Gas Laws and Kinetic Molecular Theory

Gas Laws

• Boyle's Law states that volume is inversely proportional to pressure at constant temperature (T) and amount of gas (n)
• Charles’ Law states that Volume is proportional to Temperature at constant pressure (p) and amount of gas (n)
• Avogadro's Principle states that Volume is proportional to the number of moles (n) at constant Temperature (T) and pressure (p)
• The General Gas Law relates volume, number of moles, temperature, and pressure
• This relationship is expressed as V~n.T/p, which leads to the ideal gas law equation PV = nRT
• The molar volume (Vm) can be calculated using the formula Vm = (V/n) = (RT/P)

Kinetic Molecular Theory

• Explains gas behavior through postulates and assumptions
• Gases consist of rigid particles with negligible volume compared to the total gas volume
• These particles are in constant motion, colliding with each other and container walls
• Intermolecular forces are negligible, resulting in perfectly elastic collisions
• Absolute temperature (K) is proportional to the average kinetic energy of the particles

Exam Style Questions Related to the Kinetic Molecular Theory

• The postulate that best explains why gases are easily compressed is that the volume of particles is negligible compared to the total volume occupied
• For an ideal gas, velocity (v) and the square root of temperature (T^1/2) are directly proportional
• Gas pressure is caused by gas molecules hitting the container walls

Root Mean Square Exercise

• Root mean square of N2 molecules (M = 28.02 g mol-1) at 298 K is calculated to be 515 m/s

Root Mean Square Question

• To determine the temperature at which argon has the same root-mean-square velocity as helium at room temperature is required for the question

Ideal Gas

• Achieved under conditions of high temperature (T) and low pressure (P)
• These conditions minimize attractive and repulsive forces between particles and volume of particle
• Physical parameters can be derived from the ideal gas equation PV = nRT
• Molar mass (M) = mRT/PV, density (ρ) = PM/RT, and concentration (C) = P/RT
• R Numerical Values of the Gas Constant: L-atm/mol-K =0.08206, J/mol-K = 8.314, cal/mol-K = 1.987 m³-Pa/mol-K = 8.314, L-torr/mol-K = 62.36

Unit Conversions

• Pressure conversions: 1 atm = 1.01325 x 10^5 Pa; 1 bar = 1 x 10^5 Pa; 1 millibar (mb) = 100 Pa; 1 atm = 1.01325 bar; 1 atm = 760 torr; 1 torr = 1 mm Hg.
• Volume conversions: Utilize the relationship between mL, cm³, L, and m³ with conversion factors of 1000
• Temperature conversions: K = °C + 273 and °C = K - 273.

Exam Style Question on Gases

• Calculating the form of Sulfur in gaseous form

Dalton’s Law Of Partial Pressures

• A mixture of ideal gases, exerts a pressure that’s the sum of pressures
• Total pressure (PT) is the sum of partial pressures (PA + PB + ...)
• The mole fraction (yi) is represented as (molei / moletotal) = (Pi / Ptotal)

Exam Style Question on Gases

• What are the partial pressures of oxygen and hydrogen at 27°C?

Graham’s Law Of Effusion

• The movement of gas particles through a hole
• Average molar mass of the gas mixture is M = ∑yi Mi

Exam Style Question on Gases

• What is the apparent molar mass of air?

Real Gases

• Deviations from ideal gas behavior arise from molecular interactions, especially at low temperatures and high pressures
• Deviations from ideality can be accounted for using: Modified Equations of State, Compression Factor, Fugacity, and Joule-Thomson Coefficient

Real Gases vs Attractive and Repulsive forces

• Repulsive forces are short-range, significant almost in contact, and occur at high pressure (P) and low volume (V)
• Attractive forces are relatively long-range, effective over several molecular diameters, and relevant when molecules are close but not touching

Ideal Gas Behavior vs Attractive and Repulsive forces

• Ideal gas behavior occurs at low pressures, with large volumes and minimal intermolecular forces of attraction (IMFA)
• Attractive forces dominate at moderate pressures where the separation of molecules is a few diameters, leading to increased compressibility
• Repulsive forces dominate at high pressures, with small molecule separations, reducing compressibility

Compression Factor (Z)

• Represents a gas's ability to be compressed at a given condition; for an ideal gas, Z = 1 under all conditions
• At very low pressures, Z = 1, and gases behave nearly perfectly
• At moderate pressures, Z < 1, and attraction dominates with less molar volume
• At high pressures, Z > 1, and repulsion dominates with a larger molar volume

Boyle Temperature (TB)

• The temperature at which a real gas's properties align with those of a perfect gas as pressure approaches 0

Van Der Walls Equation

• Reduction of pressure is proportional to the square of concentration (a is a positive constant)
• Pressure (p) is based on frequency/force of collisions, reduced by attractive forces
• Excluded volume (b) is total volume taken up by molecules, which is four times the volume of the particle
• Perfect gas isotherms are obtained at high temperature and large molar volumes
• Liquids and gases coexist when cohesive and dispersing effects are in balance
• Second term represents the effect of the attractive interactions
• First term arises from the kinetic energy of the molecules and their repulsive interactions
• Van der Waals loops occur when both terms have similar magnitudes
• The critical constants are related to the van der Waals coefficients
• Critical parameters are the conditions wherein the difference between gas and liquid starts to disappear

Principle of Corresponding States

• Real gases at the same reduced volume and temperature exert the same reduced pressure
• Works best with gases composed of spherical molecules

Overview of gas laws, including Boyle's Law, Charles' Law, and Avogadro's Principle. Explanation of the General Gas Law and the ideal gas law equation PV = nRT. Covers the postulates of the Kinetic Molecular Theory in explaining gas behavior.