The Maxwell-Boltzmann distribution

Question 1

In classical thermodynamics, the energy of a molecule is strictly considering transitional motion and neglecting what two types of motion.

Answer 1

Rotational and Vibraitonal.

Question 2

What assumptions are being made when determining the distribution of molecular velocity?

Answer 2

1. Molecular size is much less than the inter-molecular separation, so that we assume that molecules spend most of their time whizzing around only rarely bumping into each other. 2. We will ignore any inter-molecular forces.

Question 3

If molecules are equally likely to have all types of velocities then the name of the fraction of molecules with velocities in, say, the x direction, between v_x and v_x + dv_x, as g(v_x) dv_x is given to be what? Specifically, what is the name of g(v_x).

Answer 3

Velocity Distribution Function.

Question 4

The point of ___________ is to make variables comparable to each other. _______________ is the process of reducing measurements to a “neutral” or standard scale.

Answer 4

Normalization.

Question 5

Each molecule can exchange energy with each other due to collisions, but everything remains in equilibrium. Each molecule therfore behaves like a small system connected to a heat reservoir at temperature T, where the heat reservoir is “all the other molecules in the gas”. Hence the results of the _____________ distribution of energies.

Answer 5

Boltzmann.

Question 6

Explain the intuition behind x = 0, that is to explain why the average velocity is 0. Where x is the average of x.

Answer 6

Particles are moving in all directions with all types of velocities. Since a gas’ particles are in random motion, it is plausible that there will be about as many moving in one direction as in the opposite direction, meaning that the average velocity for a collection of gas particles equals zero.

Question 7

What is the mean velocity, v_{<strong>x</strong>}², of a collection of gas equal to?

Answer 7

k_BT / m.

Question 8

Describe the distribution function for molecular speeds?

Answer 8

Slightly skewed right.

Question 9

Give the probability distribution for a given component of velocity. Specifically, what is the probability distribution proportional to?

Answer 9

g(v_x) ∝ e^{-m(v_x·v_x) /2k_B​T}.

Question 10

What is the probability distribution of molecular speeds proportional to and what is the name given for it.

Answer 10

f(u) ∝ u² e^{-m(u·u)/2k_BT}

Maxwell-Boltzmann Distribution.

Question 11

What is u² = ?

Note: Boldness of the letter is to mean the average of that letter.

Answer 11

3k_BT / m.

<u²> = <v²> = v_x² + v_y² + v_z²,

where v_x² = v_y² = v_z² = k_BT / m.

Question 12

What is used to overcome the “directional” component of velocity and simultaneously acquire the particles’ average velocity?

Answer 12

V_rms = [u²]^1/2 = [3k_BT/m]^1/2.

This is understood to be the root mean squared speed.

Question 13

The average energy of a molecule in a gas depends only on what thermodynamic variable?

Answer 13

Temperature.

E_KE = mu²/2 = mu²/2 = 3k_BT/2.

Again the bolding represents the average of whatever is being bolded.

Question 14

An electronvolt (symbol eV, also written electron-volt and electron volt) is the amount of kinetic energy gained by a single electron accelerating from rest through an electric potential difference of one volt in vacuum. When used as a unit of energy, the numerical value of 1 eV in joules (symbol J) is equivalent to the numerical value of the charge of an electron in coulombs (symbol C). So what does 1 ev = ?

Answer 14

1 eV = 1.602176634×10⁻¹⁹ J.

W = qΔV = e^-(1 volt) = 1.60217662 × 10^-19 coulombs x 1 volt = 1.602176634×10⁻¹⁹ J = 1 eV.

Question 15

_________ _______ (also called separation energy) is the minimum energy required to disassemble a system of particles into separate parts. A bound system is typically at a lower energy level than its unbound constituents.

Answer 15

Binding energy.

Question 16

Define the equi-partition theroem.

Answer 16

In thermodynamic equilibrium at temperature T, each independent quadratic degreee of freedom contributes ½k_BT to the mean energy of a molecule.

Question 17

What is difference between intramolecular and intermolecular?

Answer 17

Intermolecular describes what happens between molecule and molecule. Intramolecular describes what happens within two atoms in a molecule.

Question 18

Define |x|.

Answer 18

In mathematics, the absolute value or modulus of a real number x, denoted |x|, is the non-negative value of x without regard to its sign. Namely, |x| = x if x is positive, and |x| = −x if x is negative, and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3.

Question 19

The _____________ is a function of the form e^{-αx^2}. It has a maximum at x = 0 and a shape that has been likened to that of a bell. It turns up in my statistical problems, often under the name of the ____________________.

Can you remember what the integral of this is equal to?

Answer 19

Gaussian; normal distribution.