Lecture Notes: Definition of Heat and Pressure in Physics 4230, Fall 2008, Study notes of Physics

Download Lecture Notes: Definition of Heat and Pressure in Physics 4230, Fall 2008 and more Study notes Physics in PDF only on Docsity! 1 Physics 4230, Fall 2008 Conceptual outline for lectures 17-19 (10/6, 10/8 and 10/10) Precise definition of heat. For an infinitesimal process, we defined heat to be Q = TdS, (1) where T is the temperature and dS is the change in entropy. This is more satisfying than the definition we adopted earlier, where we defined work, and said that heat lumps together all other energy changes. This definition says what heat is, rather than defining it by what it isn’t. We motivated this definition of heat by looking the definition of temperature, 1 T = ( ∂S ∂U ) N,V . (2) What this equation does is define T in terms of a change in entropy dS and a change in energy dU , where both changes occur as constant number N and volume V . Keeping this in mind, we can write 1 T = dS dU . (3) The crucial step (and this is really a definition) is to identify the heat with dU , so Q = dU . Given our intuition about heat and work, this is reasonable, since this is an energy change that occurs without changing V (or the number N). This then gives us the definition Q = TdS. Cartoon picture of heat and work. We considered the example of a quantum ideal gas, where the configurations (microstates) can be visualized as particles occupying a bunch of energy levels. We developed a rough cartoon picture that helps illustrate the fundamental difference between heat and work. Work corresponds to a process that shifts energy levels around without reshuffling particles among the levels. Heat, on the other hand, corresponds to a process that keeps the energy levels fixed, but shuffles the particles around among the levels (and changes the total energy). What we can say rigorously (beyond the rough cartoon) is that, if the energy levels don’t shift in a process, the work was zero. It’s harder to use this cartoon picture to make rigorous statements about heat. The reason is that heat is really defined in terms of an entropy (or multiplicity) change, and these cartoon pictures generally show just a single microstate before and after the process. Therefore, it’s hard to say what actually happened to the multiplicity. Thermodynamic Identity. We supposed the energy is a function only of entropy, volume and number of particles, U = U(S, V, N). (This is indeed true for all the examples we’ve considered so far.) Assuming for the moment that the number of particles is constant, we considered an infinitesimal change in the state of the system: dU = Q + W = TdS − PdV . (4) All we did here is take the first law of thermodynamics, and put in our definitions of heat and work. This equation is sometimes referred to as the thermodynamic identity. It’s really just a restatement of the first law for infinitesimal processes. It also holds for quasistatic processes, if we get the total change in energy by adding up all the differentials. First definition of pressure. We used the thermodynamic identity to relate T and P to partial derivatives of U : T = (∂U ∂S ) V,N (5) P = − (∂U ∂V ) S,N . (6) The first expression is just the definition of T again, just taking the inverse of both sides. The second equation can be viewed as a definition of pressure. Derivation of ideal gas law. Using our formula for the entropy of a monatomic ideal gas (Sackur-Tetrode equation), we solved for U = U(S, V, N). From this, we calculated P using the definition above, and we found P = − (∂U ∂V ) S,N = NkT V . (7) We therefore derived the ideal gas law. (Alternatively, we can view this as a check that our definition of P is consistent with the ideal gas law.) To get this result, we need to use U = (3/2)NkT , which we proved in class a while ago by plugging the ideal gas entropy into the definition of temperature.