Lecture 10

• Ideal gas model

• Ideal gas law

• Quasi-static processes: isochoric, isobaric and isothermal

• Temperature scales, absolute zero

• Phase changes, equilibrium, diagram

Temperature• temperature is related to system’s thermal energy

(kinetic and potential energy of atoms)

• measured by thermometer: small system undergoes a change upon exchagning thermal energy, e.g., length of mercury/alcohol in glass tube or ideal gas’ pressure

• Celcius/centigrade scale: boiling point (100 ) freezing point (0 )

• Fahrenheit scale: (212 and 32 )

Temperature Scales

TF = 95TC + 32!

!F !F

!C !C

Example• At what temperature does the numerical value in

degree Fahrenheit in match the numerical value in degree Celsius?

Absolute Zero and Absolute Temperature

• property changes linearly with temperature: e.g., pressure of constant-volume gas

• p = 0 for all gases at p due to collisions all motion stopped, zero thermal energy: absolute zero (lowest temperature)

• absolute temperature scale: zero point at absolute zero Kelvin scale if same unit size as Celcius scale:

T0 = !273 !C

TK = TC + 273 (no degrees for Kelvin)

Phase Changes• melting/freezing point: temperature

at which solid becomes liquid...thermal energy large enough to allow molecules to move around

• phase equilibrium: 2 phases co-exist

• condensation/boiling point: phase equilibrium between liquid and gas thermal energy too large for bonding

• phase change temperatures are pressure-dependent: freezing (boiling) point higher (lower) at lower pressure

Phase diagram• how phases and phase changes

vary with T, p

• 3 regions with phase transitions at boundaries...gas-solid (sublimation)

• critical point: liquid-gas boundary ends

• triple point: all 3 phases co-exist

• triple point of water ( ) used as reference point (reproduced with no variation) for Kelvin scale: 273.16 K 0 K fixed by gas properties

T3 = 0.01!C

• cf. Celcius scale requires 2 reference points: boiling and melting points (p-dependent)

Ideal Gases• (strong) repulsive forces between atoms

(incompressibility of solids/liquids + (weak) attractive forces (tensile strength of solids; cohesion of liquid droplets)

• solids and liquids: atomic separation

• gases: freely moving till collide (steep wall for important)

• Ideal gas model: hard non-interacting spheres, bounce on contact

• good for low density and condensation point

! req

(both mono and d-atomic gases)

average r ! req

T !

r < req.

Ideal-gas law• Experiments state variables not independent: for all gases

• Gases in sealed containers (n constant)

• Using,

• typical p, V, T: average distance between gas molecules distances over which atoms exert attractive forces

universal gas constant, R = 8.31 J/mol/K

Boltzmann's constant, k_B = R/N_A = 1.38 x 10^(-23) J/K

= p / k_B T

!

Example

• 3.0 mol of gas at a temperature of -120 degrees Celsius fills a 2.0 L container. What is the gas pressure?

Ideal gas processes (how gas changes state)• diagram: each point is unique

state of gas (T by ideal-gas law; n known for sealed container) (p, V, T)

• ideal gas process: trajectory showing intermediate states (work done depends on it)

• Quasi-static processes: slow; system in thermal equilibrium at all times

(reversible)

pV

(i) Isochoric (constant V) process

• : vertical line on diagramVf = Vi pV

(ii) Isobaric (constant p) process

• gas pressure from balancing of forces on piston

• : horizontal line on diagram

(expansion: compression if cooled)

pf = pi

p = patmos + MgA

pV

(iii) Isothermal (constant T) process• heat transfer keeps gas at same T as liquid

• : hyperbola on diagram (inverse relationship)

• location of hyperbolae (isotherms) depends on T: lower T, closer to origin

Tf = Ti and pV = nRT ! piVi = pfVf

p = nRTV = constant

V pV

Example• A rigid container holds hydrogen gas at a pressure of

3.0 atm and a temperature of 2 degrees Celsius. What will the pressure be if the temperature is raised to 10 degrees Celsius?