METR and 13 February 2004

Introduction What is thermodynamics? Study of energy exchange between a system and its surroundings In meteorology, how these changes relate to the state of the atmosphere Important for understanding the temperature distribution in the atmosphere, cloud processes and severe weather, and the general circulation

Introduction Thermodynamics considers a “system” and its “surroundings”. In meteorology System = parcel of air Surroundings = rest of the atmosphere, environment Two types of exchange between system and surroundings Energy exchange (heat, friction, work, radiation, etc) Matter exchange (movement of molecules)

Introduction Types of systems: Isolated system – no exchange of matter or energy with the surroundings Closed system – exchange of energy, but no exchange of matter with the surroundings Open system – exchange of both energy and matter

Introduction “Laws” of thermodynamics These are actually empirical relationships (based on observations) that over time and repeated experiments by scientists have been shown to hold. Zeroth law - equilibrium First law - conservation of energy Second law – entropy can never decrease

State variables Variables of state: pressure, density and temperature Pressure Pressure = force/area, with units (MKS) of Newton/meter 2 1 Nm -2 = 1 Pa (Pascal) Mean sea level pressure 1001 mb = kPa = 1001 hPa Pressure is isotropic; at any point, it is the same in all directions Dalton’s law: total pressure of a mixture of gases is equal to the sum of the partial pressures exerted by each individual gas

State variables Temperature T = average kinetic energy of molecules in a gas Depends on heat content Heat = total kinetic energy of a gas Density ρ = mass/volume, with units (MKS) of kg/meter 3 Average density of air at sea level ρ 0 = 1.23 kgm -3

State variables Temperature of an air parcel can be affected by: Radiation being absorbed or emitted Energy exchange with the surrounding air Energy exchange with the adjacent ground Evaporation or condensation of water Expansion or compression of the parcel due to vertical motion

Equation of State Want to relate variables of state to each other. Experiments have found that gases follow approximately the same equation of state over a wide range of conditions. Although the atmosphere is a mixture of gases, it behaves as though it is a single “ideal” gas: made up of a large number of molecules in random motion separation of molecules is large compared with their size molecules experience perfect elastic collisions (momentum and energy are conserved) forces between molecules are negligible (except during collisions)

Equation of State Empirical relationships Boyle’s law: If a gas is kept at a constant temperature, then pressure is inversely proportional to volume Charles’ law: When pressure is kept constant, volume is directly proportional to temperature

Equation of State Ideal gas law Equation of state for dry air p = ρ R d T where R d is the gas constant for dry air = 287 Pa K -1 m 3 kg -1 Gas constant depends on mean molecular weight of air. Now, we shall derive the equation of state and gas constant from kinetic theory of gases.

Kinetic theory of gases We shall derive the ideal gas law from first principles. You won’t need to reproduce this in a quiz or homework Consider a Cartesian coordinate system (x,y,z), with z upward, and a wall in the y-z plane with air on one side At any time, many air molecules are colliding with the wall. x z y wall air Let’s consider one molecule as it bounces off the wall. It must have experienced a force, since it changed its motion. From Newton’s Second Law, the molecule must have exerted an equal and opposite force on the wall.

Kinetic theory of gases The momentum of the molecule M x is altered by the force acting on it, so that Assuming that the collision is elastic, then the x-component of the molecule’s motion must be exactly reversed. The x-component of its momentum must be equal and opposite in sign to that prior to the collision. So M xf = -M xi = mu and the change in momentum is ΔM x = mu – (-mu) = 2mu, with m the mass of the molecule. This gives(1)

Kinetic theory of gases Now we need to extend this to an arbitrary number of molecules. Assuming random isotropic motion, at any time half the molecules are moving towards the wall. The flux of molecules striking the wall is where is the number of molecules per unit volume of air (N the total number of molecules and V the volume of the air). Combining this with (1), the total force over time t is where ½nuAt is the total no. of molecules striking the wall in time t, and 2mu is the change in momentum of a single molecule striking the wall.

Kinetic theory of gases Now divide this result by t to get the average instantaneous force on A gives where is the average square of the x-component of the velocity over all the molecules. Now, we define pressure as force per unit area, giving (2)

Kinetic theory of gases Since the molecular motion is random and isotropic, = = and Hence Putting this into the form of the ideal gas law pV = NkT gives a definition of absolute temperature T as with k = Boltzmann constant = 1.38× J K -1

Kinetic theory of gases Now, defining the gas constant and density gives the ideal gas law For dry air, R d = 287 J kg -1 K -1 = 287 Pa K -1 m 3 kg -1

Kinetic theory of gases From the definition of absolute temperature, T increases as increases. Does this mean that all molecular motion ceases at absolute zero? In practice, as T → 0, the assumptions in the kinetic theory of gases are no longer valid; collisions are not elastic, molecules are not widely spaced and intermolecular forces cannot be neglected. Another interpretation of T = 0 is, using pV = NkT, that it would be the temperature that the volume of the gas would go to zero if pressure remained constant.

Virtual temperature Water vapor has a lower molecular weight than dry air M H2O ~18, M O2 ~ 32, M N2 ~ 28 so water vapor in air makes the mean molecular weight of moist air lower than that for dry air. This means the gas constant for moist air is greater than for dry air and depends on the amount of water vapor. Rather than define a variable gas constant, we define a new temperature, the virtual temperature T v = T ( r) where r is the water vapor mixing ratio. Hence, moist air with temperature T behaves like dry air with temperature T v with gas law p = ρ R T v