The Gas Laws

u The gas laws describe HOW gases behave. u They can be predicted by theory. u The amount of change can be calculated with mathematical equations.

Avogadro’s Law u Avogadro’s law states that equal volumes of different gases (at the same temperature and pressure) contain equal numbers of atoms or molecules.

Avogadro’s Law 2 Liters of Helium 2 Liters of Oxygen u has the same number of particles as..

Avogadro’s Law u The molar volume for a gas is the volume that one mole occupies at 0.00ºC and 1.00 atm. u 1 mole = 22.4 L at STP (standard temperature and pressure). u As a result, the volume of gaseous reactants and products can be expressed as small whole numbers in reactions.

 How many moles are in 45.0 L of a gas at STP? Problem 2.01 moles

 How many liters are in moles of a gas at STP? Problem 14.2 L

 In the previous unit, we used relationships between amounts (moles) and masses (grams) of reactants and products to solve stoichiometry problems.  When the reactants and/or products are gases, we can also use the relationships between amounts (moles, n) and volume (liters, V) to solve such problems. Gas Stoichiometry

 Calculate the volume of O 2 (in liters) required for the complete combustion of 7.64 L of acetylene (C 2 H 2 ) measured at the same temperature and pressure. Problem 2 C 2 H 2 (g) + 5 O 2 (g) → 4 CO 2 (g) + 2 H 2 O(l)

 According to Avogadro’s law, at the same T and P, the number of moles of gases are directly related to their volumes. Problem 2 C 2 H 2 (g) + 5 O 2 (g) → 4 CO 2 (g) + 2 H 2 O(l) Since 2 moles C 2 H 2 = 5 moles O 2, therefore 2 L C 2 H 2 = 5 L O 2

 Assuming no change in temperature and pressure, calculate the volume of O 2 (in liters) required for the complete combustion of 14.9 L of butane C 4 H 10. Problem 2 C 4 H 10 (g) + 13 O 2 (g) → 8 CO 2 (g) + 10 H 2 O(l)

 Calculate the volume of N 2 (g) created from 60.0 g NaN 3 (s) at STP. Problem 2 NaN 3 (s) → 2 Na(s) + 3 N 2 (g)  Strategy: How is this problem different from the previous combustion examples?

 Calculate the volume of N 2 (g) created from 60.0 g NaN 3 (s) at STP. Problem 2 NaN 3 (s) → 2 Na(s) + 3 N 2 (g) g NaN 3 mol NaN 3 mol N 2 LN2LN2

Pressure u Pressure is the force applied by molecules per unit of area. u Air pressure is measured using a barometer.

Standard Atmospheric Pressure u Air pressure at higher altitudes, such as on a mountaintop, is slightly lower than air pressure at sea level.

Standard Pressure (Units) u One atmosphere is equal to 760 mm Hg, 760 torr, or kPa (kilopascals).

Pressure u Perform the following pressure conversions. a) 144 kPa = _____ atm b) 795 mm Hg = _____ atm (1.42) (1.05)

Pressure u Perform the following pressure conversions. c) 669 torr = ______ kPa d) 1.05 atm = ______ mm Hg (89.2) (798)

Ideal Gases u In this unit we will assume the gases behave ideally. u Ideal gases do not really exist, but this makes the math easier and is a close approximation.

The Ideal Gas Law u P V = n R T u Pressure times volume equals the number of moles (n) times the ideal gas constant (R) times the temperature in Kelvin.

The Ideal Gas Law u R = (L atm)/(mol K) u R = (L kPa)/(mol K) u R = 62.4 (L mm Hg)/(mol K) u The one you choose depends on the unit for pressure!

Temperature u Temperature is a measure of the average kinetic energy of the particles in a sample of matter. u Temperature must be in Kelvin (K) for the IDG Law. u K = °C + 273

Example u How many moles of air are there in a 2.0 L bottle at 19 ºC and atm? u Choose the value of R based on the pressure unit. u Since atm are use, R =

Example u Second, make sure to convert degrees Celsius to Kelvin. u How many moles of air are there in a 2.0 L bottle at 19 ºC and atm? K = + 273°C19 K = 292 K

Example P 0.983(292) n = mol u How many moles of air are there in a 2.0 L bottle at 19 ºC and atm? R=TVn (2.0) K

Example u What is the pressure in atm exerted by 1.8 g of H 2 gas in a 4.3 L balloon at 27 ºC? u Choose the value of R based on the pressure unit. u Since atm is requested, R =

Example u Second, make sure to convert degrees Celsius to Kelvin. u What is the pressure in atm exerted by 1.8 g of H 2 gas in a 4.3 L balloon at 27 ºC? K = + 273°C27 K = 300. K

Example P (300 K) u What is the pressure in atm exerted by 1.8 g of H 2 gas in a 4.3 L balloon at 27 ºC? R=T Vn (4.3 L)( atm·L/mol·K) 300. K

Example u Next, you will have to change grams to moles. u What is the pressure in atm exerted by 1.8 g of H 2 gas in a 4.3 L balloon at 27 ºC? 1.8 g 2.0 g 1 mol = 0.90 mol

Example P 0.90(300.) P = 5.2 atm u What is the pressure in atm exerted by 1.8 g of H 2 gas in a 4.3 L balloon at 27 ºC? R=TVn (4.3)(0.0821) 300. K 0.90 mol

Problem u Sulfur hexafluoride (SF 6 ) is a colorless, odorless and very unreactive gas. Calculate the pressure (in atm) exerted by 1.82 moles of the gas in a steel vessel of volume 5.43 L at 69.5 ºC. P = 9.42 atm

Problem u Calculate the volume (in liters) occupied by 7.40 g of CO 2 at STP. V = 3.77 L

Problem u A sample of nitrogen gas kept in a container of volume 2.30 L and at a temperature of 32 ºC exerts a pressure of 4.7 atm. Calculate the number of moles of gas present. n = mol

Problem u A 1.30 L sample of a gas has a mass of 1.82 g at STP. What is the molar mass of the gas? 31.4 g/mol

Problem u Calculate the mass of nitrogen gas that can occupy 1.00 L at STP g

Kinetic Molecular Theory of Gases u Gas particles are much smaller than the spaces between them. The particles have negligible volume. u There are no attractive or repulsive forces between gas molecules.

Kinetic Molecular Theory of Gases Gas particles are in constant, random motion. Until they bump into something (another particle or the side of a container), particles move in a straight line.

Kinetic Molecular Theory of Gases No kinetic energy is lost when gas particles collide with each other or with the walls of their container. All gases have the same kinetic energy at a given temperature.

Ideal Gases u There are no gases for which this is true. u Real gases behave more ideally at high temperature and low pressure.

Ideal Gases u At low temperature, the gas molecules move more slowly, so attractive forces are no longer negligible. u As the pressure on a gas increases, the molecules are forced closer together and attractive forces are no longer negligible. u Therefore, real gases behave more ideally at high temperature and low pressure.

Pressure and the Number of Molecules u More molecules mean more collisions between the gas molecules themselves and more collisions between the gas molecules and the walls of the container. u Number of molecules is DIRECTLY proportional to pressure.

Pressure and the Number of Molecules u Doubling the number of gas particles in a basketball doubles the pressure.

Pressure and the Number of Molecules u Gases naturally move from areas of high pressure to low pressure because there is empty space to move in.

1 atm u If you double the number of molecules,

u If you double the number of molecules, you double the pressure. 2 atm

u As you remove molecules from a container, 4 atm

u As you remove molecules from a container, the pressure decreases. 2 atm

u As you remove molecules from a container, the pressure decreases until the pressure inside equals the pressure outside. 1 atm

Changing the Size (Volume) of the Container u In a smaller container, molecules have less room to move. u The molecules hit the sides of the container more often, striking a smaller area with the same force.

Changing the Size (Volume) of the Container u As volume decreases, pressure increases. u Volume and pressure are INVERSELY proportional.

1 atm 4 Liters u As the pressure on a gas increases,

2 atm 2 Liters u As the pressure on a gas increases, the volume decreases.

Temperature and Pressure u Raising the temperature of a gas increases the pressure if the volume is held constant. u At higher temperatures, the particles in a gas have greater kinetic energy.

Temperature and Pressure u They move faster and collide with the walls of the container more often and with greater force, so the pressure rises.

u If you start with 1 liter of gas at 1 atm pressure and 300 K and heat it to 600 K, one of 2 things happens. 300 K

u Either the volume will increase to 2 liters at 1 atm, 300 K 600 K

300 K 600 K or the pressure will increase to 2 atm while the volume remains constant.

The Combined Gas Law u The gas laws may be combined into a single law, called the combined gas law, which relates two sets of conditions of pressure, volume, and temperature by the following equation. u P 1 V2V2 T2T2 V1V1 T1T1 P2P2 =

Example u A 15 L cylinder of gas at 4.8 atm pressure at 25 ºC is heated to 75 ºC and compressed to 17 atm. What is the new volume?

Example u First, make sure the volume units in the question match. u A 15 L cylinder of gas at 4.8 atm pressure at 25 ºC is heated to 75 ºC and compressed to 17 atm. What is the new volume? There is only one volume unit!

Example u Second, make sure the pressure units in the question match. u A 15 L cylinder of gas at 4.8 atm pressure at 25 ºC is heated to 75 ºC and compressed to 17 atm. What is the new volume? They do!

Example u Third, make sure to convert degrees Celsius to Kelvin. u A 15 L cylinder of gas at 4.8 atm pressure at 25 ºC is heated to 75 ºC and compressed to 17 atm. What is the new volume? K = + 273°C25 K = 298 K

Example u A 15 L cylinder of gas at 4.8 atm pressure at 25 ºC is heated to 75 ºC and compressed to 17 atm. What is the new volume? K = + 273°C75 K = 348 K

Example P1P1 4.8 atm 348 K 298 K V 2 = 4.9 L u A 15 L cylinder of gas at 4.8 atm pressure at 25 ºC is heated to 75 ºC and compressed to 17 atm. What is the new volume? T1T1 P2P2 = T2T2 V1V1 V2V2 (15 L) 17 atm

Problem u If 6.2 L of gas at 723 mm Hg at 21 ºC is compressed to 2.2 L at 4117 mm Hg, what is the temperature of the gas? T 2 = 594 K

Problem A sample of nitrogen monoxide has a volume of 72.6 mL at a temperature of 16 °C and a pressure of kPa. What volume will the sample occupy at 24 °C and 99.3 kPa? V 2 = 78.2 mL

Problem A hot air balloon rises to an altitude of 7000 m. At that height the atmospheric pressure drops to 300. mm Hg and the temperature cools to - 33 °C. Suppose on the hot air balloon there was a small balloon filled to 1.00 L at sea level and a temperature of 27 °C. What would its volume ultimately be when it reached the height of 7000 m? V 2 = 2.03 L

Daltons’ Law of Partial Pressures u Dalton’s law of partial pressures states that the total pressure of a mixture of gases is equal to the sum of the pressures of all the gases in the mixture, as shown below. u P Total = P 1 + P 2 + P 3 + … u The partial pressure is the contribution by that gas.

u On the next slide, determine the pressure in the fourth container if all of the gas molecules from the 1 st three containers are placed in the 4 th container. Example

2 atm1 atm3 atmatm6??

Problem u What is the total pressure in a balloon filled with air if the pressure of the oxygen is 170 mm Hg and the pressure of nitrogen is 620 mm Hg? 790 mm Hg

Example u In a second balloon the total pressure is 1.30 atm. What is the pressure of oxygen (in mm Hg) if the pressure of nitrogen is 720. mm Hg?

Example u The two gas units do not match. We must convert the 1.30 atm into mm Hg atm 1 atm 760 mm Hg = 988 mm Hg

Example u P Total = P 1 + P 2 + P 3 + … 988 mm Hg = 720 mm Hg + P oxygen 268 mm Hg = P oxygen

Problem u A container has a total pressure of 846 torr and contains carbon dioxide gas and nitrogen gas. What is the pressure of carbon dioxide (in kPa) if the pressure of nitrogen is 50. kPa? 63 kPa

Problem u When a container is filled with 3 moles of H 2, 2 moles of O 2 and 4 moles of N 2, the pressure in the container is 8.7 atm. The partial pressure of H 2 is _____. 2.9 atm

Daltons’ Law of Partial Pressures u It is common to synthesize gases and collect them by displacing a volume of water.

Problem u Hydrogen was collected over water at 21°C on a day when the atmospheric pressure is 748 torr. The volume of the gas sample collected was 300. mL. The vapor pressure of water at 21°C is torr. Determine the partial pressure of the dry gas torr

Problem u A sample of oxygen gas is saturated with water vapor at 27ºC. The total pressure of the mixture is 772 mm Hg and the vapor pressure of water is 26.7 mm Hg at 27ºC. What is the partial pressure of the oxygen gas? mm Hg