Thermodynamics Review/Tutorial - Ideal Gas Law - Heat Capacity

1. PHYS-575/CSI-655Introduction to Atmospheric Physics and ChemistryLecture Notes #3 – Part 1: Thermodynamics • Thermodynamics Review/Tutorial • - Ideal Gas Law • - Heat Capacity • - 1st & 2nd Laws of Thermodynamics • - Adiabatic Processes • - Energy Transport • Hydrostatic Equilibrium • Adiabatic Lapse Rate – DRY • Adiabatic Lapse Rate - WET • Static Stability • SLT and the Atmosphere

2. What is Thermodynamics? Thermodynamics is the study of heat and its transformation from a macroscopic point of view. "Department of Entropy" "Now, in the second law of thermodynamics..."

3. 1. Thermodynamics Tutorial Thermodynamics is the study of heat and its transformation to and from other sources of energy, from a macroscopic point of view. • Statistical Mechanics connects thermodynamics to the microscopic • world through the statistical description of an ensemble of atoms or • molecules that constitute a macroscopic system. • The transfer of heat, in turn, is driven by differences in temperature • or potential differences associated with chemical reactions. In the interest of crafting a brief tutorial for applications to the atmosphere, I have glossed over some of the finer (but yet important) points of thermodynamics. For more complete treatment: General:Fundamentals of Statistical and Thermal Physics (McGraw-Hill Series in Fundamentals of Physics) by Frederick Reif, 1965. Atmospheric:Atmospheric Thermodynamics, by C.F. Bohren and B.A. Albrecht, Oxford University Press, Oxford, 1998.

4. Defining Temperature Temperature is a measure of the mean kinetic energy of gas molecules. The temperature of an ideal monatomic gas is related to the average kinetic energy of its atoms as they move. In this animation, the size of helium atoms relative to their spacing is shown to scale under 1950 atmospheres of pressure. These room-temperature atoms have a certain, average speed (slowed down here two trillion fold).

5. Measure of Temperature Temperature of a measure of the mean kinetic energy of gas molecules.

6. Temperature Scales Temperature is a measure of the random kinetic energy of atoms and/or molecules <1/2 mv2> = 3/2 k T • The Fahrenheit Scale:  • we are most familiar with this one • water freezes at 32 degrees F. • water boils at 212 degrees F. • when we cool to the absolute lowest temperature we reach -459 degrees (this is referred to as Absolute Zero) • The Celsius Scale: • water freezes at 0 degrees C. • water boils at 100 degrees C. • Absolute Zero is at -273 degrees C. • The Kelvin Scale • Absolute Zero is 0 K • a temperature change of 1 degree K is the same as a temperature change of 1 degree C. • water freezes (or melts) at 273 K • water boils at 373 K • The Kelvin is scale is the more useful scale for our course since it refers to Absolute Zero in a direct way.

7. The Ideal Gas Law Laboratory experiments show that the pressure, volume, andtemperature of any material may be related by an Equation of State(EOS). These variables are known as State Variables. All atmospheric gasses follow an equation of state known as the Ideal Gas Law(IGL) to a very high degree of accuracy. We assume the IGL to be exact in atmospheric science. The Ideal Gas Law may be written:pV = mRTwhere p = pressure V = volume m = mass T = temperature (absolute Kelvin; K = oC + 273.15) R = gas constant The gas constant R depends upon the particular gas under consideration. Since m/V = ρ (density of the gas), the IGL may be written: p = ρRT

8. Ideal Gas Law (IGL) - continued We can also define α = 1/ρ, known as the specific volume, to write the IGL as: pα = RT Boyle’s Law:For fixed temperature, the pressure of a gas is inversely proportional to its volume, i.e., P ~ 1/V. Additional forms of the Ideal Gas Law: A mole (gram-molecular weight) of any substance is the molecular weight Mof the substance expressed in grams. For example, the molecular weight of water is 18.015 gm, so 1 mole of water is 18.015 gm of water. The number of moles (N) in a mass m (in grams) of a substance is given by: N = m/M The number of molecules in 1 moleof any substance is a universal constant called Avogadro’s number, NA. NA = 6.022 x 1023 molecules per mole.

9. Ideal Gas Law (IGL) – continued again The Ideal Gas Law for 1 mole of any gas can be written: pV = R*T Where R* is the universal gas constant = 8.3145 J K-1 mol-1. So for N moles of any gas, the IGL will be: pV = NR*T The gas constant for 1 molecule of any gas is also a universal constant known as Boltzmann’s constant, k= 1.38 x 10-23 J K-1 molecule-1 So for a gas with n gas molecules per unit volume V, the IGL is then p = nkT

10. Ideal Gas Law - Summary Ideal Gas Law: P = nkT = ρRT= RT/α PV = R*T/M P = pressure m = mass per gas particle n = number density of gas particles ρ = mn = mass density α = 1/ρ = specific volume V = volume of one mole of gas k = Boltzmann’s constant R = gas constant (gas specific) = R*/Μ M = molar mass R* = universal gas constant T = temperature

11. Heat Capacity The Heat Capacity of a material is a measure of its ability to absorb and retain heat. More precisely, the Heat Capacity is the energy (dQ) required to increase the temperature of a unit volume of any substance from T to T+dT (in Kelvin) The Heat Capacity depends upon the nature of the material and its temperature. The Heat Capacity also depends upon exactly how the energy is added. If the heat is added to a gas at constant volume the heat capacity is lower than if the heat is added at constant pressure. The reason is that heat performs work if the volume changes.

12. Heat Capacity - continued Heat capacity is related to the ability of a substance to store energy. Energy can be stored in a variety of ways. For a gas, the most obvious way to store energy is in random kinetic energy of the gas molecules. The 1/2mv2 is the kinetic energy of a molecule of mass m moving with a velocity v. There is ½ kT of energy “per degree of freedom” of the molecule. For a molecule moving in 3-dimensions, there are 3 degrees of freedom and thus the average kinetic energy is stored as 3/2kT. If there are other ways for a molecule to store energy, then the heat capacity will be higher. Thus the Heat Capacity depends upon the phase of the substance.

13. Heat Capacity of Water The Heat Capacity of water makes an excellent example. When frozen, water molecules do not have translational kinetic energy and thus its heat capacity is low. Molecules can only vibrate. Thawing requires heat and thus is a portion of its heat capacity. Upon thawing, water molecules can have kinetic energy of translation and the heat capacity increases with temperature. Evaporation requires heat and thus increases the heat capacity. http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/Thermal/gifs/HeatCapacity02.gif

14. Heat Capacity - continued As can be seen with water, the Heat Capacity is a function of only temperature. Thus we define the Internal Energy, U, of a unit volume of material to be the measure of the amount of thermal energy stored in the material. The Internal Energy thus depends only upon temperature. U = ρCvT For a gas, the distribution of speeds is a strong function of temperature. So the internal energy increases as the temperature increase. If you add heat (dq) to a gas, you can cause the internal energy (U) to increase and/or cause the gas to expand and do work on its environment.

15. Specific Heats Suppose a small quantity of heat dq is added to a unit mass of material and this causes the material to rise in temperature from T to T+dT. Then is the specific heat of the material. If the volume of the material is kept constant, then the specific heat at constant volume Cv is defined as: However, if the volume of the material is kept constant, then dq = du (heat changes internal energy and does no work on the environment) and: For an ideal gas u depends only upon temperature (T), so Cv depends only upon T.

16. Specific Heats - continued We can also define a specific heat at constant pressure Cp as: But when heat is added to a parcel of gas at constant pressure, some energy can be used in expanding the gas. So more heat must be added to a given mass of material at constant pressure to raise it to a given temperature than if the material was kept at constant volume.

17. The 3 Laws of Thermodynamics • First Law: You can’t win. • Second Law: You can’t break even. • Third Law: You can’t get out of the game.

18. The Three Laws of Thermodynamics • Conservation of Energy: Energy is neither created nor destroyed, it is merely converted from one form to another. • The Entropy of an isolated system increases when a system undergoes a spontaneous change. • The Entropy of all substances approaches zero as • the temperature (in Kelvin) approaches zero. • All substances have zero energy at absolute zero.

19. The First Law of Thermodynamics Conservation of Energy:Energy is neither created nor destroyed, it is only changed from one form to another. What is Energy? Forms of Energy: -- Gravitational Potential -- Kinetic Energy -- Chemical Energy -- Electromagnetic Energy -- Rest mass energy For any system (e.g. a specific collection of matter), the change in energy of the system is equal to the energy transferred by work plus the energy transferred by heat. Heatis the transfer of energy to or from a system associated with a temperature difference. Workis the transfer of energy to a system by the application of a force.

20. The first law of thermodynamics is the application of the conservation of energy principle to thermodynamic processes: http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/firlaw.html

21. The First Law and Work http://www.grc.nasa.gov/WWW/K-12/airplane/thermo1.html

22. http://www.fas.org/irp/imint/docs/rst/Sect14/stability2.jpg

23. Useful Forms of the First Law of Thermodynamics

24. Second Law of Thermodynamics TheEntropyof an isolated system increases when the system undergoes a spontaneous change. Entropyis the heat added (or subtracted), ΔQ, to a system divided by its temperature in Kelvin (T). It is a measure of the disorder of a system; a measure of the unavailability of a system’s energy to do work; a measure of the disorder of the molecules in a system; a measure of the number of possible states of a system. dQ is the heat absorbed in an isothermal and reversible process.

25. Implications of the Second Law The Second Law of Thermodynamics The Second Law of Thermodynamics states that it is impossible to completely convert heat energy into mechanical energy. Another way to put that is to say that the level of entropy (or tendency toward randomness) in a closed system is always either constant or increasing. • It is impossible for any process (e.g., engine), working in a cycle, to completely convert surrounding heat to work. • Dissipation will always occur. • Entropy will always increase.

26. Work and Heat Dissipation No matter how efficient the system (engine) is, dissipation will always occur. This usually appears as heat released from the system to its surroundings

27. Work and Heat http://physics.uoregon.edu/~courses/dlivelyb/ph161/heat_engine_schem.gif

28. Work and Efficiency The Second Law of Thermodynamics states that it is impossible for any heat engine to be 100 % efficient: No process is possible which results in the extraction of an amount of heat from a reservoir and its conversion to an equal amount of mechanical work.

29. Efficiency of Automobiles

30. The Second Law and Heat Dissipation for a Automobile

31. The Atmosphere as an Engine with Associated Dissipation

32. Atmospheric Circulation acts an Engine transferring heat from a hot region to a cold region.

33. Energy Flow in the Biosphere as an Engine with Dissipation Visible light contains most energy from the sun (per wavelength interval) and overlaps the region where the atmosphere is most transparent, and also is the region where most photosynthesis occurs in the biosphere.

34. Usable Energy http://trc.ucdavis.edu/biosci10v/bis10v/week2/2webimages/figure-06-03b.jpg

35. The Third Law of Thermodynamics The Entropy of all substances approaches zero as the temperature (in Kelvin) approaches zero. All substances have zero energy at absolute zero.

36. The Use of Thermodynamic Diagrams A pair of variables: (P, V) or (P,T) or (V,T) or… denote a state of the system. A P-V diagram shows the possible states that the system can have. dW = PdV = Force x Displacement

37. Parcel Concepts Below approximately 100 km altitude, air is relatively well mixed. Virtually all mixing is accomplished by the exchange of air “parcels” which have horizontal dimensions ranging from mm to the scale of the Earth. An air “parcel” of infinitesimal dimensions is assumed to be: • Thermally insulated from the environment (no energy exchange) • Moving slowly so that kinetic energy of motion is much smaller than it’s total energy. In reality, both of these assumptions are violated to some extent. But for small displacements over small time intervals they can be excellent approximations.

38. Thermodynamic Descriptions of the Atmosphere During any atmospheric process, the state of a parcel of atmospheric gas (P, V, T, S, etc) will change. The Laws of Thermodynamics determine exactly how these changes can occur.Phase Diagrams describe these changes in the state variables describing the gas.

39. Thermodynamic Descriptions of the Atmosphere During any atmospheric process, the state of a parcel of atmospheric gas (P, V, T, S, etc) will change. Any pair of variables can be used to describe the state of the system: (P,V) or (T, S) or (P, S), etc.

40. Example: Adiabatic Process - No Energy In/Out

41. Parcel Concepts: Applications of the laws of thermodynamics to air parcels First Law of Thermodynamics: dQ = dU + PdV Internal Energy: dU = CpdT Second Law of Thermodynamics: dS = dQ/T Adiabatic means dQ = 0. dQ implies dS = 0. Thus an adiabatic process is also an isentropic process.

42. Energy Transport There are three primary ways that energy is transported in planetary atmospheres. (1) Conduction: is the transfer of energy by collisions between particles (generally atoms or molecules). Also known as diffusive transport of energy. (2) Convection: is the motion of a fluid caused by density gradients which are a result of temperature differences. (3) Radiation: is the transport of energy by photons.

43. Diffusion of Mass and Heat Diffusion can be driven by concentration gradients, temperature gradients, and pressure gradients. When diffusion is produced by temperature gradients this is known as thermal conduction and leads to the transfer of heat.

44. Thermal Conductionis the transfer of heat by collisions between particles Q = heat flux (erg cm-2 s-1) dT/dz = temperature gradient in z direction κT= thermal conductivity is a measure of a material’s physical ability to conduct heat. Fick’s First Law of Diffusion The rate of change of energy per unit volume is given by: Fick’s Second Law of Diffusion

45. Thermal Conductionis the transfer of heat by collisions between particles U = internal energy = ρCPT, where ρ = mass density CP= heat capacity at constant pressure (it can also occur at constant v) T = temperature Where κD = thermal diffusivity = κT/ρCP Or

46. Convection Convectionis the motion of a fluid caused by density gradients which result from temperature differences. Examples: Boiling water Cloud formation Plate tectonics

47. Convection in the Atmosphere Convectionis the motion of a fluid caused by density gradients which result from temperature differences.

48. The Atmospheric General Circulationis a Manifestation of Convective Currents

49. Quantifying Convective Energy Transport The Convective Energy Flux can be quantified analogous to the thermal conduction flux, if the thermal diffusion coefficient is replaced by an Eddy Diffusion Coefficient, Ke. First Law of Diffusion The rate of change of energy per unit volume is given by: Second Law of Diffusion Ke= Eddy Diffusion Coefficient The key problem in convection and mixing is the choice of Ke. It is usually determined by observations of tracer motions in the atmosphere.

50. Energy Transport by Radiation c = 2.998 x 108 ms-1 speed of electromagnetic radiation λ = wavelength (wavenumber = k = 1/λ) ν = frequency, such that: Energy: h = Planck’s Constant