1. Lecture 08: A microscopic view
ENPh257: Thermodynamics
Lecture 08: A microscopic view
© Chris Waltham, UBC Physics & Astronomy, 2018

2. temperature Definitions: What you measure with a thermometer.
The property that is the same for two objects when they have been contact for long enough.
The measure of an object’s tendency spontaneously to give up energy to its surroundings.
Formally:
𝑇= 𝜕𝑈 𝜕𝑆 𝑁,𝑉
𝑈= internal energy, 𝑆 = entropy, holding constant 𝑁 = # particles, and 𝑉 = volume.
(You can change 𝑉 and 𝑆 without changing 𝑈 or 𝑇, e.g. expansion into a vacuum).
(The rate of change of 𝑈 with 𝑁 is called the chemical potential 𝜇).
© Chris Waltham, UBC Physics & Astronomy, 2018

3. Kinetic theory of gases
All gas properties can be explained in terms of a large number of small particles bouncing elastically around a container and off each other.
The idea was proposed by Lucretius in 50 BCE, picked up by Bernoulli in the 18th century, and Clausius, Maxwell and Boltzmann in the 19th.
By Greg L at the English language Wikipedia, CC BY-SA 3.0,
© Chris Waltham, UBC Physics & Astronomy, 2018

4. A microscopic view
Consider n moles of gas in a cylinder with a frictionless piston. All we know is: 𝑃𝑉=𝑛𝑅𝑇=𝑁 𝑘 𝐵 𝑇 Assume the gas consists of N (= 𝑛 𝑁 𝐴 ) non-interacting particles bouncing around in the cylinder. Consider a particle, mass 𝑚 striking a wall with normal velocity 𝑣 𝑥 and bouncing off elastically. The bounce is repeated every time interval: 𝛥𝑡=2𝐿/ 𝑣 𝑥 A L
𝑘 𝐵 = Boltzmann’s constant
𝑁 𝐴 = Avogadro’s number
𝑅 = gas constant
© Chris Waltham, UBC Physics & Astronomy, 2018

5. A microscopic view
The force on the wall is the rate of momentum change of the particle: 𝑚 2 𝑣 𝑥 2𝐿/ 𝑣 𝑥 = 𝑚 𝑣 𝑥 2 𝐿 Divide by the area A to get the pressure due to the particle: 𝑃= 𝑚 𝑣 𝑥 2 𝑉 We can identify 1 2 𝑘 𝐵 𝑇 with the mean “1-D kinetic energy” of each particle as follows: 1 2 𝑘 𝐵 𝑇= 1 2 𝑚 𝑣 𝑥 2 A L
© Chris Waltham, UBC Physics & Astronomy, 2018

6. A microscopic view
The gas is isotropic: 𝑣 𝑥 2 = 𝑣 𝑦 2 = 𝑣 𝑧 2 (Equipartition Theorem) and so: 3 2 𝑘 𝐵 𝑇= 1 2 𝑚 𝑣 2 Where 𝑣 2 is the mean square velocity of the particles, each of which have three degrees of freedom, i.e. to move in the three space dimensions. At equilibrium, each degree of freedom not in the quantum regime will have, on average, ½ 𝑘 𝐵 𝑇 of energy. A L
A note on notation:
The mean of a variable 𝑥 will often be written as 𝑥 . Thus:
𝑣 2 = 𝑣 2
© Chris Waltham, UBC Physics & Astronomy, 2018

7. Degrees of freedom
Capacity of one particle (atom, electron etc.) to gain energy in translation, rotation, vibration etc., each Cartesian direction of motion or axis of rotation counting as one.
Quantum regime: where dynamics are governed by low numbers of quanta (0, 1, 2…) of energy or angular momenta.
Temperature T is defined as before in terms of some measurable proxy, e.g. gas volume, length of liquid column etc.
By en:User:Greg L - CC BY-SA 3.0,
© Chris Waltham, UBC Physics & Astronomy, 2018

8. Boltzmann Factor
What is the probability of having an energy in a given range between 𝐸 and 𝑑𝐸? (Schroeder 6.1) Consider a single hydrogen atom (the simplest system with many energy levels): Each energy level is degenerate (consists of > 1 quantum state). The probability of the atom being in any given energy state 𝑠 is proportional the number of microstates Ω(𝑠), providing there is enough energy available in the environment to populate this state. Hence: 𝑃( 𝑠 2 ) 𝑃( 𝑠 1 ) = Ω( 𝑠 2 ) Ω( 𝑠 1 )
© Chris Waltham, UBC Physics & Astronomy, 2018

9. Boltzmann Factor
But, we know 𝑆 = 𝑘 ln Ω , although we never proved that 𝑘 is the Boltzmann constant.
𝑃( 𝑠 2 ) 𝑃( 𝑠 1 ) = exp {𝑆 𝑠 2 /𝑘} exp {𝑆( 𝑠 1 )/𝑘} = exp[ 𝑆 𝑠 2 −𝑆 𝑠 1 /𝑘 ]
In moving from one state to another: d𝑆 =𝑑𝑄/𝑇
i.e. 𝑆 𝑠 2 -𝑆 𝑠 1 =− 𝐸 𝑠 2 −𝐸 𝑠 1 𝑇
𝑃( 𝑠 2 ) 𝑃( 𝑠 1 ) = exp[ − 𝑆 𝑠 2 −𝑆 𝑠 1 /𝑘𝑇 ] = exp {−𝐸 𝑠 2 /𝑘𝑇} exp {−𝐸( 𝑠 1 )/𝑘𝑇}
© Chris Waltham, UBC Physics & Astronomy, 2018

10. Boltzmann Factor 𝑍= 𝑠 exp( −𝐸 𝑠 /𝑘𝑇 )
Define the “Boltzmann Factor”: exp( −𝐸 𝑠 /𝑘𝑇 )
The probability of being in state s is proportional to the Boltzmann factor.
The probability of being in any state is plainly 1.
Hence
𝑃 𝑠 = 1 𝑍 exp( −𝐸 𝑠 /𝑘𝑇 )
Where Z, the “Partition Function” is given by:
𝑍= 𝑠 exp( −𝐸 𝑠 /𝑘𝑇 )
© Chris Waltham, UBC Physics & Astronomy, 2018

11. Partition function
The partition function is useful for extracting macroscopic quantities like the mean energy of the system: 𝐸 = 1 𝑍 𝑠 𝐸 𝑠 exp⁡(− 𝐸 𝑠 𝑘𝑇 ) Or, if we have an explicit formula for the partition function (not proven here): 𝐸 =− 1 𝑍 𝜕𝑍 𝜕𝛽 Where 𝛽=1/𝑘𝑇, for a system in thermal equilibrium with a reservoir at temperature 𝑇.
© Chris Waltham, UBC Physics & Astronomy, 2018

12. Maxwell-Boltzmann distribution
The probability of having a certain velocity falls with increasing magnitude:
Boltzmann factor: probability of a particle having velocity 𝑣: ∝exp(− 𝑚 𝑣 2 2 𝑘 𝐵 𝑇 )
However, the number of possibilities per “volume” of velocity space rises with magnitude:
Phase space: number of vectors corresponding to velocity range 𝑣→𝑣+𝑑𝑣: ∝4𝜋 𝑣 2 𝑑𝑣
Add in a factor to ensure the average conforms to the gas laws:
𝑓 𝑣 = 𝑚 2𝜋 𝑘 B 𝑇 𝜋 𝑣 2 exp(− 𝑚 𝑣 2 2 𝑘 𝐵 𝑇 )
The mean square velocity is now given by:
𝑣 2 = 0 ∞ 𝑣 2 𝑓 𝑣 𝑑𝑣 = 3 𝑘 𝐵 𝑇 𝑚 𝑑𝑣 𝑣
© Chris Waltham, UBC Physics & Astronomy, 2018

13. Maxwell-Boltzmann distribution
𝑓 𝑣 = 𝑚 2𝜋 𝑘 B 𝑇 𝜋 𝑣 2 exp(− 𝑚 𝑣 2 2 𝑘 𝐵 𝑇 )
By The original uploader was Pdbailey at English WikipediaLater versions were uploaded by Cryptic C62 at en.wikipedia.Convert into SVG by Lilyu from Image:MaxwellBoltzmann.gif. - Originally from en.wikipedia; description page is/was here., Public Domain,
© Chris Waltham, UBC Physics & Astronomy, 2018

14. Calculating Maxwell-boltzmann
MATLAB notes:
In functions containing arrays, don’t forget the “.” before the operator to ensure element-wise operation:
e.g.
vmin = 0; % m/s
vmax = 3000; % m/s
nv = 5000;
v = linspace(0,3000,nv);
dv = v(2)-v(1);
vsq = v.^2;
NB – size of v defaults to 100 if not specified.
© Chris Waltham, UBC Physics & Astronomy, 2018