# The Ideal Gas Law. Gas Constant. Applications of the Gas law. P = ρ R T. Lecture 2: Atmospheric Thermodynamics

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1 Lecture 2: Atmospheric Thermodynamics Ideal Gas Law (Equation of State) Hydrostatic Balance Heat and Temperature Conduction, Convection, Radiation Latent Heating Adiabatic Process Lapse Rate and Stability The Ideal Gas Law An equation of state describes the relationship among pressure, temperature, and density of any material. All gases are found to follow approximately the same equation of state, which is referred to as the ideal gas law (equation). Atmospheric gases, whether considered individually or as a mixture, obey the following ideal gas equation: P = ρ R T pressure Density=m/V temperature (degree Kelvin) gas constant (its value depends on the gas considered) Gas Constant The ideal gas law can be applied to the combination of atmospheric gases or to individual gases. The value of gas constant for the particular gas under consideration depends on its molecular weight: R gas = R* / M gas where R* = universal gas constant = J deg -1 kg -1 The gas constant for dry atmospheric air is: R air = R* / M air = /28.97 = 287 J deg -1 kg -1 (M air 0.80*M N *M O2 = 0.80* *32 = 28.8) The gas constant for water vapor is: R vapor = R* / M vapor = / ρ = P v / (R v T) = 900 / (461*293) = 6.67 x 10-3 kg m -3 = 461 J deg -1 kg -1 (from Atmospheric Sciences: An introductory Survey) Applications of the Gas law Question: Calculate the density of water vapor which exerts a pressure of 9 mb at 20 C. Answer: Use the ideal gas law: P v = ρr v T and P v = 9 mb = 900 Pa (a SI unit) R v = R* / M v = 461 J deg -1 kg -1 T = ( C) = 293 K. So we know the density of water vapor is: 1

2 Virtual Temperature Moist air has a lower apparent molecular weight that dry air. The gas constant for 1 kg of moist air is larger than that for 1 kg of dry air. But the exact value of the gas constant of moist air would depend on the amount of water vapor contained in the air. It is inconvenient to calculate the gas constant for moist air. How to Calculate Virtual Temperature? It is more convenient to retain the gas constant of dry air and use a fictitious temperature in the ideal gas equation. This fictitious temperature is called virtual temperature. This is the temperature that dry air must have in order to has the same density as the moist air at the same pressure. Since moist air is less dense that dry air, the virtual temperature is always greater than the actual temperature. Where T: actual temperature p: actual (total) pressure = p d + e p d : partial pressure exerted by dry air e: partial pressure exerted by water vapor ε = R d /R v = Hydrostatic Balance in the Vertical vertical pressure force = gravitational force - (dp) x (da) = ρ x (dz) x (da) x g dp = -ρgdz dp/dz = -ρg The hydrostatic balance!! What Does Hydrostatic Balance Tell Us? The hydrostatic equation tells us how quickly air pressure drops wit height. The rate at which air pressure decreases with height ( P/ z) is equal to the air density (ρ) times the acceleration of gravity (g) (from Climate System Modeling) 2

3 Hydrostatic Balance and Atmospheric Vertical Structure The Scale Height of the Atmosphere Since P= ρrt (the ideal gas law), the hydrostatic equation becomes: dp = -P/RT x gdz dp/p = -g/rt x dz P = P s exp(-gz/rt) P = P s exp(-z/h) The atmospheric pressure decreases exponentially with height One way to measure how soon the air runs out in the atmosphere is to calculate the scale height, which is about 10 km. Over this vertical distance, air pressure and density decrease by 37% of its surface values. If pressure at the surface is 1 atmosphere, then it is 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, and so on. Different atmospheric gases have different values of scale height. A Mathematic Formula of Scale Height gas constant scale height * temperature gravity molecular weight of gas The heavier the gas molecules weight (m) the smaller the scale height for that particular gas The higher the temperature (T) the more energetic the air molecules the larger the scale height The larger the gravity (g) air molecules are closer to the surface the smaller the scale height H has a value of about 10km for the mixture of gases in the atmosphere, but H has different values for individual gases. Temperature and Pressure (from Understanding Weather & Climate) Hydrostatic balance tells us that the pressure decrease with height is determined by the temperature inside the vertical column. Pressure decreases faster in the cold-air column and slower in the warm-air column. Pressure drops more rapidly with height at high latitudes and lowers the height of the pressure surface. 3

4 hydrostatic balance Warm Core Hurricane Pressure Surface tropopause Energy (Heat) The first law of thermodynamics Z Air Temperature hurricane center The core of a hurricane is warmer than its surroundings. surface 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 ground and diminish with increasing height above ground. (from Understanding Weather & Climate and Atmospheric Sciences: An Intro. Survey) Air Pressure geostrophic balance thermal wind balance Air Motion Heat and Energy internal kinetic energy (related to temperature) internal potential energy (related to the phase) water no macroscopic kinetic/potential energy Energy is the capacity to do work. Heat is one form of energy. Heat is one form of internal energy which is associated with the random, disordered motion of molecules and atoms. Internal kinetic/potential energy are different from the macroscopic kinetic/potential energy. What Is Air Temperature? Air temperature is a measurement of the average internal kinetic energy of air molecules. Increase in internal kinetic energy in the form of molecular motions are manifested as increases in the temperature of the body. 4

5 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 is such that total energy is conserved. The change in the internal energy of a system is equal to the heat added to the system minus the work down by the system: change in internal energy (related to temperature) U = Q - W Heat added to the system Work done by the system Therefore, when heat is added to a gas, there will be some combination of an expansion of the gas (i.e. the work) and an increase in its temperature (i.e. the increase in internal energy): Heat added to the gas = work done by the gas + temp. increase of the gas H = p α + C v T volume change of the gas (from Atmospheric Sciences: An Intro. Survey) specific heat at constant volume 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) 5

6 How to Change Air Temperature? Conduction Add (remove) heat to (from) the air parcel (diabatic processes) (1) Conduction: requires touching (2) Convection: Hot air rises (3) Advection: horizontal movement of air (4) Radiation: exchanging heat with space (5) Latent heating: changing the phase of water Conduction is the process of heat transfer from molecule to molecule. This energy transfer process requires contact. Air is a poor conductor. (with low thermal conductivity) Without adding (removing) heat to (from) the air parcel (1) Adiabatic Process: Expanding and compressing air (from Meteorology: Understanding the Atmosphere) Conduction is not an efficient mechanisms to transfer heat in the atmosphere on large spatial scales. Convection Advection (from Meteorology: Understanding the Atmosphere) Convection is heat transfer by mass motion of a fluid (such as air or water). Convection is produced when the heated fluid moves away from the heat source and carries energy with it. Convection is an efficient mechanism of heat transfer for the atmosphere in some regions (such as the tropics) but is an inefficient mechanism in other regions (such as the polar regions). (from Meteorology: Understanding the Atmosphere) Advection is referred to the horizontal transport of heat in the atmosphere. Warm air advection occurs when warm air replaces cold air. Cold air advection is the other way around. This process is similar to the convection which relies on the mass motion to carry heat from one region to the other. Advection can be considered as one form of convection. 6

7 Radiation Latent Heating Radiation is heat transfer by the emission of electromagnetic waves which carry energy away from the emitting object. 680 cal/gm 80 cal/gm 600 cal/gm The solar energy moves through empty space from the Sun to the Earth and is the original energy source for Earth s weather and climate. Latent heat is the heat released or absorbed per unit mass when water changes phase. Latent heating is an efficient way of transferring energy globally and is an important energy source for Earth s weather and climate. (from Meteorology: Understanding the Atmosphere) Latent Heat of Evaporation The latent heat of evaporation is a function of water temperature, ranging from 540 cal per gram of water at 100 C to 600 cal per gram at 0 C. It takes more energy to evaporate cold water than evaporate the same amount of warmer water. Adiabatic Process If a material changes its state (pressure, volume, or temperature) without any heat being added to it or withdrawn from it, the change is said to be adiabatic. The adiabatic process often occurs when air rises or descends and is an important process in the atmosphere. 7

8 Air Parcel Expands As It Rises Air pressure decreases with elevation. If a helium balloon 1 m in diameter is released at sea level, it expands as it floats upward because of the pressure decrease. The balloon would be 6.7 m in diameter as a height of 40 km. What Happens to the Temperature? Air molecules in the parcel (or the balloon) have to use their kinetic energy to expand the parcel/balloon. Therefore, the molecules lost energy and slow down their motions The temperature of the air parcel (or balloon) decreases with elevation. The lost energy is used to increase the potential energy of air molecular. Similarly when the air parcel descends, the potential energy of air molecular is converted back to kinetic energy. Air temperature rises. (from The Blue Planet) Dry Adiabatic Lapse Rate Moist Adiabatic Lapse Rate (from Meteorology: Understanding the Atmosphere) (from Meteorology: Understanding the Atmosphere) 8

9 Concept of Stability Static Stability Static stability is referred as to air s susceptibility to uplift. The static stability of the atmosphere is related to the vertical structure of atmospheric temperature. To determine the static stability, we need to compare the lapse rate of the atmosphere (environmental lapse rate) and the dry (moist) adiabatic lapse rate of an dry (moist) air parcel. Environmental Lapse Rate The environmental lapse rate is referred to as the rate at which the air temperature surrounding us would be changed if we were to climb upward into the atmosphere. Static Stability of the Atmosphere Γe = environmental lapse rate Γd = day adiabatic lapse rate Γm = moist lapse rate Absolutely Stable Γe < Γm This rate varies from time to time and from place to place. Absolutely Unstable Γe > Γd Conditionally Unstable Γm < Γe < Γd 9

10 Absolutely Stable Atmosphere Absolutely Unstable Atmosphere Conditionally Unstable Atmosphere Day/Night Changes of Air Temperature End of Day Night (from Is the Temperature Rising?) At the end of a sunny day, warm air near the surface, cold air aloft. In the early morning, cold air near the surface, warm air aloft. The later condition is called inversion, which inhibits convection and can cause sever pollution in the morning. 10

11 Stability and Air Pollution Neutral Atmosphere (Coning) Stable Atmosphere (Fanning) Unstable Atmosphere (Looping) Stable Aloft; Unstable Below (Fumigation) Unstable Aloft; Stable Below (Lofting) (from Is the Temperature Rising?) Potential Temperature (θ) The potential temperature of an air parcel is defined as the the temperature the parcel would have if it were moved adiabatically from its existing pressure and temperature to a standard pressure P 0 (generally taken as 1000mb). θ= potential temperature T = original temperature P = original pressure P 0 = standard pressure = 1000 mb R = gas constant = R d = 287 J deg -1 kg -1 C p = specific heat = 1004 J deg -1 kg -1 R/C p = Importance of Potential Temperature Adiabatic Chart In the atmosphere, air parcel often moves around adiabatically. Therefore, its potential temperature remains constant throughout the whole process. Potential temperature is a conservative quantity for adiabatic process in the atmosphere. Potential temperature is an extremely useful parameter in atmospheric thermodynamics. (from Atmospheric Sciences: An Intro. Survey) (from The Physics of the Atmospheres) The expression of potential temperature can be modified into: T = (constant * θ) P

12 Water Vapor In the Air Saturation (from Understanding Weather & Climate) Evaporation: the process whereby molecules break free of the liquid volume. Condensation: water vapor molecules randomly collide with the water surface and bond with adjacent molecules. How Much Water Vapor Is Evaporated Into the Atmosphere Each Year? On average, 1 meter of water is evaporated from oceans to the atmosphere each year. The global averaged precipitation is also about 1 meter per year. How Much Heat Is Brought Upward By Water Vapor? Earth s surface lost heat to the atmosphere when water is evaporated from oceans to the atmosphere. The evaporation of the 1m of water causes Earth s surface to lost 83 watts per square meter, almost half of the sunlight that reaches the surface. Without the evaporation process, the global surface temperature would be 67 C instead of the actual 15 C. 12

13 by mass Measuring Air Moisture Observed Specific Humidity in unit of g/kg in unit of g/m 3 by vapor pressure in unit of % Specific.vs. Relative Humidity specific humidity 6 gm/kg saturated specific humidity 10 gm/kg saturated specific humidity 20 gm/kg Relative humidity 6/10 x 100%=60 % Relative humidity 6/20 x 100%=30 % Specific Humidity: How many grams of water vapor in one kilogram of air (in unit of gm/kg). Relative Humidity: The percentage of current moisture content to the saturated moisture amount (in unit of %). Clouds form when the relative humidity reaches 100%. Vapor Pressure The air s content of moisture can be measured by the pressure exerted by the water vapor in the air. The total pressure inside an air parcel is equal to the sum of pressures of the individual gases. In the left figure, the total pressure of the air parcel is equal to sum of vapor pressure plus the pressures exerted by Nitrogen and Oxygen. High vapor pressure indicates large numbers of water vapor molecules. Unit of vapor pressure is usually in mb. 13

14 Saturation Vapor Pressure How to Saturate the Air? Saturation vapor pressure describes how much water vapor is needed to make the air saturated at any given temperature. Saturation vapor pressure depends primarily on the air temperature in the following way: The Clausius-Clapeyron Equation Saturation pressure increases exponentially with air temperature. L: latent heat of evaporation; α: specific volume of vapor and liquid (from IS The Temperature Rising ) Two ways: (1) Increase (inject more) water vapor to the air (A B). (2) Reduce the temperature of the air (A C). Runway Greenhouse Effect Dew Point Temperature If a planet has a very high temperature that the air can never reach a saturation point Water vapor can be added into the atmosphere. More water vapor traps more heat (a greenhouse effect) The planet s temperature increases furthermore Ever more water evaporated into the atmosphere More greenhouse effect More warming More water vapor.. (from The Atmosphere) Dew point temperature is another measurement of air moisture. Dew point temperature is defined as the temperature to which moist air must be cool to become saturated without changing the pressure. The close the dew point temperature is to the air temperature, the closer the air is to saturation. 14

16 An Example Applications of Adiabatic Chart 16

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