T2

The Combined Gas Law includes Boyle’s, Charles’s, and Gay-Lussac’s Laws...

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Boyle’s Law (constant T)

P1V1 = P2V2 P1V1 = P2V2 T1 T2 Demonstrates an inverse relationship between pressure and volume:

Charles’s Law (constant P)

P1V1 = P2V2 V1 = V2 T1 T2 T1 T2 Demonstrates a direct relationship between volume and temperature:

Gay-Lussac’s Law (constant V)

P1V1 = P2V2 P1 = P2 T1 T2 T1 T2 Demonstrates a direct relationship between pressure and temperature:

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Boyle's Law The pressure on 2.50 L of anaesthetic gas is changed from 760. mm Hg to 304 mm Hg. What will be the new volume if the temperature remains constant? (A: 6.25 L) Charles's Law If a sample of gas occupies 6.8 L at 327oC, what will be its volume at 27oC if the pressure does not change? (A: 3.4 L) Gay-Lussac's Law A gas has a pressure of 50.0 mm Hg at 540 K. What will be the temperature, in oC, if the pressure is 70.0 mm Hg and the volume does not change? (A: 483oC) Combined Gas Law 1. If a gas has a pressure of 2.35 atm at 25oC, and fills a container of 543 mL, what is the

new pressure if the container is increased to 750. mL at 50.1oC? (A: 1.84 atm) 2. A sample of methane that initially occupies 250. mL at 500. Pa and 500. K is

expanded to a volume of 700. mL. To what temperature will the gas need to be heated to lower the pressure of the gas to 200. Pa? (A: 560. K)

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Ideal Gases

Real Gases

• Based on kinetic molecular theory

Because particles of real gases occupy space:

• Follows gas laws at all T and P

• Follow gas laws at most T and P

• Assumes particles: - have no volume impossible - have no attraction to each other if true, would be impossible to liquefy gases (e.g. CO2 is liquid at

≥ 5.1 atm, < 56.6oC

• At high P, individual volumes count • At low T, attractions count • The more polar the molecule, the more

attraction counts

Ideal Gas Law: PV = nRT

To describe a gas completely you need to identify: V – volume P – pressure T – temperature in K n - # of moles n = mass = m = g = moles molar mass M g mol The Ideal Gas Law:

1. Can be used to derive the combined gas law 2. Is usually used to determine a missing piece of information about a gas

(requires the ideal gas constant R)

PV = nRT P, V inversely related BOYLE P, T directly related GAY-LUSSAC V, T directly related CHARLES V, n directly related AVOGADRO R = universal gas constant Solve for R at STP: T = 0oC + 273 = 273 K; P = 1 atm;

1 mole has a volume of 22.4 L at STP R = PV = (1 atm)(22.4 L) = 0.0821 Latm nT (1 mol)(273 K) molK

depends on pressure units used See Master Reference Sheet on p. 1 of this handout

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Example Problem – Ideal Gas Law Calculate the pressure, in atmospheres, of 1.65 g of helium gas at 16.0oC and occupying a volume of 3.25 L. P = ? PV = nRT V = 3.25 L n = 1.65 g (1 mol He) = 0.412 mol He 4.00 g He R = 0.0821 Latm

molK T = 16.0oC + 273 = 289 K P = nRT = (0.412 mol)(0.0821 Latm)(289 K) V 3.25 L molK = 3.01 atm

Ideal Gas Law Practice

1. A sample of carbon dioxide with a mass of 0.250 g was placed in a 350. mL container at 127oC. What is the pressure, in kPa, exerted by the gas? (A: 54 kPa) 2. A 500. g block of dry ice (solid CO2) vaporizes to a gas at room temperature. Calculate the volume of gas produced at 25oC and 975 kPa. (A: 29.0 L CO2) 3. At what temperature will 7.0 mol of helium gas exert a pressure of 1.2 atm in a 25.0 kL tank? (A: 5.2 x 104 K) 4. What mass of chlorine (Cl2) is contained in a 10.0 L tank at 27oC

and 3.50 atm? Hint: begin by solving for n. (A: 101 g)

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Density of Gases • Measured in g/L

(compared with density of liquids and solids: g/mL)

• 1 mole of any gas = 22.4 L at STP (0oC or 273 K and 1 atmosphere)

• Practice: Derive density from the ideal gas law (RQ 14.3): PV = nRT n = m/M

Sample problems: 1. What is the molar mass of a gas that has a density of 1.28 g/L at STP?

(A: 28.7 g/mol) 2. A 0.519 g gas sample is found to have a volume of 200. mL at STP.

What is the molar mass of this gas? (A: 58.1 g/mol) 3. A chemical reaction produced 98.0 mL of sulfur dioxide gas (SO2) at STP.

What was the mass (in grams) of the gas produced? (A: 0.280 g SO2) 4. A 1.25 g sample of the gaseous product of a chemical reaction was found to have a volume of 350. mL at 20.0oC and 750. mm Hg. What is the molar mass of this gas? (A: 86.8 g/mol)

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Stoichiometry and Gases at STP Calcium carbonate reacts with phosphoric acid to produce calcium phosphate, carbon dioxide, and water. 3 CaCO3(s) + 2 H3PO4(aq) Ca3(PO4)2(aq) + 3 CO2(g) + 3 H2O(l)

1. How many grams of phosphoric acid, H3PO4, react with excess calcium carbonate, CaCO3, to produce 3.74 g of Ca3(PO4)2? (A: 2.36 g H3PO4)

2. Assuming STP, how many liters of carbon dioxide are produced when 5.74 g of H3PO4 reacts with an excess of CaCO3? (A: 1.97 L)

Stoichiometry and Gases under non-STP conditions (use binder paper to answer)

Two steps: Either stoichiometry first (known = solid or liquid), followed by the Ideal Gas Law to find the volume of a gaseous product, or use the Ideal Gas Law to find the #moles of a gaseous reactant, followed by stoichiometry to find the amount of a solid or liquid product.

Practice: If water is added to magnesium nitride, ammonia gas is produced when the mixture is heated. Mg3N2(s) + 3H2O(l) 3 MgO(s) + 2 NH3(g) 1. If 10.3 g of magnesium nitride is treated with water, what volume of ammonia gas

would be collected at 24oC and 752 mm Hg? (A: 5.03 L) 2. When you produce 16.2 L of ammonia gas at 100.oC and 802 mm Hg, how many

grams of magnesium oxide are also produced? (A: 33.7 g MgO)