Kinetic Theory of Gases & Molecular Energy

• Kinetic theory explains the behavior of matter at the molecular level

Kinetic Theory of Gases

• Gases consist of tiny particles called molecules in constant, random motion
• Gas molecules are far apart and do not interact except during collisions
• Gas pressure results from the impact of molecules on surfaces, appearing as a continuous force
• Boyle's law can be derived from the kinetic theory
• Expansion of a gas increases the space between molecules, leading to fewer impacts per area and decreased pressure

Molecular Energy

• The average kinetic energy of gas molecules is proportional to absolute temperature
• The relationship between average kinetic energy and temperature is KEave = (3/2)kT
• k, the Boltzmann constant, equals 1.38 x 10^-23 J/K
• At absolute zero (0 K), gas molecules would theoretically be at rest
• Gases at the same temperature have the same average molecular energy
• Heavier molecules move slower than lighter molecules at the same temperature
• Compressing a gas increases its temperature as molecules rebound with more energy
• Expanding a gas decreases its temperature as molecules rebound with less energy

Solids and Liquids

• Solid molecules are held together by strong inter-molecular forces, giving them a definite shape and volume
• Solid molecules are in constant motion, vibrating about fixed locations
• Liquid molecules move around each other, allowing liquids to flow, while maintaining a constant volume
• Melting a solid requires energy (heat of fusion) to overcome inter-molecular forces and create a more random liquid structure
• Boiling a liquid requires energy (heat of vaporization) to free molecules entirely into a gas

Solved Problem 20.1

• Gas molecules collide frequently, leading to complicated paths from one place to another

Solved Problem 20.2

• Evaporation occurs when faster-moving molecules overcome attractive forces and escape a liquid
• Warmer liquids evaporate more rapidly due to having more fast molecules
• Evaporation cools the remaining liquid as slower molecules are left behind

Solved Problem 20.3

• The average kinetic energy of gas at 100°C (373 K) is 7.72 x 10^-21 J
• Calculated from KEav = (3/2) kT = (3/2) (1.38 x 10^-23 J/K) (373 K)

Solved Problem 20.4

• The average velocity of oxygen molecules at 100°C is 540 m/s
• Calculated from Vav = sqrt(3kT/m)

Solved Problem 20.5

• A tank with 1 g of hydrogen at 0°C contains 16 times more molecules than an identical tank with 1 g of oxygen
• Hydrogen gas exerts 16 times greater pressure due to the higher number of molecules impacting the container walls
• Average molecular energies arethe same in both gases, as their temperatures are the same
• Hydrogen molecules have four times greater average velocity than that of oxygen molecules

Relative Humidity

• Humidity is the amount of water vapor in the air
• Air is saturated when it holds the maximum possible water vapor for the current temperature
• Higher temperatures allow for greater water vapor density at saturation
• Relative humidity indicates the degree of saturation
• 0% relative humidity: perfectly dry air
• 50% relative humidity: air contains half the maximum possible water vapor
• 100% relative humidity: air is saturated

Atoms and Molecules

• Elements are fundamental substances, over 100 are known
• Elements cannot be transformed by chemical or physical means, but can combine into compounds
• Atoms compose elements, while molecules compose gaseous compounds
• Molecules consist of specific arrangements of atoms
• Atomic mass units (u) measure the masses of atoms and molecules
• 1 atomic mass unit equals 1.660 x 10^-27 kg
• The mass of a molecule is the sum of the masses of its constituent atoms

The Mole

• A mole is the amount of a substance with a mass equal to its molecular mass in grams
• It is not possible to count atoms or molecules in laboratory or industry
• If two samples of different substances have masses in the same proportion as their molecular masses, then the number of molecules will be the same
• A mole of water (H2O) has a mass of 18 g, since its molecular mass is 18 u
• The SI unit for the amount of a substance corresponding to a mole is 1 mol
• Avogadro's number (N = 6.023 x 10^23 molecules) is the number of molecules in a mole
• To find the number of molecules in a sample, multiply the number of moles by Avogadro's number

Molar Volume

• Equal gas volumes at the same temperature and pressure contain equal molecule numbers and moles
• Standard Temperature and Pressure (STP) is 0°C (273 K) and 1 atm
• The molar volume of a gas is 22.4 L at STP; 1 mole of any gas occupies this amount
• At STP, gas volume is V= (moles of gas) x (molar volume)

Universal Gas Constant

• Based on the ideal gas law, pV/T is constant for a gas
• The universal gas constant, R, can be calculated using the molar volume at STP
• R = 0.082 atmL/(molK) or 8.31 J/(mol*K) in SI units
• The ideal gas law: pV = nRT

Example 3

• Molecular Formulas: 2C+4H equals (2) (12.01) u + (4) (1.008) u which gives you 28.05u=28.05 g/mol density at STP
• One mole of C2H4 therefore has a density at STP, which is p= m/V equals 28.05g/22.4L equals 1.25g/L