PHYSICS 231 Lecture 26: Ideal gases

PHYSICS 231Lecture 26: Ideal gases Remco Zegers Walk-in hour: Thursday 11:30-13:30 am Helproom PHY 231

Ideal Gas: properties • Collection of atoms/molecules that • Exert no force upon each other • The energy of a system of two atoms/molecules cannot be reduced by bringing them close to each other • Take no volume • The volume taken by the atoms/molecules • is negligible compared to the volume they • are sitting in PHY 231

Potential Energy Rmin 0 -Emin R Ideal gas: we are neglecting the potential energy between The atoms/molecules Potential Energy Kinetic energy 0 R PHY 231

Number of particles: mol 1 mol of particles: 6.02 x 1023 particles Avogadro’s number NA=6.02x1023 particles per mol It doesn’t matter what kind of particles: 1 mol is always NA particles PHY 231

What is the weight of 1 mol of atoms? Name Number of protons Z X A molar mass PHY 231

Weight of 1 mol of atoms 1 mol of atoms: A gram (A: mass number) Example: 1 mol of Carbon = 12 g 1 mol of Zinc = 65.4 g What about molecules? H2O 1 mol of water molecules: 2x 1 g (due to Hydrogen) 1x 16 g (due to Oxygen) Total: 18 g PHY 231

Example A cube of Silicon (molar mass 28.1 g) is 250 g. A) How much Silicon atoms are in the cube? B) What would be the mass for the same number of gold atoms (molar mass 197 g) PHY 231

Boyle’s Law ½P0 2V0 T0 P0 V0 T0 2P0 ½V0 T0 At constant temperature: P ~ 1/V PHY 231

Charles’ law P0 2V0 2T0 P0 V0 T0 If you want to maintain a constant pressure, the temperature must be increased linearly with the volume V~T PHY 231

Gay-Lussac’s law P0 V0 T0 2P0 V0 2T0 If, at constant volume, the temperature is increased, the pressure will increase by the same factor P ~ T PHY 231

Boyle & Charles & Gay-LussacIDEAL GAS LAW PV/T = nR n: number of particles in the gas (mol) R: universal gas constant 8.31 J/mol·K If no molecules are extracted from or added to a system: PHY 231

Example An ideal gas occupies a volume of 1.0cm3 at 200C at 1 atm. A) How many molecules are in the volume? B) If the pressure is reduced to 1.0x10-11 Pa, while the temperature drops to 00C, how many molecules remained in the volume? PHY 231

And another! An airbubble has a volume of 1.50 cm3at 950 m depth (T=7oC). What is its volume when it reaches the surface (water=1.0x103 kg/m3)? PHY 231

A small matter of definition Ideal gas law: PV/T=nR PV/T=(N/NA)R n (number of mols)= N (number of molecules) NA (number of molecules in 1 mol) Rewrite ideal gas law: PV/T = NkB where kB=R/NA=1.38x10-23 J/K Boltzmann’s constant PHY 231

From macroscopic to microscopic descriptions: kinetic theory of gases 1) The number of molecules is large (statistical model) 2) Their average separation is large (take no volume) 3) Molecules follow Newton’s laws 4) Any particular molecule can move in any direction with a large distribution of velocities 5) Molecules undergo elastic collision with each other 6) Molecules make elastic collisions with the walls 7) All molecules are of the same type For derivations of the equations, see the textbook PHY 231

Pressure Number of Molecules Mass of 1 molecule Averaged squared velocity Number of molecules per unit volume Volume Average translation kinetic energy PHY 231

Microscopic Macroscopic Temperature ~ average molecular kinetic energy Average molecular kinetic energy Total kinetic energy rms speed of a molecule M=Molar mass (kg/mol) PHY 231

example What is the rms speed of air at 1atm and room temperature? Assume it consist of molecular Nitrogen only (N2)? PHY 231

And another... What is the total kinetic energy of the air molecules in the lecture room (assume only molecular nitrogen is present N2)? PHY 231

PHYSICS 231 Lecture 26: Ideal gases