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Thermodynamics Basic Concepts

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Thermodynamics Basic Concepts Thermodynamics Basic concepts • Thermodynamics is a phenomenological description of macroscopic systems with many degrees of free- dom. Its microscopic justication is provided by statistical mechanics. • Equilibrium is a state of an undisturbed system that does not change over time in any observable way - at least within time periods that are much longer than the time intervals of observation. A macroscopic system left to itself will spontaneously approach its equilibrium state. • An isolated system cannot exchange energy or anything else with its environment. However, a system can be coupled to the environment in various ways. Dierent types of coupling enable dierent types of equilibrium. • The coupling to the environment allows us to control only a very small number of the system's degrees of freedom. The controllable and measurable properties of a system are various state variables, quantities such as temperature, pressure, volume, energy, charge, magnetization, etc. Most degrees of freedom 23 (whose number can be of the order of Avogadro's number NA ∼ 10 ) are far beyond our ability to control, measure or even mathematically handle. • Nevertheless, thermodynamics, formulated by mathematical relationships between state variables (laws of thermodynamics), is suciently detailed to enable quantitative predictions of the macroscopic sys- tem dynamics. Historically, the development of thermodynamics led to applications such as engines and refrigerators. • The most studied system in thermodynamics is gas. An ideal gas is a collection of many non-interacting identical particles, which can't collide with one another but bounce from the walls of a container. Realistic gases at very low densities (rare collisions), or with very weak interactions (insignicant collisions) can be approximated by an ideal gas. Other systems of interest are uids (interacting gases and liquids), solids (crystals and amorphous), electrons in materials, magnets, plasmas, nuclear matter, biological systems, etc. Thermodynamics studies both classical and quantum systems. Thermal equilibrium and temperature • 0th law of thermodynamics: If two systems are separately in equilibrium with a third system, then they are also in equilibrium with each other. • The rst implication is that two systems can be in equilibrium with respect to each other, meaning that if such two systems are in contact with each other (and isolated from everything else) then their properties will not change over time. • Second, if two systems A and B are separately in equilibrium with a third system C, then they have something in common: being in equilibrium with C. This common property can be expressed as a state variable that separately characterizes the states of each system A and B. • The universal state variable that characterizes the equilibrium of any system is temperature. Two systems in equilibrium with each other (A and B) have the same common temperature T (this is an alternative statement of the 0th law). Dierent kinds of equilibrium (mechanical, chemical, etc.) may be characterized by additional state variables (pressure, chemical potential, etc. respectively). • We still have no denition of temperature. From our everyday experience, we regard temperature as a measure of how hot something is. More practically, we can measure temperature of a system by putting some uid in contact with it, waiting until the uid and the system equilibrate, and measuring the volume by which the uid expands or contracts. The change of uid's volume is roughly proportional to the change of its temperature, but depends also on the choice of uid. We must pick a particular uid and decide what reference points and scale to use in order to rigorously dene temperature. 1 • It has been noticed that heated water mixed with ice maintains its cold temperature until all ice is melted. Similarly, heated boiling water maintains its hot temperature until all water evaporates. This provides one convenient temperature scale, Celsius: water and ice in equilibrium have a reproducible temperature T = 0o C, while water and steam in equilibrium have a reproducible temperature T = 100o C (by denition). • The modern scale for temperature has been established at the 10th General Conference on Weights and Measures in 1954. The SI unit of temperature is Kelvin. The reproducible temperature of the triple point of ice-water-gas system is dened to be 273:16 K, while 0 K (absolute zero) is the lower bound for temperature of any system. The water triple point is a much better reference than freezing or boiling temperatures, because the former is an equilibrium between all three phases of water that occurs only at a particular pressure, while freezing/boiling critical temperatures actually depend on the ambient pressure. The reference of 273:16 K is chosen in order to make the temperature change of one Kelvin (K) identical to the temperature change of one Celsius (determined at standard atmospheric pressure). • In addition to a scale, one must also specify a concrete reference procedure for measuring temperature, which will provide a linear scale (by denition). For this purpose, one uses an ideal gas. It has been experimentally observed that the quantity pV=T (p is pressure, V is volume and T is temperature in Kelvins measured by a typical accurate thermometer) is a constant in any equilibrium state of any suciently dilute (approximately ideal) gas. Therefore, for two equilibrium states at the same pressure: V1 V2 V2 = ) T2 = T1 T1 T2 V1 Since one temperature (T1) can be determined by achieving equilibrium with a reference system (water at its triple point), any other temperature can be found by measuring volume in an ideal gas expansion experiment. This observed linear relationship for an ideal gas is then promoted to the denition of temperature, so that, by denition, accurate thermometers measure temperature and not some arbitrary function of temperature. The conservation of energy • A fundamental property of any system is energy. Internal energy E is the part of the system's total energy that is stored in its own degrees of freedom and depends only on the state (and not environment) variables. For any system of particles, internal energy is the total kinetic energy of all particles (measured in the center-of-mass frame) together with the potential energy of interactions between the particles. • A system can exchange energy with its environment in dierent ways. The general exchange of energy is work W done on the system by the environment (or work −W done on the environment by the system). Normally, we will not be concerned with work that accelerates the center of mass or changes the potential energy of the system due to external forces. Hence, all work will go toward internal energy. • The amount of work dW done on the system by external forces in an innitesimal time interval is: X dW = JidXi i as long as the system is not thrown far from equilibrium (so its state can still be faithfully represented by state variables such as temperature, pressure, etc.). Here, Ji are quantities that represent forces and Xi are the corresponding state variables that represent responses (displacements) of the system to the applied force. For example: Length change dL due to an external (tension) force F : dW = F dL Surface area change by dA due to a surface tension S: dW = SdA Volume change by dV due to internal pressure p: dW = −pdV 2 Magnetization change by dM in an external magnetic eld B: dW = BdM Electric polarization change by dP in an external electric eld E: dW = EdP The number of particles change by dN at a chemical potential µ: dW = µdN • There are processes in which energy is exchanged but we cannot identify the detailed corresponding change of the system's state (typically due to its microscopic complexity). The energy exchanged in such processes is called heat. For example, if two solid objects at dierent temperatures are placed in contact with each other, the hotter object will transfer heat to the colder one until both objects reach the same temperature (equilibrium). This need not lead to any change of volume, the number of particles, etc, which we would need to observe in order to recognize this energy exchange as work according to one of the above formulas. The change of the system is microscopic and hidden from our view (the distribution of particle velocities changes). • A system is said to evolve adiabatically if it does not exchange heat with its environment. • 1st law of thermodynamics: Energy is conserved in any process. If the system takes heat dQ from its environment, and work dW is done on it by external forces, then its internal energy changes by dE according to: dQ + dW = dE The sign convention is such that a positive value indicates energy added to the system. Response functions • Response functions are easy-to-measure quantities that depend on state variables and thus characterize the state of a system. • Heat capacity: how much energy (heat) dQ does it take to change the temperature of the system by dT under various conditions... dQ CV = ··· measured at given volume V = const dT V dQ Cp = ··· measured at given pressure p = const dT p • Force constant: a thermodynamic generalization of the spring constant 1 dX RT = ··· the change of displacement due to applied force, per unit volume V dJ T 1 dV κT = − ··· compressibility (pressure changes volume) V dp T 1 dM χT = ··· magnetic susceptibility (magnetic eld changes magnetization) V dB T • Thermal responses: 1 dV αp = ··· expansivity of a gas at constant pressure V dT p Ideal gas • Joule's free expansion experiment: If an ideal gas expands through empty space (without having any environment to exchange heat or work with), then its initial and nal temperature are found to be the same. But, according to the 1st law of thermodynamics: dE = dW + dQ ! 0 in this case, so internal energy E does not change even though the volume of the gas changes.
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