Gases and Gas Laws by Owen Borville October 23, 2025
Properties of Gases: Gases differ from solids and liquids in several ways: (1) A sample of gas assumes both the shape and volume of the container. (2) Gases are compressible. (3) The densities of gases are much smaller than those of liquids and solids and are highly variable depending on temperature and pressure (4) Gases form homogenous mixtures (solutions) with one another in any proportion.
Kinetic molecular theory explains how the molecular nature of gases gives rise to their macroscopic properties. Kinetic molecular theory follows these basic assumptions: (1) A gas is composed of particles that are separated by large distances. The volume occupied by individual molecules is negligible. (2) Gas molecules are constantly in random motion, moving in straight paths, colliding with perfectly elastic collisions. (3) Gas molecules do not exert attractive or repulsive forces on one another. (4) The average kinetic energy of gas molecules in a sample is proportional to the absolute temperature: Ek (avg) ∝ T
Assumptions: Gases are compressible because molecules in the gas phase are separated by large distances. Pressure is the result of the collisions of gas molecules with the walls of their container. Decreasing volume increases the frequency of collisions. Pressure increases as collision frequency increases. Heating a sample of gas increases its average kinetic energy.
Gas molecules must move faster. Faster molecules collide more frequently and at a greater speed. Pressure increases as collision frequency increases. Ek (avg) ∝ T
Total Kinetic Energy of a mole of gas is equal to: (3/2)RT
The average kinetic energy of one molecule is Ek = 1/2 mv^2 where m is the mass, v^2 is the mean square speed.
For one mole of gas: Na(1/2mv^2) = 3/2RT where Na is Avogadro's number. M is the molar mass. Rearrange and take the square root (m x Na = M).
Urms = √3RT/M
Root Mean Square (rms) speed (Urms) is the speed of a molecule with the average kinetic energy in a gas sample. Urms is directly proportional to temperature.
Urms = √3RT/M where M is molar mass in kg/mol, R is the gas constant (8.314 J/Kmol), T is temperature.
Urms is inversely proportional to the square root of M.
Urms = √3RT/M
When two gases are at the same temperature, it is possible to compare the Urms values of the different gases.
[Urms(1)/Urms(2)] = √(M2/M1)
Diffusion is the mixing of gases as the result of random motion and frequent collisions.
Effusion is the escape of gas molecules from a container to a region of vacuum.
Graham's Law = the rate of diffusion or effusion of gas is inversely proportional to the square root of its molar mass.
Rate ∝ (1/ √M)
Pressure of gas is defined as the force applied per unit area: pressure = (force)/(area)
The SI unit of force is the newton (N), where 1 N = 1 kg*m/s^2
The SI unit of pressure is the pascal (Pa), defined as 1 N per square meter 1 Pa = 1 N/m^2
Units of pressure commonly used in science are:
Pressure at sea level = 1 atm = 101,325 Pa (standard atmosphere)
mmHg = Barometer measurement = 1 mmHg = 133.322 Pa
torr = Torricelli, inventor of barometer = 1 torr = 133.322 Pa (1 torr = 1/760 atm)
bar = 1 bar = 1 x 10^5 Pa (100,000 Pa) used in meteorology (1 bar = 750 torr)
1 atm = 760 torr = 1.01325 barr
Barometer is an instrument used to measure atmospheric pressure. A barometer consists of a long glass tube, closed at one end, filled with Hg (mercury). Tube of Hg is inverted so no air enters the tube. Open end is submerged into a container of Hg. Some Hg flows out of the tube, creating an empty space. Height of the Hg is 760 mm.
Standard atmospheric pressure (1 atm) was originally defined as the pressure that would support a column of mercury exactly 760 mm high.
P = h*d*g
h = height in meters
d - density in kg/m^3
g = is the gravitational constant (9.8 m/s^2)
1 atm = 101,325 Pa = 760 mmHg = 760 torr = 1.01325 bar = 14.7 psi
Manometers are devices used to measure pressures other than atmospheric pressure. Closed tube manometers are used to measure pressures lower than atmospheric pressure. Open tube manometers are used to measure pressures higher than atmospheric pressure.
Gas Laws:---------------------------------------------------------------
Boyles Law (Pressure - Volume Relationship) states that the pressure of a fixed amount of gas at constant temperature is inversely proportional to the volume of gas. V∝ (1/P)
P1V1=P2V2 at constant temperature
Charles-Gay-Lussac's Law (Charles Law) states that the volume of a gas maintained at constant pressure is directly proportional to the absolute temperature of the gas. V∝ T and (V1/T1) = (V2/T2) at constant pressure.
Avogadro's law (Amount-Volume Relationship) states that the volume of a sample of gas is directly proportional to the number of moles in the sample at constant temperature and pressure. V∝ n and (V1/n1) = (V2/n2) at constant temperature and pressure.
When cooling gas at constant volume, pressure decreases. When heating gas at constant volume, pressure increases.
When cooling gas at constant pressure, volume decreases. When heating gas at constant pressure, volume increases.
The presence of additional molecules of gas causes an increase in pressure. The dependence of volume on amount of gas at constant temperature and pressure is demonstrated when gas is removed, volume decreases. When gas molecules are added, volume increases.
Combined Gas Law (Pressure-Temperature-Amount-Volume Relationship) The combined gas law can be used to solve problems when any or all of the variables change.
(P1V1)/(n1T1) = (P2V2)/(n2T2)
Ideal Gas Equation: The gas laws can be combined into a general equation that describes the physical behavior of all gases.
V∝ 1/P => Boyle's Law
V ∝ T => Charles's Law
V ∝ n => Avogadro's Law
V ∝ nT/P
V = RnT/P =>
PV = nRT
R is the proportionality constant, called the gas constant.
The ideal gas equation describes the relationship among the four variables P, V, n, and T.
PV = nRT
An ideal gas is a hypothetical sample of gas whose pressure-volume-temperature behavior is predicted accurately by the ideal gas equation.
The gas constant (R) is the proportionality constant and its value and units depend on the units in which P and V are expressed. Such as:
R = 8.314 J/K*mol
R = 0.08206 L*atm/K*mol
R = 62.36 L*torr/K*mol
Standard Temperature and Pressure (STP) is a special set of conditions where: Pressure is 1 atm and temperature is 0 degrees Celsius (273.15 K).
The volume occupied by one mole of an ideal gas is then 22.41 L
V = [(1 mol)(0.08206 L*atm/K*mol)(273.15 K)]/1 atm = 22.41 L
It is possible to solve algebraically for variables other than those that appear explicitly in the ideal gas equation.
PV = nRT
(n/V) = (P/RT)
M x n/V = P/RT x M
d = PM/RT
d is the density in g/L
M is the molar mass in g/mol
What pressure would be required for helium at 25 degrees C to have the same density as carbon dioxide at 25 degrees C and 1 atm.?
d (CO2)= PM/RT = (1 atm)(44.01 g/mol)/(o.o8206 Latm/molK)(298.15K) = 1.7966 g/L
1.7966 g/L = (P(He))(4.003 g/mol)/(0.08206 Latm/molK)(298.15 K)
P(He) = 11.0 atm
Factors That Cause Deviation from Ideal Gas Behavior: At high pressure, molecules are close together and individual volume becomes significant. At low temperatures, molecules are moving slower and any intermolecular forces become significant.
Van der Waals Equation is useful for gases that do not behave ideally: [P + an^2/V^2](V-nb) = nRT
P is the experimentally measured pressure and V is the container volume. The equation shows the corrected pressure and volume terms, respectively. The pressure exerted by the real gas is often lower than the value predicted by the ideal gas equation.
Gas Mixtures: Dalton's Law of Partial Pressures: When two or more gases are placed in a container, each gas behaves as though it occupies the container alone.
1 mol of N2 in a 5.o0 L container at 0 degrees C exerts a pressure of 4.48 atm, according to the ideal gas equation. The addition of 1.00 mol of O2 in the same container exerts an additional 4.48 atm of pressure, according to the ideal gas equation.
Therefore the total pressure of the mixture is the sum of the partial pressures (Pi):
Ptotal = Pn2 + Po2 = 4.48 atm + 4.48 atm = 8.96 atm
Dalton's Law of Partial Pressures states that the total pressure exerted by a gas mixture is the sum of the partial pressures exerted by each component of the mixture.
Ptotal = ∑ Pi Each component of a gas mixture exerts a pressure independent of the other components. The total pressure is the sum of the partial pressures.
Mole Fractions: The relative amounts of the components in a gas mixture can be specified by using mole fractions: Xi = ni/ntotal, where Xi is the mole fraction, ni is the moles of a certain component, and ntotal is the total number of moles. The mole fraction of a mixture component is always less than 1. The sum of mole fractions for all components of a mixture is always 1. Mole fractions are dimensionless.
P and n are proportional to each other at a specified T and V: Xi = Pi/Ptotal
Xi x ntotal = ni
Xi x Ptotal = Pi
Reactions with Gaseous Reactants and Products: Calculating the Required Volume of a Gaseous Reactant:
2CO(g) + O2(g) => 2CO2 (g) The ratio of combination of carbon monoxide to oxygen is 2:1 in moles or volume (all gas phase).
2Na(s) + Cl2(g) => 2NaCl(s) Use the ideal gas equation for 1 gas reactant.
Convert the given mass to moles, use the balanced equation to determine the stoichiometric amount, and then use the ideal gas equation to convert moles to liters.
Determining the Amount of Reactant Consumed Using Change in Pressure: Although there are no empirical gas laws that focus on the relationship between n and P, it is possible to rearrange the ideal gas equation to find the relationship: PV = nRT => n = P x V/RT
The change in pressure in a reaction vessel can be used to determine how many moles of gaseous reactant are consumed. Δn = ΔP x V/RT
The volume of gas produced by a chemical reaction can be measured using laboratory equipment. When gas in a tube is collected over water in this manner (as it is heated), the total pressure is the sum of two partial pressures: Ptotal = P(collected gas) + P(H2O)
The vapor pressure of water is known at various temperatures and can be found in reference tables.
To determine the mass of a gas produced at a certain temperature and pressure when a certain volume of the gas is collected over water, use Dalton's law of partial pressures to determine the partial pressure of the gas, then use the ideal gas equation to find the moles of the gas, and use the molar mass of the gas to convert to mass. Atomic pressure is given in atmospheres, while the vapor pressure of water is calculated in torr.
Properties of Gases: Gases differ from solids and liquids in several ways: (1) A sample of gas assumes both the shape and volume of the container. (2) Gases are compressible. (3) The densities of gases are much smaller than those of liquids and solids and are highly variable depending on temperature and pressure (4) Gases form homogenous mixtures (solutions) with one another in any proportion.
Kinetic molecular theory explains how the molecular nature of gases gives rise to their macroscopic properties. Kinetic molecular theory follows these basic assumptions: (1) A gas is composed of particles that are separated by large distances. The volume occupied by individual molecules is negligible. (2) Gas molecules are constantly in random motion, moving in straight paths, colliding with perfectly elastic collisions. (3) Gas molecules do not exert attractive or repulsive forces on one another. (4) The average kinetic energy of gas molecules in a sample is proportional to the absolute temperature: Ek (avg) ∝ T
Assumptions: Gases are compressible because molecules in the gas phase are separated by large distances. Pressure is the result of the collisions of gas molecules with the walls of their container. Decreasing volume increases the frequency of collisions. Pressure increases as collision frequency increases. Heating a sample of gas increases its average kinetic energy.
Gas molecules must move faster. Faster molecules collide more frequently and at a greater speed. Pressure increases as collision frequency increases. Ek (avg) ∝ T
Total Kinetic Energy of a mole of gas is equal to: (3/2)RT
The average kinetic energy of one molecule is Ek = 1/2 mv^2 where m is the mass, v^2 is the mean square speed.
For one mole of gas: Na(1/2mv^2) = 3/2RT where Na is Avogadro's number. M is the molar mass. Rearrange and take the square root (m x Na = M).
Urms = √3RT/M
Root Mean Square (rms) speed (Urms) is the speed of a molecule with the average kinetic energy in a gas sample. Urms is directly proportional to temperature.
Urms = √3RT/M where M is molar mass in kg/mol, R is the gas constant (8.314 J/Kmol), T is temperature.
Urms is inversely proportional to the square root of M.
Urms = √3RT/M
When two gases are at the same temperature, it is possible to compare the Urms values of the different gases.
[Urms(1)/Urms(2)] = √(M2/M1)
Diffusion is the mixing of gases as the result of random motion and frequent collisions.
Effusion is the escape of gas molecules from a container to a region of vacuum.
Graham's Law = the rate of diffusion or effusion of gas is inversely proportional to the square root of its molar mass.
Rate ∝ (1/ √M)
Pressure of gas is defined as the force applied per unit area: pressure = (force)/(area)
The SI unit of force is the newton (N), where 1 N = 1 kg*m/s^2
The SI unit of pressure is the pascal (Pa), defined as 1 N per square meter 1 Pa = 1 N/m^2
Units of pressure commonly used in science are:
Pressure at sea level = 1 atm = 101,325 Pa (standard atmosphere)
mmHg = Barometer measurement = 1 mmHg = 133.322 Pa
torr = Torricelli, inventor of barometer = 1 torr = 133.322 Pa (1 torr = 1/760 atm)
bar = 1 bar = 1 x 10^5 Pa (100,000 Pa) used in meteorology (1 bar = 750 torr)
1 atm = 760 torr = 1.01325 barr
Barometer is an instrument used to measure atmospheric pressure. A barometer consists of a long glass tube, closed at one end, filled with Hg (mercury). Tube of Hg is inverted so no air enters the tube. Open end is submerged into a container of Hg. Some Hg flows out of the tube, creating an empty space. Height of the Hg is 760 mm.
Standard atmospheric pressure (1 atm) was originally defined as the pressure that would support a column of mercury exactly 760 mm high.
P = h*d*g
h = height in meters
d - density in kg/m^3
g = is the gravitational constant (9.8 m/s^2)
1 atm = 101,325 Pa = 760 mmHg = 760 torr = 1.01325 bar = 14.7 psi
Manometers are devices used to measure pressures other than atmospheric pressure. Closed tube manometers are used to measure pressures lower than atmospheric pressure. Open tube manometers are used to measure pressures higher than atmospheric pressure.
Gas Laws:---------------------------------------------------------------
Boyles Law (Pressure - Volume Relationship) states that the pressure of a fixed amount of gas at constant temperature is inversely proportional to the volume of gas. V∝ (1/P)
P1V1=P2V2 at constant temperature
Charles-Gay-Lussac's Law (Charles Law) states that the volume of a gas maintained at constant pressure is directly proportional to the absolute temperature of the gas. V∝ T and (V1/T1) = (V2/T2) at constant pressure.
Avogadro's law (Amount-Volume Relationship) states that the volume of a sample of gas is directly proportional to the number of moles in the sample at constant temperature and pressure. V∝ n and (V1/n1) = (V2/n2) at constant temperature and pressure.
When cooling gas at constant volume, pressure decreases. When heating gas at constant volume, pressure increases.
When cooling gas at constant pressure, volume decreases. When heating gas at constant pressure, volume increases.
The presence of additional molecules of gas causes an increase in pressure. The dependence of volume on amount of gas at constant temperature and pressure is demonstrated when gas is removed, volume decreases. When gas molecules are added, volume increases.
Combined Gas Law (Pressure-Temperature-Amount-Volume Relationship) The combined gas law can be used to solve problems when any or all of the variables change.
(P1V1)/(n1T1) = (P2V2)/(n2T2)
Ideal Gas Equation: The gas laws can be combined into a general equation that describes the physical behavior of all gases.
V∝ 1/P => Boyle's Law
V ∝ T => Charles's Law
V ∝ n => Avogadro's Law
V ∝ nT/P
V = RnT/P =>
PV = nRT
R is the proportionality constant, called the gas constant.
The ideal gas equation describes the relationship among the four variables P, V, n, and T.
PV = nRT
An ideal gas is a hypothetical sample of gas whose pressure-volume-temperature behavior is predicted accurately by the ideal gas equation.
The gas constant (R) is the proportionality constant and its value and units depend on the units in which P and V are expressed. Such as:
R = 8.314 J/K*mol
R = 0.08206 L*atm/K*mol
R = 62.36 L*torr/K*mol
Standard Temperature and Pressure (STP) is a special set of conditions where: Pressure is 1 atm and temperature is 0 degrees Celsius (273.15 K).
The volume occupied by one mole of an ideal gas is then 22.41 L
V = [(1 mol)(0.08206 L*atm/K*mol)(273.15 K)]/1 atm = 22.41 L
It is possible to solve algebraically for variables other than those that appear explicitly in the ideal gas equation.
PV = nRT
(n/V) = (P/RT)
M x n/V = P/RT x M
d = PM/RT
d is the density in g/L
M is the molar mass in g/mol
What pressure would be required for helium at 25 degrees C to have the same density as carbon dioxide at 25 degrees C and 1 atm.?
d (CO2)= PM/RT = (1 atm)(44.01 g/mol)/(o.o8206 Latm/molK)(298.15K) = 1.7966 g/L
1.7966 g/L = (P(He))(4.003 g/mol)/(0.08206 Latm/molK)(298.15 K)
P(He) = 11.0 atm
Factors That Cause Deviation from Ideal Gas Behavior: At high pressure, molecules are close together and individual volume becomes significant. At low temperatures, molecules are moving slower and any intermolecular forces become significant.
Van der Waals Equation is useful for gases that do not behave ideally: [P + an^2/V^2](V-nb) = nRT
P is the experimentally measured pressure and V is the container volume. The equation shows the corrected pressure and volume terms, respectively. The pressure exerted by the real gas is often lower than the value predicted by the ideal gas equation.
Gas Mixtures: Dalton's Law of Partial Pressures: When two or more gases are placed in a container, each gas behaves as though it occupies the container alone.
1 mol of N2 in a 5.o0 L container at 0 degrees C exerts a pressure of 4.48 atm, according to the ideal gas equation. The addition of 1.00 mol of O2 in the same container exerts an additional 4.48 atm of pressure, according to the ideal gas equation.
Therefore the total pressure of the mixture is the sum of the partial pressures (Pi):
Ptotal = Pn2 + Po2 = 4.48 atm + 4.48 atm = 8.96 atm
Dalton's Law of Partial Pressures states that the total pressure exerted by a gas mixture is the sum of the partial pressures exerted by each component of the mixture.
Ptotal = ∑ Pi Each component of a gas mixture exerts a pressure independent of the other components. The total pressure is the sum of the partial pressures.
Mole Fractions: The relative amounts of the components in a gas mixture can be specified by using mole fractions: Xi = ni/ntotal, where Xi is the mole fraction, ni is the moles of a certain component, and ntotal is the total number of moles. The mole fraction of a mixture component is always less than 1. The sum of mole fractions for all components of a mixture is always 1. Mole fractions are dimensionless.
P and n are proportional to each other at a specified T and V: Xi = Pi/Ptotal
Xi x ntotal = ni
Xi x Ptotal = Pi
Reactions with Gaseous Reactants and Products: Calculating the Required Volume of a Gaseous Reactant:
2CO(g) + O2(g) => 2CO2 (g) The ratio of combination of carbon monoxide to oxygen is 2:1 in moles or volume (all gas phase).
2Na(s) + Cl2(g) => 2NaCl(s) Use the ideal gas equation for 1 gas reactant.
Convert the given mass to moles, use the balanced equation to determine the stoichiometric amount, and then use the ideal gas equation to convert moles to liters.
Determining the Amount of Reactant Consumed Using Change in Pressure: Although there are no empirical gas laws that focus on the relationship between n and P, it is possible to rearrange the ideal gas equation to find the relationship: PV = nRT => n = P x V/RT
The change in pressure in a reaction vessel can be used to determine how many moles of gaseous reactant are consumed. Δn = ΔP x V/RT
The volume of gas produced by a chemical reaction can be measured using laboratory equipment. When gas in a tube is collected over water in this manner (as it is heated), the total pressure is the sum of two partial pressures: Ptotal = P(collected gas) + P(H2O)
The vapor pressure of water is known at various temperatures and can be found in reference tables.
To determine the mass of a gas produced at a certain temperature and pressure when a certain volume of the gas is collected over water, use Dalton's law of partial pressures to determine the partial pressure of the gas, then use the ideal gas equation to find the moles of the gas, and use the molar mass of the gas to convert to mass. Atomic pressure is given in atmospheres, while the vapor pressure of water is calculated in torr.