Thermodynamic States The ideal gas law relates the pressure (P), volume (V), number of moles (n) and temperature (T) of a gas. PV = nRT In SI units, R = 8.31 J/(mol*K). This law is approximate. It only applies to cases where the temperature is well above the condensation point and the volume of the molecules is much less than the volume of the container. Thermodynamic Processes ΔU = Q – W The first law of thermodynamics is a statement of conservation of energy. A system can exchange energy with its environment with heat (Q) or work (W, also denoted as Ws). This results in changes in its internal or thermal energy (U, also denoted as Eth). By convention, heat coming into the system is considered positive and heat going out of the system is considered negative. Work done by the system is considered positive, while work done on the system is considered negative (note that this is the opposite of the classical mechanics sign convention for work). W = ∫PdV (± the area under the curve in the PV diagram) always. For an isochoric process, W = 0. For an isobaric process, W = P∫dV = PΔV. For an isothermal process, the ideal gas law can be used to find P as a function of V and substituted into the equation for work. W = ∫(nRT/V)dV = nRT*ln(Vf/Vi). The definitions of Cp and Cv can be used to obtain a formula for heat during isobaric and isochoric processes. For an isobaric process, dQ = nCpdT. If Cp and n are constant (usually the case, at least approximately), then Q = nCpΔT. Likewise, for an isochoric process, dQ = nCvdT. If Cv and n are constant (usually the case, at least approximately), then Q = nCvΔT. This formula for heat during an isochoric process can be used with the first law and the formula for work to obtain expressions for the change in internal energy for any process. For an isochoric process, W = 0, so ΔU = Q = nCvΔT. This formula for ΔU can be generalized to all processes since internal energy is path independent. Physical Situation Name State P V T ΔU Q W Variables Insulated Add Adiabatic PVγ = Up Down Up nCvΔT 0 -nCvΔT < 0 Sleeve weight to compression constant; >0 piston TVγ-1 = constant Insulated Remove Adiabatic PVγ = Down Up Down nCvΔT 0 -nCvΔT > 0 Sleeve weight expansion constant; <0 from TVγ-1 = piston constant Heat gas Lock Isochoric V fixed; Up Fixed Up nCvΔT nCvΔT > 0 0 piston PαT >0 Cool gas Lock Isochoric V fixed; Down Fixed Down nCvΔT nCvΔT < 0 0 piston PαT <0 Heat gas Piston Isobaric P fixed; Fixed Up Up nCvΔT nCpΔT > 0 PΔV > 0 free to expansion VαT >0 move Cool gas Piston Isobaric P fixed; Fixed Down Down nCvΔT nCpΔT < 0 PΔV < 0 free to compression VαT <0 move Immerse Add Isothermal T fixed, Up Down Fixed nCvΔT nRT*ln(Vf/Vi) nRT*ln(Vf/Vi) gas in weight to compression PV = =0 <0 <0 large bath piston constant Immerse Remove Isothermal T fixed, Down Up Fixed nCvΔT nRT*ln(Vf/Vi) nRT*ln(Vf/Vi) gas in weight expansion PV = =0 >0 >0 large bath from constant piston Unknown Unknown No Name PV = ? ? ? nCvΔT ΔU + W ∫PdV = ± area nRT under curve in PV diagram Tips: Know which formulas are specific to a particular process and which are true for any process. See the next section for notes on Cp, Cv and γ. Heat Capacities Cv, the molar heat capacity at constant volume (zero work), can be estimated using the number of degrees of freedom multiplied by ½R. For a monatomic gas, there are three degrees of freedom from the translation of the particles in three dimensions, so C v = 3/2*R. For a diatomic gas, there are five degrees of freedom from the three directions of translation and two axes of rotation. The third possible axis does not have a significant rotational kinetic energy and is therefore insignificant. Therefore, Cv = 5/2*R. For solids, there are three degrees of freedom from translation and three from vibration, so Cv = 3R (Dulong-Petit). All three formulas are theoretical and classical, and generally give reasonable agreement with empirical evaluations. Deviations from these formulas can be explained with quantum mechanics which is beyond the scope of this course. Cp, the molar heat capacity at constant pressure, can be calculated for an ideal gas. For an isobaric process where n and Cp are constant, Q = nCpΔT. The change in internal energy can be calculated with the general formula ΔU = nCvΔT. W = P∫dV = PΔV by the definition of work. Using the ideal gas law, PΔV = nRΔT. ΔU = Q – W by the first law of thermodynamics. Combine the above formulas to obtain an expression for Cp. ΔU = Q – W nCvΔT = nCpΔT – nRΔT Cp = Cv + R The ratio of heat capacities is denoted by the letter gamma (γ), and is defined by the following formula: γ = Cp/Cv For a monatomic ideal gas, γ = (3/2*R + R)/(3/2*R) = 5/3. For a diatomic ideal gas, γ = (5/2*R + R)/(5/2*R) = 7/5. Gas Cv Cp γ Examples monatomic 3/2*R 5/2*R 5/3 He, Ne, Ar diatomic 5/2*R 7/2*R 7/5 H2, N2, O2 Thermodynamic Cycles A cycle must have a total ΔU = 0. Therefore ΣQ = ΣW by the first law. Normally, it is useful to separate the values of Q that are positive from those that are negative. Positive values represent heat input into the system and negative values represent heat output from the system. The total work for a cycle can be calculated graphically with the area enclosed in the PV diagram. If the cycle is clockwise, then the device is a heat engine and W > 0. If the cycle is counter-clockwise, then the device is a refrigerator/air conditioner or heat pump and W < 0. Heat Engines A heat engine has a characteristic called efficiency, e (also denoted η). e = ΣW/ΣQH The summation of work includes all processes, regardless of sign. The summation for heat includes only heat input from the “hot reservoir” which in this case includes only the processes with positive heat. There is a theoretical upper limit on the efficiency of an engine operating between two temperature extremes. This is the Carnot efficiency. ecarnot = 1 – Tc/Th Heat Pumps, Refrigerators, and Air Conditioners If a cycle has a total work less than zero, then the device might be a refrigerator/air conditioner or a heat pump. These devices have heat input from a lower temperature system and have heat output to a higher temperature system. The physical construction for these two devices can be exactly the same, but the use and desired outcomes are different. With a refrigerator/air conditioner, the goal is to transfer heat from the colder system. With a heat pump, the goal is to transfer heat to the hotter system. With any of these devices, the energy input is in the form of work. This work is typically done by a compressor (you pay for the energy to run this). A refrigerator/air conditioner has a characteristic called the coefficient of performance, C.O.P. or K. K = ΣQc/|ΣW| The summation for heat includes only the heat input from the “cold reservoir” which in this case includes only the processes with positive heat. The summation of work includes all processes, regardless of sign. There is a theoretical upper limit on the coefficient of performance for a refrigerator/air conditioner operating between two temperature extremes. This is the Carnot coefficient of performance and is based on the second law of thermodynamics. Kcarnot = Tc/(Th – Tc) A heat pump also has a characteristic called the coefficient of performance, C.O.P. or K. K = ΣQh/ΣW The summation for heat includes only the heat output to the “hot reservoir” which in this case includes only the processes with negative heat. The summation of work includes all processes, regardless of sign. There is a theoretical upper limit on the coefficient of performance for a heat pump operating between two temperature extremes. This is the Carnot coefficient of performance and is based on the second law of thermodynamics. Kcarnot = Th/(Th – Tc) Entropy Entropy, S, is often characterized as a measure of disorder. The second law of thermodynamics states that for an isolated system (no energy or matter exchanged with external agents), the total entropy of the system cannot decrease: ΔS ≥ 0 There are two general methods for calculating entropy. The first is useful when there are exchanges of energy in the form of heat. The second is useful when there is mixing of particles: ΔS = ∫(dQ/T) ≈ Q/T S = k*ln(W) S = entropy Q = heat T = temperature k = Boltzmann’s constant = 1.38E-23 J/K W = number of possible microscopic states consistent with the macroscopic state Misuses of the Laws of Thermodynamics There are commonly held views among the general public that the laws of thermodynamics falsify both cosmic and biological evolution. These views are not supported by data. One view unsupported by data is that the Big Bang couldn’t possibly be correct because it violates the first law of thermodynamics. The claim is that there is obviously a lot of energy now and there couldn’t be any energy before the Big Bang. The total energy apparently increased thus violating the first law of thermodynamics. For this alleged violation to be true, we must know both the total energy of the universe both before and after the Big Bang and show that they are different (putting aside objections that there is no such thing as “before the Big Bang”). No one has calculated the energy of the universe “before the Big Bang.” But even if one assumes that the energy is zero “before the Big Bang”, a calculation of the total energy of the universe based on classical physics also yields a total energy of zero! This was shown in a paper written by E. Tryon in a 1973 article in the journal Nature (and not refuted to date). How can that be so with all the stuff moving around (kinetic energy), light energy, etc.? It turns out that the negative gravitational potential energy balances out the positive energy and the net sum is zero. Even if we assume the total energy of the universe before the Big Bang is zero, there is no proven violation of the first law of thermodynamics. Another view unsupported by data is that the second law of thermodynamics prohibits the evolution of chemicals to simple organisms or of simple organisms to more complex organisms, an apparent decrease in entropy. There are two important objections to the "life violates the second law" argument. First, there is no calculation that shows that complex life forms are lower entropy than less complex forms or non-living things. Hand waving metaphors are no substitute for a proper calculation. Second, life forms are not closed systems. They continuously exchange energy and particles with their environments, so the second law has nothing to say about them unless you include their environments in the calculation. Evolution has not been proven to violate the second law of thermodynamics.