pchem 2

isolated system

no exchange of anything

microcanonical ensemble- NVE kept constant

closed system

exchange of energy but not matter

canonical ensemble- NVT kept constant

• heat bath

open system

energy and matter is exchanged

state variables

independent of path,

Volume, Pressure, Temperature, Entropy, Enthalpy, Internal energy, mols

path variable

path dependent

heat and work

process

converting 1 state into another

extensive property

depends on system size, it depends on how much is in the system

• mass, volume, density

intensive property

independent of system size

• density, temperature, pressure

internal energy

total kinetic and potential energy in a system

denoted by U

U=<E>=1/QSum of Eie^-BEi

kinetic theory assumptions

• large number of particles- all the particles must be identical and moving randomly everywhere. This makes sure that we can find the average behavior and predict with it

• point particles- size is negligible compared to distance. This is like normal conditions when gas particles are really far apart

• elastic collisions- no energy is lost during collisions. This makes sure that the conservation of energy in a system is true—ensures pressure stay constannt

• no long range interactions - this assumes that there is no attractive or repulsive forces. This makes the potential energy negligible

• Folows classical mechanics- F=ma or F=dp/dt.

Total pressure equation microscopic

P=1/V sum (1/3 mv²) or P=Nm<v²>/3V

• We derived it by looking at the amount of collisions on each of the walls

• we used newton’s 2nd law to derive the average force on the wall and took it over the area to get pressure.

• make sure to account for x, y, and z direction

isotropic movement

distance of the velocity is the same in all directions, making sure that there are no interactions and all the particles are the same

Kinetic energy

<Ekin>=1/2 m<v²> or 3/2 kBT

Average value of <v²>

• shows that temperature is directly proportional to velocity and kinetic energy

  • <v²>=3kBT/m

Maxwell Boltzmann Speed distribution

• tells you the probability density for finding a molecule between V and V+dv.

• allows you to find the most probable speed

• It is a probaility distribution so it should add up to 1

trend for the v

• vmp < <V> < vrms

Boltzmann Curves

• x-axis - v (speed), y-axis - probability density

• trends- KE=1/2 mv²

  • lighter gases have less mass, so they will have higher speeds leading to broader peaks and a shifted right curve

  • heavier gases have more mass, they will have lower speeds, leading to the curve to shift left, and the curves will be more pointed.

  • lower temperature means that particles are moving slower so this shifts the curve left, creating narrower and taller peaks

  • higher temperature- particles move faster, leading to the curve to shift right and making broader peaks

Boyle

holds T and N constant

negative exponential relationship between volume and pressure

Cb/V = P

Boyle goes back and forth, P inc is V dec

Charles law

Holds pressure and mols constant

shows that there is a positive linear slope between temperature and volume

Cozy heat expands, heat gas causes volume expansion

V=CT

Gay lussac

• holds volume and mols constant

• showed that there is a positive linear relationship between temperature and pressure

• gas pressure goes up with heat

• P=CT

avagadro’s law

hods pressure and temperature constant

positive linear relationship between number of particles and volume

more particles needs more space

real gases requires

• minimal interactions to be ideal

• to do this, you want to

  • keep pressure low- lower pressure makes the collisions elastic. If it is too high pressure, it’ll break

  • keep density low- this makes them point particles

  • keep temperature high- keeps their interactions short

volume differential

• shows that V can be derived from P,n,T

• derives Kb = PV/NT

microscopic assumptions

• particles have no volume - this makes sure that they are below the volume imit

• particles do not interact (allows elastic colisions)

• colisions are elastic

macroscopic assumptions

• low Pressure - allows no interactions to occur, you lack space for it

• high volume - more space means less interactions

• low P(N/V) - increase volume very high = less int per particle

• T high- reduces time for interactions

N vs n

N- num of molecules

n- num of mols

average distance

• proportional to volume and temperature

• is inverse to number of mols and pressure

real gases

• have finite molecular size and intermolecular forces

• may be at high pressure which brings the molecules too close together, increasing colision frequency

• may be at low temperature- less KE

Z

• compressibility factor

• tells you whether your gas acts ideal or not

• closer to 1 = more ideal

• =PV/nT or PV/NkbT

trends in Z

• less than 1 - your pressure is medium, so your Vreal < Videal

  • attractive interactions will dominate

  • caused by h-bonding, van der waals

• greater than 1- your pressure is really high so Vreal >Videal

  • repulsive interactions dominate

  • caused by sterics

Corresponding States Principle

• all the states collapse onto the same curve, if you know one state, you know them all

• at critical temperature, all of them are expected to have a Z=3/8.

• leads you to use the reduced variables- makes all gases look the same

van der waals equation of states

• links interactions to EOS

• has 2 variables a- stickiness, b- size

super critical fluid trends

• T=Tc- you will have a root on your Isotherm graph

• T<Tc- you will have liquid at low pressure, gas at high pressure, and a mixure between

• T>Tc- you will have a supercritical fluid, transition disappears

statistical average

• tells you how much you’re expecting this value

• it weighs your outcomes based on probability

• <x> = sum xipi

ensemble

set of microstates consistent with macroscopic constraints

fundamental postulate for ensembles

for an isolated system—microcanonical ensemble—since the energy is conserved, every accessible microstate is equally probable

• p =1/M

grand canonical ensemble

open system

u, V, T held constant

canonical ensembles

• heat bath makes T negligible because the environment’s N is so large. Since E= 3/2 NkbT, we can rearrange to T= 2E/3NKb, when you have such a high N, your T goes to 0.

• No matter the E absorbed, T doesn’t change

• Your energy may fluctuate around that point

  • these fluctuations tell you about the density of the energy states around it

ratio between energy states

• energy is relative

• these ratios are multiplicative

• if they are well behaved, you use e^BdE

Boltzmann distribution equation

e^-BE / Q

partition function

Q - sum of all the possible microstates

B

when T is low, 1/kBT increases B→ 1/kb

when T is high, 1/kbT decreases → e^0 =1

trends with microstates and temperature

• low T- only samples 1 microstate (100%)

• high T- samples both microstates, making them equaly likely - converges to 50%

trends on the T vs Q graphs

• gap- tels you about how strongly energy difference infuences population distribution.

• increase energy gap- you need more energy to get between each of the states, so the ground state is prefered. This makes the probabilty gap larger

• decrease energy gap- more states can be sampled, all the states look the same, so it is more likely for our system to try many states. This makes the energy gap smaller

• higher T- gap decreases -there is enough E to populate the states equally, so the differences between the states blur, making the gap decrease since all states are equally likely.

• lower T- gap increases- there is little E- making it so the states that are lower energy are preferred- specifically ground state. So, the population is really imbalanced, creatng a larger gap.

thermal contact

state where 2 systems an exchange energy

chemical contact

state where matter exchange occurs

mechanical equilibruim

net force is 0th

thermal equilibrium

net energy is 0

chemical equilibrium

no net matter exchanged

Partia derivative of dn(q)/dB

= -1/Q (dQ/dB) N,V

steps to calculate internal energy

1. calculate the ln of the partition function

2. take partial derivatives of dln/db

3. Plug into Ep/Q

Internal energy plot

• your energy will be centered around an average of the energies in the system

• when you have low T- your B wil be high so e^-BdE goes to 0, this means that your system will perfer the ground state.

• when you have high T- your B will be low so e^-BdE goes to 1, which means you can sample both states.

heat capacity

how much energy is needed to increase temp of object by 1 degrees at a fixed N and V

• Cv = dU/dt = variance of E/kbT²

• when T increases - C decreases

• When variance inreases - C increases

bandwidth of variance trends

• broad - many systems

  • high density

  • each increase in temperature increases energy more

  • your variance tells you that the interaction in the system is fragile

  • disordered

• narrow- 1 system

  • you need more energy to populate the states

  • not as many interaction because the states aren’t accessible

  • high degeneracy because energy changes doesn’t change state

  • ordered

canonical vs microcanonical concepts

• microcanonical applies the fundemental theorem - all the accessible states are equally possible, you don’t use the partition function or Boltzmann curves.

• The energy for microcanonical is fixed, energy for canonical is not

• For canonical- your states probability depends on Boltzmann and partition functions.

• Canonical energy is used to calculate heat capacity.