Gas Laws

• Empirical gas laws describe pressure, temperature, volume, and the number of moles in determining a gas’s state
• The quantity mainly focused on is number of moles

Gas Law Relationships

• Volume and pressure are inversely proportional
• Volume and temperature are directly proportional
• Volume is directly proportional to the number of moles present

Learning Gas Laws

• There are three components that vary in gas law equations: pressure, temperature, and volume
• When one of these components remains constant, gas laws predict the relationship between the other two
• It is necessary to deduce which components are varying and which one stays the same when faced with a calculation problem
• This is so you can decide which formula to use

Mnemonic Device

• The mnemonic Paid TV Can Be Good helps determine which formula to use
• Write the letters PTV and CBG to use the mnemonic
• P, T, and V stand for pressure, temperature, and volume
• C, B, and G represent Charles’, Boyle’s, and Gay-Lussac’s laws
• Constant variables are directly above letters in the bottom row

Important Conversion

• Temperature values must be expressed in Kelvin
• Celsius to Kelvin: °K = ° C + 273
• Celsius to Fahrenheit: ° F = 1.8( ° C) + 32
• Fahrenheit to Celsius: ° C = (°F - 32)/ 1.8
• Units for volume or pressure don’t matter as long as they are consistent on both sides of the equation

Boyle's Law

• Boyle’s Law describes the Volume-Pressure Relationship
• The volume of a fixed gas sample is inversely related to pressure if the temperature is constant, and the mass is fixed
• As pressure increases, volume decreases, and vice versa
• The mathematical statement of Boyle’s Law is P₁V₁ = P₂V₂
• Boyle’s Law involves four quantities, so three variables must be known to solve a problem

Example Question

• In Boyle’s Law, what is the volume if a gas occupies 2 liters at a pressure of 2 atm, and the pressure is 4 atm? • P₁ = 2 atm, V₁ = 2 liters, P₂ = 4 atm
• By increasing the pressure on a gas, its volume decreases

Boyle's Law Applications

• A large volume of gas is released from a pressurized cylinder.
• A bag valve mask increases and pressure and decreases volume when squeezed
• Spontaneous inspiration causes intrapulmonary pressure to fall and the volume increases

Charles's Law

• Charles’s Law describes the Volume-Temperature relationship
• The volume of a gas is proportional to its absolute temperature as long as its pressure and quantity are held constant
• Temperatures must be expressed in Kelvin
• °C = K – 273.15

Key Points

• Describes the relationship between volume and temperature
• Volume is directly proportional to the absolute temperature
• Volume increases when temperature increases
• Charles’s Law answers if you change the temperature, what volume will the gas occupy?

Example

• Using Charles’s Law, for gas at 150 Kelvin with a volume of 1 Liter, what is the volume at 200 Kelvin? • The initial volume is V₁ and the initial temperature is T₁ Solve for V₂ (T₂ changes) • V₁T₂ = V₂T₁ -> (1L)(200K) = (V₂)(150K) -> (V₂) = (1L) (200K)/(150K) -> V₂ = 1.33 L

Gay-Lussac's Law

• Gay-Lussac's law describes the direct relationship that exists between pressure and temperature
• Pressure and temperature are directly proportional, and as temperature increases, pressure increases
• Increasing pressure raises the temperature, and vice versa

Sample Question

• If a gas is at 250 Kelvin at a pressure of 120 mmHg, what will the pressure be if you heat it to 312 Kelvin using Gay-Lussac's Law? • P1T2=P2T1 • First pressure(P₁) = 120 mmHg, first temperature (T₁) = 250 Kelvin, second temperature(T₂) = 312 Kelvin, solve for P₂ (120 mm Hg)(312 K)= (P₂)(250 K) -> P₂ = (120 mm Hg)(312 K)/(250 K) = 149.76 mm Hg

Applications

• Shows a direct relationship between pressure and temperature, meaning pressure increases with temperature
• Explains the decrease in pressure in a N₂O tank with Gay-Lussac’s Law
• Temperature decreases with pressure (Joule-Thompson)

Combined Gas Law

• The preceding three laws can be formulated together • P₁V₁/T₁ = P₂V₂/T₂ or P₁V₁T₂ = P₂V₂T₁
• It is the only gas law needed to calculate a gas state change due to changing parameters (volume, pressure, temperature, or moles)
• Drop from the equation if one of the parameters does not change

Avogadro's Law

• Avogadro’s Law determines the Volume-Mole Relationship Describe the relationships between the amount of gas, and the volume of gas
• Volume is directly proportional to the number of molecules
• Law is expressed as V1/n₁=V2/n2
• One mole of ANY gas at standard temperature (0 degrees C) and standard pressure (1 ATM) occupies a volume of 22.4 liters (newer measurement is 22.71 L)
• Two moles of a gas equals 44.8 liters and 0.5 equals 11.2 liters

Example Application

• With Avogadro's Law: if the N₂O tank weighs 7 kg and an empty E cylinder is 5.9 kg, how many liters are present? • V₁/n₁ = V₂/n₂ -> 22.4/1 = V₂/25 • 1.1 kg or 1100 g/44 g (molecular weight of N₂O) = 25 moles • 22.4 L/1 = V₂/25 • V₂ = 560 L N₂O

Ideal Gas Law

• The ideal gas law is a state function that describes a gas’s state under a given set of standard conditions (STP)
• The Ideal gas law takes the form PV = nRT, where: • P = Pressure • V = Volume • n = Moles • T = Absolute temperature • R = A constant (constant (R) =PV/nT)
• To completely describe the state of an ideal gas using the ideal gas law: the pressure, volume, number of moles of gas, and absolute temperature must be specified
• With three of the four variables, the unknown variable can be calculated

Ideal vs Real

• Describes the behavior of an ideal gas under all conditions
• As long as the pressure isn’t too high and the temperature isn’t too low, real gases will approximate ideal behavior

Kinetic Molecular Theory

• Describes molecular behavior in its various states combining Newton’s laws of motion and the bulk properties of gases
• Molecules have no volume (gases have particles whose volume is negligible compared to the gas’s volume)
• Gas molecules exert no force on each other unless they collide
• There are no attractive or repulsive forces between particles
• All collusions are elastic
• Collisions do not decrease the energy of the system with the walls of the container
• Gas molecules move in a constant and random motion
• Gas temperature depends on the average kinetic energy
• Gas energy is entirely kinetic

Kinetic Theory (cont.)

• Sample gas molecules may show a range of kinetic energies, but the average KE only depends on temperature
• Distribution of Kinetic Energies • The sum of kinetic and potential energies of molecules in a sample of gas is the internal energy of the sample • Sample temperature is directly proportional to the average Kinetic Energy
• The average KE for a gas depends on the temperature and not the identity of the gas
• Different gases may have the same average KE if their temperatures are the same

Ideal vs Real Gases

• Ideal gases obey the ideal gas law at all temperatures and pressures
• Real gases deviate from ideal behavior most strongly at high pressures and/or low temperatures
• Real gases most closely approximate ideal gas at high temperatures and low pressures

Kinetic Molecular Theory Failure

• Basic tenets of fail when: *Gases consist of particles which have a negligible volume • The volume occupied by a gas is greater than that of the individual molecules, and the volume of the gas molecules themselves is not zero
• This tenet starts to fail when real gasses have larger pressures, because at higher pressures, more gas molecules are crowded into the same sample volume.
• Kinetic Molecular Theory gas molecules are always in constant, random motion, which will always apply in real gasses
• Although molecules may have a range of kinetic energies, their average depends only on temperature
• This is an excellent model for real gasses
• Attractive or repulsive forces between the gas particles do not exist so all collisions are elastic
• This tenet begins to fail with real gasses at lower temperatures, when the energy balance must overcome attractive forces

Graham's Law of Effusion

• Effusing is the movement of a gas through a smaller opening
• The rate of effusion is inversely proportional to the square root of the molecular mass
• The rate of effusion depends on the speed of the molecules
• KE is constant for all gasses
• Average KE is the same for two different gases at the same temperature
• Lower mass gasses have a higher speed

Van der Waals Equation

• Corrects for deviations from ideal gasses
• Solved for P to get: - The Van der Waals equation is based on the ideal gas law
• The new terms a and b stand for constants
• Term “a” is attractions that reduce the observed pressure in the gas sample
• Term "b" accounts for the gasses volume and becomes more important as there are more moles of the gas