The Ideal Gas Law Lecture 2: Atmospheric Thermodynamics An equation of state describes the relationship among pressure, temperature, and density of any material. Ideal Gas Law (Equation of State) All gases are found to follow approximately the same equation Hydrostatic Balance of state, which is referred to as the “ideal gas law (equation)”. Atmospheric gases, whether considered individually or as a Heat and Temperature mixture, obey the following ideal gas equation: Conduction, Convection, Radiation Latent Heating P = ρ R T Adiabatic Process Lapse Rate and Stability pressure Density=m/V temperature (degree Kelvin) gas constant (its value depends on the gas considered) ESS55 ESS55 Prof. Jin-Yi Yu Prof. Jin-Yi Yu Gas Constant Applications of the Gas law The ideal gas law can be applied to the combination of atmospheric gases or to individual gases. Question: Calculate the density of water vapor which exerts a pressure of 9 mb at 20°C. The value of gas constant for the particular gas under consideration depends on its molecular weight: Answer: R = R* / M gas gas ρ where R* = universal gas constant = 8314.3 J deg-1 kg-1 Use the ideal gas law: Pv= RvT and Pv = 9 mb = 900 Pa (a SI unit) The gas constant for dry atmospheric air is: -1 -1 -1 -1 Rv = R* / Mv = 461 J deg kg Rair = R* / Mair = 8314.3/28.97 = 287 J deg kg ≅ (Mair 0.80*MN2 + 0.20*MO2 = 0.80*28 + 0.2*32 = 28.8) T = 273 + 20 (°C) = 293 K. So we know the density of water vapor is: The gas constant for water vapor is: ρ = P / (R T) = 900 / (461*293) = 6.67 x 10-3 kg m-3 Rvapor = R* / Mvapor = 8314.3/18.016 v v = 461 J deg-1 kg-1 (from Atmospheric Sciences: An introductory Survey) ESS55 ESS55 Prof. Jin-Yi Yu Prof. Jin-Yi Yu 1 Virtual Temperature How to Calculate Virtual Temperature? Moist air has a lower apparent molecular weight that dry air. Î The gas constant for 1 kg of moist air is larger than that for 1 kg of dry air. Î But the exact value of the gas constant of moist air would depend on the amount of water vapor contained in the air. Î It is inconvenient to calculate the gas constant for moist air. It is more convenient to retain the gas constant of dry air and use a fictitious temperature in the ideal gas equation. Where T: actual temperature Î This fictitious temperature is called “virtual temperature”. p: actual (total) pressure = pd + e Î This is the temperature that dry air must have in order to has the same pd: partial pressure exerted by dry air density as the moist air at the same pressure. e: partial pressure exerted by water vapor ε Î Since moist air is less dense that dry air, the virtual temperature is = Rd/Rv = 0.622 always greater than the actual temperature. ESS55 ESS55 Prof. Jin-Yi Yu Prof. Jin-Yi Yu Hydrostatic Balance in the Vertical What Does Hydrostatic Balance Tell Us? vertical pressure force = gravitational force The hydrostatic equation tells us how - (dP) x (dA) = ρ x (dz) x (dA) x g quickly air pressure drops wit height. dP = -ρgdz ÎThe rate at which air pressure decreases with dP/dz = -ρg height (∆P/ ∆z) is equal to the air density (ρ) times the acceleration of gravity (g) The hydrostatic balance !! (from Climate System Modeling) ESS55 ESS55 Prof. Jin-Yi Yu Prof. Jin-Yi Yu 2 Hydrostatic Balance and Atmospheric Vertical Structure The Scale Height of the Atmosphere One way to measure how soon the air runs out in the Since P= ρRT (the ideal gas law), the hydrostatic equation atmosphere is to calculate the scale height, which is becomes: about 10 km. dP = -P/RT x gdz Over this vertical distance, air pressure and density Î dP/P = -g/RT x dz decrease by 37% of its surface values. Î P = P exp(-gz/RT) s If pressure at the surface is 1 atmosphere, then it is Î P = P exp(-z/H) s 0.37 atmospheres at a height of 10 km, 0.14 (0.37x0.37) at 20 km, 0.05 (0.37x0.37x0.37) at 30 km, The atmospheric pressure and so on. decreases exponentially with height Different atmospheric gases have different values of scale height. (from Meteorology Today) ESS55 ESS55 Prof. Jin-Yi Yu Prof. Jin-Yi Yu A Mathematic Formula of Scale Height Temperature and Pressure gas constant temperature * Hydrostatic balance tells us gravity that the pressure decrease with height is determined by the temperature inside the vertical scale height column. molecular weight of gas Pressure decreases faster in the The heavier the gas molecules weight (m) Î the smaller the scale cold-air column and slower in height for that particular gas the warm-air column. The higher the temperature (T) Î the more energetic the air molecules Î the larger the scale height Pressure drops more rapidly The larger the gravity (g) Î air molecules are closer to the surface Î with height at high latitudes and the smaller the scale height lowers the height of the pressure H has a value of about 10km for the mixture of gases in the atmosphere, surface. but H has different values for individual gases. (from Understanding Weather & Climate) ESS55 ESS55 Prof. Jin-Yi Yu Prof. Jin-Yi Yu 3 Warm Core Hurricane Energy (Heat) Pressure Surface tropopause The first law of thermodynamics Z Air Temperature h y d t r h o er s m t a al t w i c in b d surface a b hurricane center a l a la n n The core of a hurricane is warmer than its surroundings. c c e e The intensity of the hurricane (as measured by the depression of pressure surface) must decrease with height. Thus, a warm core hurricane exhibits its greatest intensity near the Air Pressure geostrophic balance Air Motion ground and diminish with increasing height above ground. ESS55 ESS55 (from Understanding Weather & Climate and Atmospheric Sciences: An Intro. Survey) Prof. Jin-Yi Yu Prof. Jin-Yi Yu Heat and Energy internal kinetic energy What Is Air Temperature? (related to temperature) Energy is the capacity to do internal potential energy work. (related to the phase) Air temperature is a measurement of the average Heat is one form of energy. internal kinetic energy of air molecules. Heat is one form of internal energy which is associated with the random, disordered Increase in internal kinetic energy in the form of motion of molecules and molecular motions are manifested as increases in atoms. the temperature of the body. Internal kinetic/potential water energy are different from the macroscopic kinetic/potential energy. no macroscopic kinetic/potential energy ESS55 ESS55 Prof. Jin-Yi Yu Prof. Jin-Yi Yu 4 (from Atmospheric Sciences: An Intro. Survey) The First Law of Thermodynamics This law states that (1) heat is a form of energy that (2) its conversion into other forms of energy Therefore, when heat is is such that total energy is conserved. added to a gas, there will be some combination of an The change in the internal energy of a system is expansion of the gas (i.e. the work) and an increase in its equal to the heat added to the system minus the temperature (i.e. the increase work down by the system: in internal energy): ∆ U = Q - W Heat added to the gas = work done by the gas + temp. increase of the gas ∆H = p ∆α + C ∆T change in internal energy v Heat added to the system Work done by the system (related to temperature) ESS55 ESS55 Prof. Jin-Yi Yu volume change of the gas specific heat at constant volume Prof. Jin-Yi Yu Heat and Temperature Specific Heat Heat and temperature are both related to the internal kinetic energy of air molecules, and therefore can be related to each other in the following way: Q = c*m*∆T Heat added Mass Temperature changed Specific heat = the amount of heat per unit mass required to raise the temperature by one degree Celsius (from Meteorology: Understanding the Atmosphere) ESS55 ESS55 Prof. Jin-Yi Yu Prof. Jin-Yi Yu 5 How to Change Air Temperature? Conduction Add (remove) heat to (from) the air parcel (diabatic processes) Conduction is the process of heat transfer from molecule to (1) Conduction: requires touching molecule. (2) Convection: Hot air rises This energy transfer process (3) Advection: horizontal movement of air requires contact. (4) Radiation: exchanging heat with space (5) Latent heating: changing the phase of water Air is a poor conductor. (with low thermal conductivity) Without adding (removing) heat to (from) the air parcel Conduction is not an efficient (from Meteorology: Understanding the Atmosphere) mechanisms to transfer heat in (1) Adiabatic Process: Expanding and compressing air the atmosphere on large spatial scales. ESS55 ESS55 Prof. Jin-Yi Yu Prof. Jin-Yi Yu Convection Advection Convection is heat transfer by mass Advection is referred to the motion of a fluid (such as air or water). horizontal transport of heat in the atmosphere. Convection is produced when the heated fluid moves away from the heat Warm air advection occurs when warm air replaces cold air. source and carries energy with it. Cold air advection is the other way around. Convection is an efficient mechanism This process is similar to the of heat transfer for the atmosphere in (from Meteorology: Understanding the Atmosphere) some regions (such as the tropics) but convection which relies on the mass motion to carry heat from is an inefficient mechanism in other one region to the other.