probability stuff Flashcards
general p(state) equation
p(state) = 1/z * e^( -E(state)/KT )
E(state) is any energy (kinetic, potential, rotational, etc)
K: Boltzmann constant
z: normalisation constant
p(state): probability density
state: the variable in the energy equation. (position, x, velocity, v, etc)
how do you find the normalisation constant
z = ∫e^( -E(s)/KT ) d(s)
limits are usually from minus to plus infinity but could be different based on the question
how do you calculate the expectation value
<E> = ∫E(x) * p(x) dx
</E>
what’s the equipartition theorem
Each degree of freedom that contributes quadratically to the total energy of a system in thermal equilibrium, contributes ½KT average energy per particle.
K: Boltzmann constant
T: temperature
how many degrees of freedom are in a monatomic and diatomic molecule’s energy?
monatomic: only has kinetic energy. In 3 dimensions , x,y,z. so it has 3 degrees of freedom
diatomic: has 7 total DoF
kinetic energy. in 3 dimensions, provides 3 DoF
rotational energy. has 2 axes of rotation. provides 2 DoF
vibrational energy. the compression/expansion of the bond has a kinetic and elastic energy, provides 2 DoF
what does ‘frozen out’ mean
at low temperatures some of the energies that contribute to the degrees of freedom of a diatomic molecule don’t apply.
how is the maxwell-Boltzmann speed distribution derived?
use E= ½mv²
calculate p(v) (the probability speed distribution)
start the volume integral, dV = v²sin(θ)dvdθdφ for a sphere
(the limits are 0 –> infinity for the dv integral)
sin(θ)dθdφ becomes 4π.
hence you have the integral:
∫ 4πp(v)v² dv
The maxwell boltzmann speed distribution is just whats inside the integral (4πp(v)v²)
how do you find the mean speed, most probably speed and rms speed for a maxwell boltzmann distribution
most probably: dp(v)/dt = 0
mean speed: <v> = ∫v*p(v) dv</v>
rms speed: <v²> = ∫v²*p(v) dv
then square rooted.
