Week 4: Statistical Physics (19 C) Flashcards
State and view of Thermodynamics by 17th C?
• Natural philosophers had theorized about the science of heat since antiquity
• By 17th century, the first rigorous empirical study of heat was underway, enabled by invention of many measuring instruments through 17th and 18th century
E.g. Thermometer, Manometer (pressure) and Calorimeter (heat transfer)
• At the time, the study of heat was centered on the study of gasses (pneumatic chemistry)
Two main theories:
• Caloric Theory (dominated 18thC)
- Kinetic Theory
What were the three main Thermodynamic breakthroughs of the 1800’s ?
- James Joule - Mechanical Equivalence of Heat
- Hermann von Helmholtz - Conservation of Force (Energy)
- Sadi Carnot - Reflection on the Motive Power of Heat
- reversible process is limit of steam engine efficiency
James Joule?
An English brewer, scientist, mathematician.
Mechanical Equivalence of Heat (1845 – 1847)
- heat (temperature of water) is proportional to work applied to it
- only possible because of his brewing background
• Experiment: Joule filled a tank with water, and placed wooden paddles within
By connecting paddles to weights, he could drive the paddles and push the water around by lowering the weights
Joule measured the temperature in the water, and compared it to the energy spent expended by lowering the weights
• Discovery: Temperature of water increases proportionally with the energy expended into the paddles!
Demonstrated direct relationship between mechanical work and heat
• Joule believed mechanical work was literally heat – it was just a unit conversion
• Context: Accurate repeated measurements of a fluid’s temperatures were not easy to perform. Very likely Joule was only able to perform this experiment due to his background as a brewer, where exact fluid temperatures were critical
Hermann von Helmholtz
German physicist.
Conservation of Force (1846)
• Idea that energy is conserved came from Hermann von Helmholtz
Note: Helmholtz considered “force” what is today known as “energy”
• Believed various forms of force were interchangeable
Forms included kinetic, potential, electromagnetic, physiological
Sadi Carnot
French military engineer/physicist.
“Reflection on the Motive Power of Heat” (1824)
• Through logical & mathematical theory, he found that a reversible engine maximizes η
E.g. Mixing hot and cold water is a non-reversible process
• Impossible through any means to separate it back into hot & cold
Theorized a reversible steam engine called the “Carnot Cycle”
Formalization of Thermodynamics?
Rudolph Clausius (German) &; William Thomson (Lord Kelvin; Scottish) Independently give first formal formulation
Two laws:
1. Energy is Conserved
2. Entropy (ΔS) Increase
No thermal process produces more mechanical work than a reversible process
Clausius introduces the idea of entropy in 1865 (S)
• If a process is reversible, ΔS = 0 (else ΔS > 0)
o S is non-decreasing, like an arrow of time
State and view of Thermodynamics by 19th C?
Thanks to Clausius & Thompson, thermodynamics had a formal formulation. However there were two attitudes towards it:
- Two laws are fundamental axioms
Not uncommon; Einstein subscribed to this chain of thought
Laws should be used as baseline of all science - Mechanical foundation for two laws should be found
Scientists tried to reduce thermodynamics to Newtonian Physics, which was more fundamental
Thermodynamics should be explainable as the interactions of particles
Daniel Bernoulli?
Swiss mathematician/physicist - related to Jacob of before
Tried to provide kinetic explanation of thermo:
- When particles are compressed, they give a “mechanical push” back, all resisting against the flow; this explains pressure
- When particles move quickly, their Energy (and therefore Temperature) are high
Caloric Theory?
Caloric Theory was the dominant theory of the 18th century
Explained heat with the idea of the “caloric”, an imponderable
Hot air was caused by a high presence of caloric in that air
Fire transfers caloric to the surroundings, heating them
Had many supporters such as Laplace and Lavoisier
Provided a physical and tangible interpretation of heat
Kinetic Theory? What theories were based on it?
Kinetic Theory was a long-standing theory, existing from antiquity
-All objects constituded of microscopic particles
All phenomena (including heat) due to their interactions
e.g. Fire energizes particles, setting them rapidly in motion
Heat is not as directly quantifiable (as caloric theory presents it)
Experienced a resurgence in 19th C, as physicists searched for a mechanical foundation for thermodynamics:
David Bernoulli:
- explained pressure using particle’s giving a mechanical push back
- temperature as particle’s rapid motion
Rudolph Clausius
-derived Ideal gas law by considering particles with unique direction but same speeds
Refined by JCM to give Maxwell distribution
-then further by Boltzmann in Transport theory
Rudolph Clausius
German Physicist
- formalized (along with William Thompson) thermo
- described gas as collection of particles moving at the same speed but different direction
- thought reasonable since temp was constant throughout
- considered collisions and the momentum change required upon collision
- derived IDEAL GAS LAW
Many (e.g. JCM) took issue with assumption on molecular velocities.
How did JCM see Rudolph Clausius’ theory? What conclusion did he reach?
James Clerk Maxwell (the glorious Scott) took issue with Clausius’ assumption of uniform speeds
• Published Illustration of the Dynamic Theory of Gases (1860)
• Explained thermodynamic phenomena from mechanical physics
• His view on particles drew inspiration from Quetelet’s notion of “average man”
Temperature is a common characteristic, a mean – but it is does not capture the varying individual energies of the particles
By giving molecular velocity a statistical distribution according to just their speed, he found that it followed a Gaussian distribution!
- Combining three axis distributions together gives MAXWELL DISTRIBUTION
- only worked in steady state (Boltzmann would revise with transport theory)
How did Ludwig Boltzmann build on JCM’s work?
Austrian physicist.
Wanted to explore how a physical state progressed towards steady state
Explored this with his TRANSPORT THEORY (1869-1872)
When not at steady state, distribution of molecules depends not only the magnitude of velocity, but also speed and time
Using Newtonian mechanics, he found the time evolution of f(V_x,V_y,V_z,x,y,z,t)
Described f with Boltzmann’s Transport Equation
Implications: Defined ‘H’ function for his transport equation
H always decreases with time
Like a negative entropy; demonstrates mechanical foundation of irreversibility
Entropy can be derived from Newtonian mechanics!
What were the flaws with transport theory? How were they addressed?
Came from two angles, showing that Newtonian mechanics were incompatible with the second law (increasing entropy) of thermodyanmics:
- JCM & “Maxwell’s Demon”
- Newtonian mechanics allowed you to perform irreversible processes. These would decrease Entropy! - Johann Josef Loschmidt
- Newton’s laws of motion are time symmetric
- Any Newtonian mechanical process is reversible!
Boltzmann addresses by providing statistical interpretation of second law:
- abandon mechanical equations of motion, focus on “microstates” capturing the energy levels of each particle
- microstates are equiprobable, and a states probability is determined by the number of states leading to it
Core Idea: Existence of physical states as described by Maxwell’s distribution will have very high probability of occurring
o Other physical states are all theoretically possible – but with low probability
• This allowed Boltzmann to explain the existence of Newtonian reversible processes, while countering that they were impossibly rare
