Thermodynamics of Moist Air

Transcription

1 Thermodynamics of Moist Air Phase transitions: what are the equilibrium conditions for a mixed-phase system, e.g. a mixture of water and water vapor? Consider a system consisting of liquid water and H 2 O (v) undergoing a change in state, vapor liquid vapor liquid denotes liquid phase denotes vapor phase

2 Define Hence Since both states are stable equilibrium states, G 1 =G 2, and therefore n 1 ʹ ʹ g ʹ ʹ + n 1 ʹ ʹ g ʹ ʹ ʹ ʹ = n ʹ 2 ʹ g ʹ ʹ + n ʹ 2 ʹ ʹ g ʹ ʹ ʹ But since the system is closed ( n 1 ʹ ʹ n ʹ 2 ʹ ) g ʹ ʹ = ( n ʹ 2 ʹ ʹ n 1 ʹ ʹ ʹ ) g ʹ ʹ ʹ Therefore n 1 ʹ ʹ + n 1 ʹ ʹ ʹ = n ʹ 2 ʹ + n ʹ 2 ʹ ʹ ʹ ʹ n 2 = ʹ ʹ n 1 + ʹ ʹ ʹ n 1 ʹ ʹ ʹ n 2

3 At equilibrium, the specific Gibbs free energy is the same for both phases At the triple point, Lets look at g between 2 phases in more detail: c vapor liquid g f b,e d a T From earlier discussions, Where s is the specific entropy of the vapor phase. Hence the slope of vs. T gives the specific entropy. Likewise for the liquid phase Note the slopes are <0, since larger values of g correspond to lower T s. The difference between entropies is Since vapor liquid

4 Clausius-Clapeyron Equation This equation is very important since it describes the variation in pressure with temperature, for a system consisting of two phases in equilibrium. For the case of a liquid/vapor or vapor/solid system, this pressure is the saturation vapor pressure. For a system in equilibrium, vapor ʹ ʹ ʹ liquid ʹ ʹ Suppose the system undergoes a small change to a new pressure and temperature, p+dp, T+dT such that the new state is also a stable state. Each phase changes by amounts. Then for each phase: Since the new state is also an equilibrium state,

5 Or, s, v are specific quantities! CLAUSIUS-CLAPEYRON EQUATION Since =latent heat for vapor-liquid interface For vapor-liquid system For vapor-liquid system For vapor-ice system Geometrical Interpretation P SOLID S-L L-V Triple point S-V LIQUID Critical point gives the slope of the equilibrium line between the 2 phases involved. (on P-V-T diagram.) T VAPOR

6 Looking again at the Clausius-Clapeyron Equation we have three interfaces, vapor liquid solid vapor liquid solid Since always, the slope of the equilibrium line between the 2 phases involved is determined by the difference in specific volumes. 1. vapor liquid vapor solid ] ALWAYS For all materials!

7 2. for a substance that expands upon freezing (like water). Then P S L Triple V T substance contracts upon freezing. (MOST SUBSTANCES!)

8 Integrate the Clausius-Clapeyron Eq n/ to get e s as a function of T. For a vapor/liquid system, Assume GAS LAW Therefore after integration

9 Example Calculate the change in the melting point of ice if the pressure is increased from 1 to 2 atm. The specific volumes of ice and water at 0 C are The latent heat of melting is Increase in pressure of 1 atm decreases the melting point of ice by Ice is unusual in this respect: the specific volume of the liquid form of most materials is greater than the specific volumes of the solid; therefore the melting point increases with increasing pressure. This applies for most substances, but NOT water.

10 Better approximations: recognize that L = L(T) When the individual heat capacities are of the form, Then, (1) Integration constant found from experimental pairs of p,t. For vaporization, first approximation (2) Constant L v

11 Second approximation; heat capacities constant so that is a linear function of T, (3) This is Magnus formula T( C) Eq n (2) Eq n (3) Obs For sublimation, assuming =constant, VAPOR/ICE transition:

12 Clausius-Clapeyron Equation What is p at 100 C? ice Implications for depositional growth of ice crystals? Why?

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16 quick summary so far We found that a criterion for (vapor-liquid) phase equilibrium is the specific Gibbs free energy has the same value in the liquid and vapor phases (or for any two phases in equilibrium). We derived the Clausius-Clapeyron equation starting with small changes in each phase in equilibrium: dp dt = 23 T( v ʹ ʹ ʹ v ʹ ʹ ) 23 Tv ʹ ʹ ʹ = 23 T 2 R v / p Finally after integration we found We made arguments about some of the slopes of lines in phase diagrams and how to compute them. Let s look at water s phase diagram more closely

17 Phase diagram in P-T space close to the critical point, small changes in pressure or temperature result in large changes in density (volume) The heat of vaporization is zero at and beyond the critical point, so there is no distinction between the two phases. Above the critical temperature a liquid cannot be formed by an increase in pressure. We would find an inflection on a p-v diagram: (from wikipedia)

18 Phase diagrams fill P-V-T space Note the axes used in each!

19 PvT Diagram for Water helios.augustana.edu/~lc/ph313/pp/5pure1s.ppt

20 PvT Diagram for Water Increase in pressure of 1 atm decreases the melting point of ice by deg C Ice is unusual in this respect; the specific volume of the liquid form of most materials is greater than the specific volumes of the solid; therefore the melting point increases with increasing pressure. This applies for most substances, but NOT water. helios.augustana.edu/~lc/ph313/pp/5pure1s.ppt

21 Specific volumes Taken at saturated vapor pressure (P increases with T here) Interactive 3D phase diagrams using Jmol. A. Herráez, R.M. Hanson and L. Glasser (2009) J. Chem. Educ. 86: 566 url: Jmol/plots/phase-diagrams/

22 Revisiting the adiabatic parcel We already derived that, for dry air, the rate of change with height of the temperature of an air parcel is given by dt dz = gˆ c p We found that g / c ˆ p for dry air is C / 100m. For moist air, we need to correct the heat capacity (now for a two-component system): c ˆ ʹ p = (1 w v )ˆ c pair ˆ c pwater vapor + w v ˆ c pwater vapor > c ˆ pair so c ˆ ʹ p > c ˆ pair From the last inequality, can see that the rate of decrease of T with z is smaller for a moist parcel than for a dry one (i.e., moist air should be warmer than dry at the same height more buoyant) The parcel cools with height. At some temperature (height), the saturation vapor pressure of water at that temperature equals the partial pressure of water in the parcel condensation occurs heat is released! dt ΔH v is the latent heat of vaporization per g of water w v is the mass of water vapor / mass of dry air in the parcel volume m is the mass of air dq = ΔH v mdw v dz = gˆ + ΔH v c p c ˆ p dw v dz

23 Revisiting the adiabatic parcel dt dz = gˆ + ΔH v c p c ˆ p dw v dz If we start out with enough water vapor to make w v correspond to saturated conditions, enough condenses out at each height to maintain 100% RH locally The equation says that the rate of cooling of a rising parcel inside a cloud is less than that for dry air The saturation pressure of water increases strongly with temperature (it means at warmer temperatures, w v tends to be higher) The wet adiabatic lapse rate is not a constant independent of z In warm tropical air, the wet adiabatic lapse rate is roughly 1/3 of the dry adiabatic lapse rate In cold polar regions, not much difference between the two

24 Saturated water mixing ratio (tan dashed) P k Saturated adiabats (blue dashed) T

25 Equivalent Potential Temperature, θ e θ e is the temperature a parcel of air would reach if all the water vapor in the parcel were to condense, releasing its latent heat, and the parcel was brought adiabatically to a standard reference pressure, usually 1000 hpa (1000 mbar). r = mixing ratio of water vapor mass per mass of dry air (g/kg or in 10-3 Kg/Kg) Saturated air can be unstable even though its potential temperature increases with height: if the warming due to condensation is enough that a parcel of saturated air which is displaced upward (and would otherwise cool to below the ambient temperature) ends up warmer (lighter) than the surrounding air, it will continue to rise. Potential temperature is a temperature adjusted for potential warming due to adiabatic compression; equivalent potential temperature factors in potential warming due to condensation as well. For saturated air, or for air which is likely to be lifted high enough to reach saturation, it is the equivalent potential temperature which must increase with height in order to ensure stability. (will revisit again.) 25