## Activity: Introducing entropy

Contemporary Challenges 2022 (4 years)
This lecture introduces the idea of entropy, including the relationship between entropy and multiplicity as well as the relationship between changes in entropy and heat.

This lecture has a student group activity embedded in it. Should the Monte Carlo video be a separate "activity", so it could be done live instead of as a pre-class video?

See this video for a Monte Carlo simulation showing how the multiplicity tends to increase.

As we have seen, the multiplicity $\Omega$ of a material is of considerable interest, because the combined multiplicity of a system plus its surroundings tends to increase. We have also seen that for an Einstein solid the multiplicity tends to grow very rapidly (beyond exponentially) with system size for around $10^{23}$ atoms (a few grams of any material), the multiplicity would be closer to $10^{10^{20}}$, which is a bit much. In addition, to combine the multiplicity of two systems, we have to multiply rather than add.

Instead of working with multiplicity, we far prefer the entropy $S$ \begin{align} S \equiv \text{(constant)} \ln \Omega \end{align} where the constant is called Boltzmann's constant and in SI units is $k_B=1.38\times 10^{23}\text{ J/K}$. (It is also possible to choose units such that temperature is expressed in energy units, and $k_B=1$. I tend to do this in my own research.)

The natural log function ensures $S$ is not astronomically large, and even more importantily it makes entropy an additive quantity. If we have two subsystems with multiplicity $\Omega_1$ and $\Omega_2$, then the combined system has multiplicity \begin{align} \Omega_{\text{combined}} &= \Omega_1\cdot\Omega_2 \\ S_{\text{combined}} &= k_B\ln(\Omega_{\text{combined}}) \\ &= k_B\ln(\Omega_1\cdot\Omega_2) \\ &= k_B\ln\Omega_1 + k_B\ln\Omega_2 \\ &= S_1 + S_2 \end{align} If you're hazy on the behavior of logarithms, this wouldn't be a bad time to review those properties. Most crucially: \begin{align} \ln(ab) &= \ln a + \ln b \\ \ln\left(\frac{a}{b}\right) &= \ln a - \ln b \\ \ln(a^b) &= b \ln a \end{align}

### Analyzing a real system's entropy

Imagine dropping a hot block of metal into a room-temperature tub of water. How would you answer if I asked you to find $\Omega_{\text{metal,initial}}$, $\Omega_{\text{metal,final}}$, $\Omega_{\text{water,initial}}$, and $\Omega_{\text{water,final}}$? You'd be stumped. I'd be stumped, if I tried to compute these multiplicities.

Does this mean that entropy is unhelpful in practice? Absolutely not! Entropy was actually discovered before the connection between it and multiplicity, and changes in entropy are readily measurable! We can't easily measure the absolute entropy (although it is possible and is actually done), but changes in entropy are often quite straightforward to measure.

The key is the property that \begin{align} \Delta S &= \frac{Q}{T} \end{align} where $Q$ is the energy entering the object through heating (positive means it gains energy), and $T$ is the temperature of the object we are considering, provided $Q$ is small enough that we can neglect changes in temperature. $T$ must be measured on an absolute scale (e.g. Kelvin).

Why does this equation work? It's related to the definition of temperature.

• face Entropy and Temperature

face Lecture

120 min.

##### Entropy and Temperature
Thermal and Statistical Physics 2020

These lecture notes for the second week of Thermal and Statistical Physics involve relating entropy and temperature in the microcanonical ensemble, using a paramagnet as an example. These notes include a few small group activities.
• accessibility_new Using Arms to Represent Time Dependence in Spin 1/2 Systems

accessibility_new Kinesthetic

10 min.

##### Using Arms to Represent Time Dependence in Spin 1/2 Systems
Quantum Fundamentals 2023 (2 years)

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, use their left arms to demonstrate time evolution in spin 1/2 quantum systems.
• assignment Carbon monoxide poisoning

assignment Homework

##### Carbon monoxide poisoning
Equilibrium Absorbtion Thermal and Statistical Physics 2020

In carbon monoxide poisoning the CO replaces the $\textsf{O}_{2}$ adsorbed on hemoglobin ($\text{Hb}$) molecules in the blood. To show the effect, consider a model for which each adsorption site on a heme may be vacant or may be occupied either with energy $\varepsilon_A$ by one molecule $\textsf{O}_{2}$ or with energy $\varepsilon_B$ by one molecule CO. Let $N$ fixed heme sites be in equilibrium with $\textsf{O}_{2}$ and CO in the gas phases at concentrations such that the activities are $\lambda(\text{O}_2) = 1\times 10^{-5}$ and $\lambda(\text{CO}) = 1\times 10^{-7}$, all at body temperature $37^\circ\text{C}$. Neglect any spin multiplicity factors.

1. First consider the system in the absence of CO. Evaluate $\varepsilon_A$ such that 90 percent of the $\text{Hb}$ sites are occupied by $\textsf{O}_{2}$. Express the answer in eV per $\textsf{O}_{2}$.

2. Now admit the CO under the specified conditions. Fine $\varepsilon_B$ such that only 10% of the Hb sites are occupied by $\textsf{O}_{2}$.

• face Boltzmann probabilities and Helmholtz

face Lecture

120 min.

##### Boltzmann probabilities and Helmholtz
Thermal and Statistical Physics 2020

These notes, from the third week of Thermal and Statistical Physics cover the canonical ensemble and Helmholtz free energy. They include a number of small group activities.
• group Equipotential Surfaces

group Small Group Activity

120 min.

##### Equipotential Surfaces

Students are prompted to consider the scalar superposition of the electric potential due to multiple point charges. First a single point charge is discussed, then four positive charges, then an electric quadrupole. Students draw the equipotential curves in the plane of the charges, while also considering the 3D nature of equipotentials.

format_list_numbered Sequence

This sequence starts with an introduction to partial derivatives and continues through gradient. While some of the activities/problems are pure math, a number of other activities/problems are situated in the context of electrostatics. This sequence is intended to be used intermittently across multiple days or even weeks of a course or even multiple courses.
• face Thermal radiation and Planck distribution

face Lecture

120 min.

##### Thermal radiation and Planck distribution
Thermal and Statistical Physics 2020

These notes from the fourth week of Thermal and Statistical Physics cover blackbody radiation and the Planck distribution. They include a number of small group activities.
• accessibility_new Using Arms to Visualize Complex Numbers (MathBits)

accessibility_new Kinesthetic

10 min.

##### Using Arms to Visualize Complex Numbers (MathBits)
Lie Groups and Lie Algebras 23 (4 years)

Arms Sequence for Complex Numbers and Quantum States

Students move their left arm in a circle to trace out the complex plane (Argand diagram). They then explore the rectangular and exponential representations of complex numbers by using their left arm to show given complex numbers on the complex plane. Finally they enact multiplication of complex numbers in exponential form and complex conjugation.
• assignment Quantum harmonic oscillator

assignment Homework

##### Quantum harmonic oscillator
Entropy Quantum harmonic oscillator Frequency Energy Thermal and Statistical Physics 2020
1. Find the entropy of a set of $N$ oscillators of frequency $\omega$ as a function of the total quantum number $n$. Use the multiplicity function: $$g(N,n) = \frac{(N+n-1)!}{n!(N-1)!}$$ and assume that $N\gg 1$. This means you can make the Sitrling approximation that $\log N! \approx N\log N - N$. It also means that $N-1 \approx N$.

2. Let $U$ denote the total energy $n\hbar\omega$ of the oscillators. Express the entropy as $S(U,N)$. Show that the total energy at temperature $T$ is $$U = \frac{N\hbar\omega}{e^{\frac{\hbar\omega}{kT}}-1}$$ This is the Planck result found the hard way. We will get to the easy way soon, and you will never again need to work with a multiplicity function like this.

• format_list_numbered Arms Sequence for Complex Numbers and Quantum States

format_list_numbered Sequence

##### Arms Sequence for Complex Numbers and Quantum States
“Arms” is an engaging representation of complex numbers in which students use their left arms to geometrically represent numbers in the complex plane (an Argand diagram). The sequence starts with pure math activities in which students represent a single complex number (using prompts in both rectangular and exponential forms), demonstrate multiplication of complex numbers in exponential form, and act out a number of different linear transformation on pairs of complex numbers. Later activities, relevant to spin 1/2 systems in quantum mechanics, explore overall phases, relative phases, and time dependence. These activities can be combined and sequenced in many different ways; see the Instructor's Guide for the second activity for ideas about how to introduce the Arms representation the first time you use it.

Learning Outcomes