## Raoults law revisited

Now consider an ideal solution with its saturated vapour. The equilibrium condition for each component in the solution and the vapour is given by Eq. 8.9. For the solution, the chemical potential for each component is gi,l g0i,l(p,T) + RiT ln (ni n), (8.26) where we added the subscript l to make explicit that we refer to the solution. The vapour is also considered ideal so we can use Eq. 8.17. However, in the present case it is convenient to write the chemical potentials for the ideal vapour,...

## The Clausius Clapeyron equation

Consider a closed cylinder with a liquid and its vapour, but nothing else, coexisting.29 All the time some of the liquid molecules will evaporate to vapour, while some of the vapour molecules will condense to liquid. This situation will equilibrate when excess molecules are leaving the fluid to go into the vapour, the pressure of the vapour will increase and consequently the number of molecules moving back into the liquid will increase, reducing the excess pressure until equilibrium is...

## Radiativeconvective equilibrium

It is of interest to consider the radiative transfer problem in an atmosphere which is uniform in the horizontal. We are then only interested in the vertical flux of radiation in either the upward or downward direction. We will consider a wide-band approximation, with the long-wave and short-wave fluxes in separate bands. As a vertical coordinate we will use the effective long-wave optical depth S between the surface and some level in the atmosphere. It is related to the geometrical height z by...

## Cp 1 qvs Cpd qvsCpv qi Cpi cpd 1 dv 1 qvs

This improved expression for the heat capacity can be used to find a more accurate equation for the adiabatic lapse rate. For modest water content, the heat capacity is accurately represented by Eq. 6.6. The moist-adiabatic lapse rate depends on the total water content of the parcel, except in its approximate form, Eq. 6.7. Therefore it cannot be expressed as a function of pressure and temperature alone. To overcome this problem we often consider pseudo-adiabatic processes where any liquid...

## Vertical structure in thermodynamic diagrams

Here we will take a graphical approach to the analysis of vertical structure, to complement the formal approach in the previous sections. We will use a tephigram, to illustrate this. A tephigram is a Ts diagram which has been adjusted for meteorological use. Appendix C discusses the precise structure of a tephigram as well as the relationship between tephigrams and skewT-logp diagrams, the other commonly used thermodynamic diagram. The analysis below works essentially the same for either of...

## Dry static energy and Bernoulli function

For a parcel moving hydrostatically in a geopotential field, the specific enthalpy varies according to dh T ds + v dp T ds - dfa. (4.48) We can move the differential dfa to the left-hand side to form the differential of the combination This expression is called the generalized enthalpy or the static Bernoulli function. The first order variation in generalized enthalpy is For an ideal gas with isobaric heat capacity cp we have dh cp dT, so we can integrate the expression for dh + dfa to h* CpT +...

## Buoyancy

Vertical stability (or parcel stability) of the atmosphere depends on the weight of a parcel relative to its environment. If it is warmer and lighter, it will experience an upward buoyancy force in accordance with Archimedes' law if it is cooler and heavier, it will experience a downward buoyancy force. According to Archimedes' law the upward buoyancy force F on a fluid parcel is with pp and pe the densities of the parcel and environment, respectively, and Vp the volume of the parcel. The...

## Potential temperature

In atmospheric science it is common usage to map the specific entropy of a parcel onto a temperature scale called the potential temperature. It is defined as follows The potential temperature 0 is the temperature a fluid parcel would have if brought at constant entropy to a reference pressure p0. From this definition it follows that the potential temperature is a function of the specific entropy of the fluid parcel (p0 is a fixed parameter). Like the specific entropy, the potential temperature...

## Global energy budget and the greenhouse effect

The Sun's radius is about Rs 6.96 x 108m and its surface temperature is about TS 5780K. The Sun can be approximately treated as a black body. From Stefan's law we can then find the total radiative energy output of the Sun. When this radiation has reached the Earth's orbit (average orbital radius rE 149.5 x 109m) this energy is used to irradiate a much larger area, instead of 4nR2. So the radiative energy flux per unit area S0 at the distance of the Earth can be found from The flux S0 is called...

## General applications

Now that we have reviewed the first and second laws of thermodynamics we can start to look at some of their basic consequences. This will provide us with relationships that will be used throughout the rest of the book and that are also applicable outside the field of atmospheric physics. In this chapter we only consider simple substances. Simple substances have only one constituent, or are accurately described as if made up of one constituent, as opposed to compound substances where variations...

## Adiabatic lapse rate

The hydrostatic equation can be used to relate the specific entropy profile to the temperature profile. Considering specific entropy a function of p and T we find Figure 4.3 Top panel Mean wintertime geopotential thickness of the 1000-500 hPa layer in decametres. Bottom panel Same as top panel but with the longitudinal mean removed to emphasize the deviations from the dominant pole-to-equator thickness gradient. Thin contours represent negative values. Contour values are , 75, -25, 25, 75, m....

## Thermodynamics of a photon gas

Thermodynamics can be applied to a gas of photons, particles that mediate the electromagnetic field. In this section we will derive the Stefan-Boltzmann law and the Wien displacement law from thermodynamic arguments only.54 In Chapter 1 we used a kinematic argument to derive how the pressure in a gas can be written as an average of the momentum transfer in the direction of a wall over all particles that collide with the wall, see Eq. 1.11. For a photon gas this equation needs to be...

## Entropy budget for saturated air

Following similar arguments leading to Eq. 5.32, the specific entropy s for a parcel consisting of dry air, vapour (concentration by mass qv), and liquid (concentration by mass qi) equals s (1 - qw) Sd + qvSv + qisi, (6.10) with, as before, qw qv + qi the total water concentration. At equilibrium we have with L the latent heat of evaporation. Equilibrium means that the vapour is in a Clausius-Clapeyron equilibrium with the liquid in the parcel. In other words, the vapour is always saturated, qv...

## Homogeneous nucleation the Kelvin effect

Naively, we would expect vapour to start to condense and thus form cloud droplets if the relative humidity is above 100 . However, we have to be careful the saturation vapour pressure as calculated from the Clausius-Clapeyron equation is only valid for flat water surfaces. Cloud droplets have strongly curved surfaces and as a consequence the surface tension needs to be incorporated in the free energy budgets leading to the Clausius-Clapeyron equation. The surface tension effect is named after...

## Ideal gas mixtures and ideal solutions

For ideal gas mixtures we can write down explicit expressions for the chemical potentials. In order to calculate the chemical potentials in an ideal gas mixture we will first calculate its total Gibbs function. We will take it that for an ideal gas mixture the individual components contribute independently to the extensive variables as each component, by definition of the ideal gas, is not influenced by the presence of the other components. For an ideal gas mixture the total internal energy U...

## Calculation of saturated vapour pressure

The Clausius-Clapeyron equation needs to be integrated to find the vapour pressure at a certain temperature. To do this, assume that the vapour is an ideal gas, so we have esvv RvT, with Rv the specific gas constant for the vapour (in the case of water vapour Rv 461.5Jkg-1 K-1). Further assume that vv > vi, which is true away from the critical temperature. The Clausius-Clapeyron equation then becomes Further assuming that L is constant (this is not a very good assumption, see below) this...

## Adiabatic lapse rate for moist air

The adiabatic lapse rate for a saturated parcel is expected to be different from that of an unsaturated parcel. If we lift a saturated parcel, it will expand and cool down, leading to vapour condensation. This condensation will release latent heat, which will partially offset the cooling. The adiabatic lapse rate for a saturated parcel is therefore lower than that for an unsaturated parcel. First consider moist but unsaturated air. There is no liquid water present in the parcel and the entropy...

## Derivation of the Planck law

One cannot help but be impressed by the thermodynamic arguments leading to Eq. 9.104, or the equivalent Eq. 9.105. These equations contain the Stefan-Boltzmann law and the Wien displacement law. However, thermodynamics alone cannot take us any further. The derivation of the Planck law itself requires quantum mechanical concepts. We present its derivation here as it is one of the classic results in physics. This by now canonical derivation is based on statistical mechanics, which employs the...

## Thermodynamic diagrams

The thermodynamic state of a simple substance is determined by two variables. That means that a two-dimensional diagram with a particular thermodynamic variable on each of the two coordinate axes will be sufficient to describe any thermodynamic state of the substance. The pV diagrams in Chapter 1 and 2 are examples. As we saw in Chapter 2, the area on a pV diagram corresponds to (work) energy. The area inside a closed cycle C on a pV diagram is which also equals the total work performed by an...

## The Stefan Boltzmann and Wien displacement laws

The Stefan-Boltzmann law expresses how much total radiative energy flux a black body emits the Wien displacement law expresses at what wavelength the black body has its maximum emission. These laws can be derived from thermodynamic arguments as well as from the Planck law. The former method is covered in section 9.8. Here we discuss how both laws follow from the Planck law. The Planck law can be integrated over all wavelengths to get the total emission B of a black body, r , r 2nhc2 1 B Bi dl...

## Ideal gases

In this chapter we introduce the concept of an ideal gas, a gas of non-interacting molecules. An ideal gas is an accurate model of dilute gases such as the atmosphere. We further introduce the notion of macroscopic variables, amongst them such familiar ones as temperature or pressure. These macroscopic variables must be related to some property of the microscopic state of the molecules that make up the substance. For example, for the systems we consider here, temperature is related to the mean...

## Khler theory

Combining the Kelvin and Raoult effects, we get an expression for the ratio of the saturated vapour pressure over a flat surface of pure water, as derived in Section 5.1, to the saturated vapour pressure of a solution in a spherical droplet of radius r at temperature T. This saturation ratio is b Zvi iiJ 4.3 x 10-6 (tMs s) m3. (7.29b) A plot of S as a function of radius is called a K hler curve. Figure 7.3 shows an example of a K hler curve highlighting the Kelvin and Raoult effects. The K hler...

## Moist static energy

In Section 4.5 we saw how the generalized enthalpy can be used to define the energy budget of a parcel moving under hydrostatic balance. Here we will examine how to define the energy budget for a moist air parcel. Equation 4.50 is generally valid for an air parcel that only exchanges heat and work with its environment. In particular we can think of the air parcel as consisting of dry air, water vapour, and liquid water (and possibly ice we shall ignore this here for simplicity). The total mass...

## Humidity variables

In atmospheric physics several different variables are used to describe the amount of water vapour in the air. These tend to be used in different contexts and here we put the most commonly encountered variables together. The first variable is the concentration by mass of water vapour, usually called the specific humidity q, with pv and pd the local densities of the vapour and the dry air respectively. It was introduced in section 1.3 to define the virtual temperature. The specific humidity is...