http://sites.science.oregonstate.edu/portfolioswiki/ 2020-01-27T00:58:03-08:00 http://sites.science.oregonstate.edu/portfolioswiki/ http://sites.science.oregonstate.edu/portfolioswiki/lib/images/favicon.ico text/html 2012-07-16T10:52:25-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:eelec:eelec3rdthermolaw?rev=1342461145 Lecture: Third Law of Thermodynamics (20 minutes) text/html 2016-07-11T11:20:06-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:eelec:eeleccarnoteff?rev=1468261206 Lecture: Carnot Efficiency (30 minutes) Lecture notes from Dr. Roundy's 2014 course website: A heat engine is a device that accepts energy in the form of heat from something hot, and uses that energy to do work. The Kelvin formulation states that this cannot be all that a heat engine does. In fact, a heat engine will also “waste” energy by heating something cool. Thus the energy you put in from the “hot place” will not all get used to do useful work. text/html 2016-07-08T11:41:50-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:eelec:eelecdiaidealgas?rev=1468003310 Lecture on Finding the Internal Energy of a Diatomic Ideal Gas (20 minutes) FIXMEFIXME Lecture notes from Dr. Roundy's 2014 course website: Let's consider a diatomic ideal gas, such as nitrogen. In this case, the energy levels of a single molecule are given by the sum of the translational kinetic energy, rotational kinetic energy and vibrational energy---both kinetic and potential: $$E_{n_xn_yn_zn_vlm}^{(1)} = \frac{\hbar^2 \pi^2 \left(n_x^2 + n_y^2 + n_z^2\right)}{2mL^2} + \frac{\hbar^2 l(l+1)… text/html 2016-07-07T14:15:42-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:eelec:eelecdulongpetit?rev=1467926142 Lecture: Dulong and Petit Rule (5 minutes) Ice Calorimetry Lab Lecture notes from Dr. Roundy's 2014 course website: In 1819, shortly after Dalton had introduced the concept of atomic weight in 1808, Dulong and Petit observed that if they measured the specific heat per unit mass of a variety of solids, and divided by the atomic weights of those solids, the resulting per-atom specific heat was essentially constant. This is the Dulong-Petit law, although we have since given a name to that constant… text/html 2016-07-08T15:21:04-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:eelec:eelecenergyconstraint?rev=1468016464 Lecture: Energy Constraints (?? minutes) FIXME Lecture notes from Dr. Roundy's 2014 course website: If the energy of a system is actually constrained (as it generally is), then we should be applying a second constraint, besides the one that allows us to normalize our probabilities. $$\mathcal{L} = -k_B\sum_iP_i\ln P_i + \alpha k_B\left(1-\sum_i P_i\right) + \beta k_B \left(U - \sum_i P_i E_i\right)$$ where $\alpha$ and $\beta$ are the two Lagrange multipliers. We want to maximize this, so we … text/html 2016-07-11T10:41:15-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:eelec:eelecfairent?rev=1468258875 Lecture: Fairness and Entropy (5 minutes) Lecture notes from Dr. Roundy's 2014 course website: Given two uncorrelated systems, AA and BB, we can show that the fairness of the combined system is equal to the sum of the fairnesses of the two separate systems. This means that FF is extensive. $$\mathcal{F}_A = -k \sum_i P_i \ln P_i$$ $$\mathcal{F}_B = -k \sum_i P_i \ln P_i$$ $$\mathcal{F}_{AB} = -k \sum_{ij} P_{ij} \ln\left( P_{ij} \right)$$ $$= -k \sum_{ij} P_iP_j \ln\left( P_iP_j \right)$$ $$= … text/html 2016-07-08T15:17:15-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:eelec:eelecfairintenergy?rev=1468016235 Lecture: Relating Internal Energy and Fairness (15 minutes) Lecture notes from Dr. Roundy's 2014 course website: $\newcommand\myderiv[3]{\left(\frac{\partial #1}{\partial #2}\right)_{#3}}$ Let's talk a bit about fairness. We used the fairness to find the probabilities of being in the various eigenstates, by assuming that the ``fairest'' distribution would prevail. If you bring two separate systems together and allow them to equilibrate, then you would expect that the net fairness would either … text/html 2016-07-08T13:45:28-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:eelec:eelecfairnessfunc?rev=1468010728 Lecture: Fairness (10 minutes) Lecture notes from Dr. Roundy's 2014 course website: The primary quantity in statistical mechanics is the probability $P_i$ of finding the system in eigenstate ii. Once we know the probability of each eigenstate for any given state, we will be able to compute every thermodynamic property of the system. text/html 2016-07-08T15:50:12-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:eelec:eelecfirstlaw?rev=1468018212 Lecture: First Law of Thermodynamics (10 minutes) Lecture notes from Dr. Roundy's 2014 course website: The first law of thermodynamics simply states that energy is conserved. But it is useful to look at those two non-state variables work and heat. Both are changes in energy of a system, so we can write the first law as $$\Delta U=Q+W$$ where $U$ is the internal energy of the system, $Q$ is the energy added to the system by heating, and $W$ is the work done by the system (or the energy removed … text/html 2016-07-07T14:17:18-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:eelec:eelecheatcapacity?rev=1467926238 Lecture: Heat Capacity (10 minutes) Lecture notes from Dr. Roundy's 2014 course website: As we learned last week, heat capacity is amount of energy required to raise the temperature of an object by a small amount. $$C \sim \frac{đ Q}{\partial T}$$ $$đ Q = C dT \text{ At constant what?}$$ If we hold the volume constant, then we can see from the first law that $$dU = đQ - pdV$$ since $dV=0$ for a constant-volume process, $\newcommand\myderiv[3]{\left(\frac{\partial #1}{\partial #2}\right)_{… text/html 2016-07-08T12:26:33-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:eelec:eeleclagrangemultip?rev=1468005993 Lecture: Lagrange Multipliers (?? minutes) FIXME Lecture notes from Dr. Roundy's 2014 course website: Usually, (analytically) we maximize functions by setting their derivatives equal to zero. So we could maximize the fairness by $$\frac{\partial\mathcal{F}}{\partial P_i} = 0$$ $$= -k_B (\ln P_i + 1)$$ Using the formula for the fairness function, what can this tell us about $P_i$? It doesn't make much sense at all... it means $P_i = e^{-1}$.$\ddot\frown$ There is a problem with this, which is… text/html 2016-07-08T15:40:56-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:eelec:eelecmaxwelllegendre?rev=1468017656 Lecture: Legendre Transformations (5 minutes) Lecture notes from Dr. Roundy's 2014 course website: On your devices, measure the following two derivatives and describe how they are related: [SWBQ] $\newcommand\myderiv[3]{\left(\frac{\partial #1}{\partial #2}\right)_{#3}}$ $$\myderiv{F_1}{x_2}{x_1} \qquad\qquad \myderiv{F_2}{x_1}{x_2}$$ Hiding one string and fixing one weight text/html 2016-07-07T14:36:09-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:eelec:eelecmaxwellrel?rev=1467927369 Lecture: Maxwell Relations (?? minutes) FIXME Lecture notes from Dr. Roundy's 2014 course website: In the Interlude, we learned that mixed partial derivatives are the same, regardless of the order in which we take the derivative, so $$\left(\frac{\partial \left(\frac{\partial f}{\partial x}\right)_y}{\partial y}\right)_x=\left(\frac{\partial \left(\frac{\partial f}{\partial y}\right)_x}{\partial x}\right)_y$$ $$\frac{\partial^2 f}{\partial x \partial y}=\frac{\partial^2 f}{\partial y \partial… text/html 2016-07-08T15:20:27-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:eelec:eelecpartitionfunc?rev=1468016427 Lecture: Partition Function (?? minutes) FIXME Lecture notes from Dr. Roundy's 2014 course website: The partition function is a particularly useful quantity. Physically, it is nothing more than the normalization factor needed in order to compute probabilities, but in practice, finding that normalization is typically the hardest part of a calculation---once you have found all the energy eigenvalues, that is. text/html 2016-07-11T11:23:00-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:eelec:eelecrevieweneigenv?rev=1468261380 Lecture: Reviewing Several Energy Eigenvalues (10 minutes) Lecture notes from Dr. Roundy's 2014 course website: I'm going to quickly review and introduce the energy eigenvalues for some simple quantum mechanical problems. For each of the following, I will sketch out the potential, then sketch the wavefunctions and the spacing of the energy levels. text/html 2012-08-27T11:23:01-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:eelec:eelecrubberbandstretch?rev=1346091781 Lecture: Snapping a Rubber Band (15 minutes) text/html 2016-07-08T14:50:02-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:eelec:eelecsecondlaw?rev=1468014602 Lecture: The Second Law of Thermodynamics (5 minutes) Lecture notes from Dr. Roundy's 2014 course website: Second Law and Entropy If you drop a hot chunk of metal into a cup of water, which way will energy be transferred by heating? What is the rule that governs this? text/html 2016-07-08T14:25:24-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:eelec:eelecstatthermocompare?rev=1468013124 Lecture: Introduction to the Statistical Approach (?? minutes) FIXME Lecture notes from Dr. Roundy's 2014 course website: A statistical approach So far in this class, you have learned classical thermodynamics. Starting next week, we will be studying statistical mechanics. Thermodynamics may look ``theoretical'' because it involves a lot of math, but ultimately it is an experimental science. Thermodynamics puts severe (and interesting) constraints on equations of state, but can never tell us … text/html 2016-07-07T14:34:55-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:eelec:eelecthermoidentity?rev=1467927295 Lecture: The Thermodynamic Identity (?? minutes) FIXME Lecture notes from Dr. Roundy's 2014 course website: The internal energy is clearly a state function, and thus its differential must be an exact differential. $$dU = \text{ ?}$$ $$= đQ - đW$$ $$ = đQ - pdV \text{ only when change is quasistatic}$$ text/html 2016-07-11T11:45:08-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:eelec:eelecthermoterms?rev=1468262708 Lecture: Thermodynamic Terms (7 minutes) Naming Thermodynamic Variables Lecture notes from Dr. Roundy's 2014 course website: We begin with the now-familiar thermodynamic identity $$dU=TdS-pdV$$ Remember in the Interlude I talked about what if one of the weights were hidden in the black box, so you could not change it, or measure its position? Now we get to see why. text/html 2016-07-08T14:40:17-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:eelec:eelecthermowork?rev=1468014017 Lecture: Work in Thermodynamics (5 minutes) Lecture notes from Dr. Roundy's 2014 course website: We can find the change in potential energy of our system by measuring the work done by our weights as they move up and down. If we consider that $x_1$ and $x_2$ increase as the weights move down, the work done by the $x_1$ weight is given by: $$W_1 = \int F_1 dx_1$$ Similarly, the work done by the $x_2$ weight is given by: $$W_2 = \int F_2 dx_2$$ Taken together, we can see that the change in the … text/html 2016-07-18T10:17:15-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:eelec:eelecweightedavg?rev=1468862235 Lecture: Weighted Averages (?? minutes) FIXME Lecture notes from Dr. Roundy's 2014 course website: Most thermodynamic quantities can be expressed as weighted averages over all possible eigenstates (or microstates). For instance, the internal energy is given by: by: $$U = \sum_i P_i E_i$$ Note that this will probably not be an eigenvalue of the energy, but that's okay. The energy eigenvalues are so close for the total energy of a macroscopic object that we couldn't distinguish them anyhow. Any…