What are excess thermodynamic functions
Experimental Physics II: Thermodynamics
1 Experimental Physics II: Thermodynamics Second Attempt Summer Semester 09 William Hefter 11/09/2009 Contents 1 Temperature, Heat and Work Insert for the mathematical notation First principle of thermodynamics Thermal expansion Heat transfer Ideal gas: Kinetic gas theory Basic terms Statistical mechanics Ideal gas: Adiabatic processes 7 4 Ideal gas: summary of the processes 7 5 Real gas: an-der-waals equation 8 6 Entroy and the second principle of thermodynamics For the calculation of S circular processes Appendix 1: Total differentials 12 8 Appendix 2: Enthaly, thermodynamic potentials and Joule-Thomson -Process 13 1
2 Thermodynamics 1 Temperature, heat and work Kelvin scale: Absolute scale, defined by helium pressure. 0K ˆ = 273, 15 C Celsius scale: Scale adapted to everyday needs, using the melting and boiling point of water. Heat is thermal energy, or more precisely, heat is energy transferred between them due to a temperature difference between two bodies. Convention: dq> 0 if heat is added to the system. dq <0 if heat is extracted from the system. Thermal capacity denotes the amount of heat required ro temperature increase, i.e. C = dq [C] = J / K. Material-specific quantities or those under constant conditions are more practical: Specific heat capacity, related to mass: dq = mc [c] = J kg K Specific heat capacity, related to the amount of substance (also molar heat capacity): Heat capacity at constant volume: () dq c = dq = nc mol [c mol] = J mol K [c] = JK Latent heat During phase transitions, a substance absorbs or gives off heat without changing its temperature. The energy is expended to e.g. gradually evaporating (earth evaporation energy) or is gradually withdrawn, whereby the substance solidifies (crystallization energy). The temperature does not change. This energy is called the latent heat L w with the unit J kg. For example, when water is frozen, the temperature remains at 0 C until all of the water is frozen, and only then continues to drop. Work A gas can do work when there is a change in volume, e.g. if a gas lifts a cylinder cover through expansion, work dw = F ds = A ds = d is done for an infinitesimal path ds. This result also applies very generally, from which it follows: ˆ W = ˆ 2 dw = d 1 So only the pressure function of the system has to be known. Analogous to heat, the second applies
3 Convention: dw> 0 if work is being done on the system. dw <0 when the system is doing work. 1.1 Inset on the mathematical notation It is important to understand the notations for changing a function A; A and don't get mixed up there. da denotes an infinitesimal change in a function. For some functions the infinitesimal element is also written as δa and then denotes an inexact differential for which the integral δa is not path-independent. Here should only be used there. A is a difference and simply the sum of many infinitesimal changes, so A = da. E.g. the expression for the heat change dq = C (T) in general with a temperature-dependent heat capacity. The difference Q can only be given for temperature-independent C as Q = C T. Another note on the notation of articular derivatives in thermodynamics: () for example means the change (derivative) of the volume with the pressure at constant temperature. The specification of the constant quantities only makes sense for articular derivatives. T 1.2 First principle of thermodynamics The internal energy U of a system denotes the total energy of a system, the change in internal energy is composed (with a constant number of parts) from the heat dq given off or absorbed by the system and the work performed on the system or that of the System performed work dw. First sentence: du = dq + dw U = Q + W The internal energy is a state function (important!), So its differential du is exact, the change in the internal energy of a system during a process does not depend on the process itself, but only from the start and end point in the diagram. In contrast, Q and W are not state functions; you cannot assign any heat or work content to a system. The internal energy of a system generally depends on the state variables, and T and contains the complete thermodynamic information. The following results from the conventions for the signs of heat and work: Important special cases: du> 0, when energy is supplied to the system. du <0, when energy is withdrawn from the system. Adiabatic processes: A process is carried out so quickly or so thermally insulated that no heat is exchanged with the environment: dq = 0 du = dw. Isochoric change of state: (constant volume) d = 0 du = dq. Circular process: The initial state is always restored, i.e. du = 0 dq = dw. 3
4 1.3 Thermal expansion The thermal expansion coefficient of a system is given by α = 1 () T (i.e. how does the volume change when the temperature changes, under constant pressure?) The factor 1 is used for normalization. If the expansion coefficient is known, the change in the volume is infinitesimal: d = α and with a constant α: = α T 1.4 Heat transfer There are three types of heat transfer: conduction, convection and radiation. Thermal conduction Two materials are connected by a thermal conductor of thickness L, contact area A and a coefficient of thermal conductivity λ. Then the following applies for the transferred heat output: P = dq dt and for a constant temperature gradient: = λa dx P = dq dt = λa T hot T cold L with the coefficient of thermal conductivity λ [λ] = W m K convection heat transfer through buoyancy in materials of different densities that arose from a temperature difference. In contrast to heat conduction, matter is transported here. Thermal radiation Every body with T> 0K radiates energy in the form of electromagnetic radiation. The spectrum of this radiation is given by Planck's law of radiation. If this is integrated over the entire frequency spectrum, one finds the Stefan-Boltzmann law P rad = σɛat 4 K with the Stefan-Boltzmann constant σ = 5,, the area A of the radiating body, m 2 K 4 its temperature TK and the emissivity ɛ [0, 1]. For ɛ = 1 there is a black body. According to Kirchoff's radiation law, the degree of emission and absorption are the same. A black body is a detected radiation and an absorber at the same time (our sun is a very good example). Similarly, a body absorbs radiation from an environment with a temperature T according to W P abs = σɛat 4 Umg 4
5 The difference between the radiated and absorbed power is therefore: P = σɛa (t 4 KT 4 Umg) The maximum of the radiation curve - the wavelength that has the most radiation from the body - shifts with the temperature according to Wien's law of displacement λ max = 2898, 7 µmk T 2 Ideal gas: Kinetic gas theory 2.1 Basic terms Avogadro number Number of particles ro Mol: NA = 6, mol Amount of substance Amount of mol of a sample: n = NNA = m M Ideal gas is defined as dot-shaped molecules no interaction of the particles among each other except for elastic collisions From the observations = const. (Boyle-Marriotte) and / T = const. (Gay-Lussac) follows (for low densities) ideal gas law = nrt = Nk BT with R = J 8, 31 mol K k B = R = 1, JNAK general gas constant Boltzmann constant For ideal gases this means: work at isothermal processes (T = const.): W = ˆ 2 1 d = = nrt ln 2 1 ˆ 2 1 nrt d = nrt [ln] 2 1 work with isochoric processes (= const.) is still: W = 0 work for isobaric processes (= const.): W = 5
6 2.2 Statistical Mechanics The particles of a gas of temperature T do not all have the same speed. The actual speeds are statistically distributed; this grant is the Maxwell grant, a probability distribution. With the Maxwell certificate, some useful quantities can easily be calculated; One of the most important is the mean speed square, also known as rms speed (rootmean square): v rms = v2 3RT = M = 3kB T m the energy of the ideal gas The mean square of the velocity v rms can be used to calculate the mean kinetic energy of a gas particle as well as the sum of all kinetic energies, the internal energy: Kinetic energy of an ideal gas particle: E kin = 1 2 mv2 rms = 3 2 k BT Internal energy of an ideal gas: U = NE kin = 3 2 nrt = 3 2 Nk BT The last equation allows one important conclusion: the internal energy of a gas depends only on its temperature. Specific heat capacity of the ideal gas In the case of a gas, it is advisable to consider amounts of substance instead of a total mass, which is why specifically here refers to the amount of substance. If you consider an ideal gas in a fixed volume (d = 0 dw = 0), the specific heat capacity at constant volume (for monatomic gases) is the change in internal energy with the temperature ro mol, i.e. C = 1 n (dq) = 1 n () du dw d = 0 = 1 n () du C = 3 2 R At constant pressure the work is W = and therefore C = 1 () dq = 1 () () du dw = 1 3 2nR + dnnn (= 1 3 nrt 2nRT + d) () = 1 5 2nRT nn thus there is also a relation for the specific heat capacity at constant pressure (for monoatomic gases): 6
7 C = C + R = 5 2 R Important: The change in the internal energy is always du = nc (only dependent on C!) Because with a non-constant volume the excess energy is converted into work. Degrees of Freedom of the Ideal Gas So far, ideal monatomic gases have been dealt with. Diatomic or polyatomic gases have more than just three translational degrees of freedom; at higher temperatures they are stimulated to rotate and vibrate, which in turn require or save energy. With an increasing number of degrees of freedom (these are activated at certain temperatures), the thermal capacity of a gas also increases. Maxwell's uniform distribution theorem provides a statement on this: Every quadratic degree of freedom has an energy of 1 2 k BT on average. Thus it follows for the heat capacities at f degrees of freedom: C = f 2 R C = () f R Example ideal gas: octagonal, 3 translational degrees of freedom C = 3 2 R known. 3 Ideal gas: Adiabatic processes For very rapid changes of state or for those that take place completely thermally isolated from the environment, Q = 0 can be set. The adiabatic equations apply: γ = const. or 1 γ 1 = 2 γ 2 T γ 1 = const. T 1 γ 1 1 = T 2 γ 1 2 with the adiabatic coefficient γ = C C The work in adiabatic processes can only come from the internal energy! E.g. the white steam in a champagne bottle immediately after opening. The air expands suddenly and cools down, which leads to condensation of moisture. 4 Ideal gas: summary of the processes The following table can now be compiled from all the previous sections: 7
8 Figure 1: State changes in the - diagram The following applies to all processes: du = dq + dw du = nc State changes of the ideal gas path const. Designation Infinitesimal element 1 isobaric dq = nc Q = nc T (for C const.) Dw = d W = 2 T isothermal du = 0 dw = nrt d W = nrt ln 2 1 (since T const.) 3 γ, T γ 1 adiabatic du = dw dq = 0 4 isochor du = dq dq = nc Q = nc T (for C const.) 5 Real gas: an-der-waals-equation In real gases the gas particles have a finite extension and they attract / repel Interaction. The preservation of these gases can be described to a good approximation by the an-der-waals equation for real gases: () + n2 2 a (bn) = nrt This corresponds to the ideal gas equation with correction factors a and b, a describes the internal pressure, thus the interaction of the particles with one another (which increases the pressure); b describes the intrinsic volume of the particles (the effective volume decreases). 8th
9 Figure 2: --Diagram of a real gas. For pressures, volumes and temperatures below the so-called critical point, this equation predicts unhysical preservation (volume increase with pressure increase, see figure). In fact, the gaseous and liquid phases coexist in this area and the pressure remains constant because the substance is liquefied. In the diagram these are the straight lines below the critical point. This is defined by () c, c, tc = (2 2) c, c, tc = 0 which results in c = a 27b 2 c = 3nb T c = 8a 27bR 6 Entroie and the second main principle of thermodynamics coffee is always colder, never warmer; Pocorn does not shrink back to the corn kernel and buildings do not rebuild by themselves: time has a direction. So there has to be a quantity that has not been preserved and which increases over time - the entroie S. The absolute entroie as an abstract quantity is a measure of the number of states that a system can assume, i.e. its disorder. The change in entropy of a system is more interesting: S = ˆf i dq T The second principle of thermodynamics makes a statement about S: S 0 The change in entropy in a closed system is never negative! 9
10 Or the pigsty always grows on its own. Important: In contrast to the energy, the entropy is not preserved! Individual sub-processes can very well have a negative change in entropy (see above), but never a complete system. For an irreversible process, S then a reversible replacement process must be found. For a cycle, ˆf i S = 0. For reversible processes in a closed system, the entropy of an ideal gas is also a state function and only depends on the start and end point; thus calculable dq T dq = du dw = nc + d = nc + nrt d S = ˆT ˆT dq T = ˆ nc T + nr d S = nc ln T + nr ln + const. (t 0, 0) 6.1 For the calculation von S In the case of irreversible processes, the connection line between the initial and final state in the - diagram is not known. Since S only depends on the start and end point, the process can be replaced by a reversible process that connects the same two points in the - diagram and for which the state functions are known. Example: Two copper sheets (c c Ku f er = 386 J kg K) with T = 20 C or T = 60 C and a mass of m = 1.5 kg each are brought together so that the equilibrium of spatially constant temperature is established. You can replace this process with two reversible processes by simply placing the two copper sheets on two heating bars with exactly 40 C each: 10
11 These processes are easy to describe; for both the following applies: dq = mc S = from which follows for the entire change in entropy ˆ T2 T 1 mc T = mc ln T 2 T 1 S = SL + SR = mc (ln 313K 333K) 313K + ln 293K = 35, 86 JK + 38, 23 JK = 2, 4 JK 6.2 Circular processes In circular processes, the same path is always followed in the - diagram. The enclosed area corresponds to the work done, so W = d. The entropy is equal to zero for one revolution, there is a state variable, i.e. S = dq T = 0. The Carnot process is such a circular process and is intended here to illustrate and investigate circular processes. It is an ideal circular process and as such has no losses, e.g. by friction. Figure 3: Carnot process: ab isothermal expansion, bc adiabatic expansion, cd isothermal compression, da adiabatic compression This process is characterized by two heat baths with the temperatures T w and T k, in which the two isothermal processes with exchange of the heat quantities Q w > 0 and Q k <0 and two adiabats. The thermal energy absorbed during one cycle is dq = Q w Q k. From this the following can be calculated: Work: Since there is a circular process, du = 0 for a full cycle and thus dw = dq = Q k Q w Entroie: Here only the isothermal processes (dq = 0) contribute: S = S w + S k = Q w T w 11 Q k T k
12 Efficiency: In general, thermal efficiency can be defined as (roughly benefit through costs) getting energy W total ɛ = = energy paid for Q w In the Carnot cycle, W total = dw = Q w Q k and thus results for the Efficiency of the Carnot cycle with the temperature baths T w and T k: ɛ = 1 T k T w With the help of the last definition, the second main principle can also be formulated differently: There is no machine that works eriodically between the temperatures T w and T k, which has a higher efficiency than a Carnot machine with these temperatures. Refrigeration machine and heat volume Circular processes can also be operated the other way round. In this case, work is done on the machine that is used to convert heat from the cold into the warm bath, i.e. Q k is absorbed and Q w is emitted. A refrigeration machine has the task of cooling the cold heat bath further.In this case, a more sensible definition of efficiency is this: ɛ cool = dissipated heat expended work = Q k W applied A heat room, on the other hand, is intended to further heat the warm bath, i.e. the same process, viewed differently, here the applied heat ɛ warm = Work expended = Q w W elaborate For a Carnot refrigeration machine, we get ɛ cool, c = Q k Q w Q k It is a backward-working Carnot machine, which is why we, taking into account the previous section, for the coefficient of performance of the Carnot refrigeration machine can write: ɛ cool, c = T k T w T k 7 Appendix 1: Total differentials A total differential is the differential of a function of several ariables and describes the change in the function with a small - infinitesimal - change in its ariables. It results from the chain rule. For example, the total differential of the internal energy is U (S,), the variables of which are the entropy and the volume: And it is known that what results from du = (US) () U = TS ds + du = TdS d 12 () U d S () U = S
13 This can be used to make some useful calculations, e.g. to derive the (non-specific) heat capacity at constant pressure: du = dq + dw = dq d dh = du + d + d = dq + d The latter line comes from Appendix 2. It follows further: () () () dq du + d ie dc = = = = () HT The latter expression can e.g. be useful for simplifying formulas. 8 Appendix 2: Enthaly, thermodynamic potentials and Joule-Thomson process Analogous to internal energy, there are other state functions that contain the entire thermodynamic information of a system. These are the thermodynamic potentials. Depending on which variables can be controlled from the outside, it makes sense to consider a different potential. Here the enthaly H = U + is to be named, the variables of which are the enthaly and the pressure. It is preserved in the Joule-Thomson process. Its total differential is: dh = (HS) ds + (H) S d = du + d + d = TdS + d In the Joule-Thomson process, a gas with the volume 1 and the pressure 1 is pressed through a throttle into the volume 2, the pressure behind the throttle decreasing to 2. For a real gas, the temperature can either decrease or increase, depending on whether the initial temperature is below or above the so-called inversion temperature. The Joule-Thomson coefficient makes a statement about this: µ JT = () T = [T 1 H C ()] T 13
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