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}
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 Lecture
120 min.
paramagnet entropy temperature statistical mechanics
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 Kinesthetic
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assignment Homework
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.
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 \(\textsf{O}_{2}\).
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 Lecture
120 min.
ideal gas entropy canonical ensemble Boltzmann probability Helmholtz free energy statistical mechanics
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 Small Group Activity
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face Lecture
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Planck distribution blackbody radiation photon statistical mechanics
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 Kinesthetic
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arms complex numbers Argand diagram complex plane rectangular form exponential form complex conjugate math
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 Homework
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: \begin{equation} g(N,n) = \frac{(N+n-1)!}{n!(N-1)!} \end{equation} 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 \(N-1 \approx N\).
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 \begin{equation} U = \frac{N\hbar\omega}{e^{\frac{\hbar\omega}{kT}}-1} \end{equation} 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.
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