8d Flashcards
whats ni
number of particles in the energy level ‘i’
we want to find the relationship of ni and N
aka the number of particles in energy level, ‘i’ and the total number of particles , how is this done §
by looking at q
what is q
partition coefficient
works as a distribution normalisation at a given temp, T for a given set of energies, Ei.
kinda also describes a system in terms of micro states.
where a microstate is basiclaly just particles in different energy levels.
q =
sum of: pi e( -Ei / KT)
this describes a system in terms of microstates.
kinetic theory of gases gives us what equation
1/2 m<v> ^2 = 3/2 KbT</v>
movement in every direction contributes how much
1/2 KBT
what does < > mean
average velocity of all particles
means that several energy levels can be occupied or visited.
<1/2 m v^2 > = what other 2 equations
1/2 m <v^2>
3/2 KBT
q helps us find what
Q
what can eb found usuig Q
u can find S and U
entropy and internal energy
N ,, total number particals equals what
it equals the sum of ni
boltzman population equation for non degenerative energy levels =
Ni/Nj = exp(-Ei/KT) / exp (-Ej/KT)
= exp -(Ei-Ej) / KT
boltzman distribution for degenerative energy levels
Ni/Nj = gi/gj exp ( -(Ei-Ej) /KT) )
what is Q
Q is the molar partition function
q doesnt relate to the number of particles there is ,, but it does relate to what
it relates // contains the different energy levels.
q = molecular or molar
molecular
Q is molecular or molar
molar
Q = q^N relates to what
the molar partition function when u have a distinguishable set of particles
aka in a solid or liquid ig.
an assembled set of atoms
Q = q^N / N! relates to what
it relates to particles that are indistinguishable
aka not assmebled such as in a gas etc
Q = q^N / N! is used for indistinguishable particles and canhelp us find what
it can help us find the specific heat from a statistical approach
energy levels in a system can eb what
they can be spaced out largely or close together
what does thr spacing between energy levels and the enrgy gap between them alter
it alters the amount of partices that will populate the larger energy levels.
aka the ones further away fro mthe ground state
if energy levels are close together,, what is the effect
q will be larger
bc more energy levels will be populated
theres enough thermal energy to promote a bunch of energy levels
if energy levels are far apart from eachother,, what does this do
it makes it harder to populate the higher energy leveks, ,the ones further away fro mthe ground state,, bc theres not enough thermal energy to do so.
larger q =
close together energy levels
lots of higher energy levels will also be populated!!
small q
further away energy levels
most of the energy levels will be populated near the groudn state!!
when energy levels are spaced further apart and particles cant be prototed to themm,, whats the relationship between Ei and Kt
Ei»_space;» KT
which is why they cant be promoted,, bc even with an increase in temp,, the Ei is still too large to be overcome.
ni when Ei»_space; Kt is what
around 0
ni when u increase T
ni is no longer similar to 0.
bc Ei is more populated
bc. EI»_space; KT but increasing T makes KT more similar to Ei.
allowing it to be populated as increase in T means more thermal energy for promotion.
what does q do then
measures how much particles are spread over energy levels that are not the ground state
q values can range from what to what,, and what influences its value
it can range from 0 –> 1 or very large numbers
temp influences its value
the larger the temp ,, the larger the value of q.
large separattion is noemally seen between what
diff vibrational states of diatomic molecules
a smaller separation is normally seen between what
translational energy levels // DOF
okay so describe a graph where diff 🔺E are plotted.
y axis = fractional occupation
x axis = energy level
small 🔺E = flatish slope / line ,, theres a low fractional occupation across a range of different enrgy levels
large 🔺E = steep slope. lower energy levels = larger fractional occupation,, larger enrgy levels have basically no fractional occupation!!
so this means that for large 🔺E,, higher energy levels arent rlly populated!! just the ones closer to the ground state are populated.
S long eqution
S = k(N ln(N) - alphaN - betaU)
e(alpha) = what
n0
= N/q
alpha =
ln(N) - ln(q)
beta =
- 1/KT
S for a distinguishable particle
S = k ln( q^N) + U/T
in a crystalline solid
high U valuse must be assocated with what
the population of higher energy levels of translational or vibrational nature.
how does u link to q,, and what do we want to remove from thr equation
we want to remove Ei fro mthe equation by substituting in the q values depending on if its a distinguishable or undistinguishable particle.
U = (NKT^2 / q) (dq/dT)
= KT^2 d ln(Q) / dT
okay so yes q, is the partition function,, but what acc is it
it contains all the info that. describes the equilibrium state of a system
whats this all about in terms of particles
how the particles are distributed over energy levels + whether those particles are localised (crystal) or delocalised (gas)
translational partition function =
mv = nh / 2L
kinetic energy =
1/2mv^2
= n^2 h^2 / 8mL^2
En from schrodinger =
En = n^2 pi^2 h.dash^2 / 2mL^2
n = 1,2,3 etc
what is particle in a box about
a particle being confined by an infinate potential well of width L.
q trans 1D for 1D motion =
sum of exp ( -n^2 h^2 / 8mL^2 KT)
energy levels are closely spaced tho so
exp ( -n^2 h^2 / 8mL^2 KT) dn
x^2 = ( -n^2 h^2 / 8mL^2 KT)
= root ( 2pi m K T) L / h
for 3D motion,, each x, y ,z contributes what ?? the same quantity or different
it contributes the same quantity !!
so we just cube it.
so ( root[ 2 pi m KT] L / h ) ^3
L ^3 is what in particle in a box
L ^3 is thr volume of the box!!
what happens to q as volume of box increases
q increases as the volume of the box increases. bc it determines energy levels in the particle in a box approach.
how do we find U due to translational energy states.
U = NKT ^2 3/2 1/T = 3/2 NKT = 3/2 RT
bc gas particles are indistinguishable making the Q and q relationship for 1 mole of gas molecules: (Q trans) 3d = 1/n! ((q trans) 3d) ^N.
