# Thermodynamics

## Vapor pressure

We start with a discussion of the equilibrium gas pressure in a chamber in diathermal contact with a reservoir at a fixed temperature, T0. The chamber is to have a volume that is adjustable, as shown in Figure 5.1. In the following let the chamber have no air present - only the gas from evaporation of the liquid. We are to choose a volume V such that there is some liquid present at the bottom of the chamber (we say here the bottom of the chamber, but otherwise we ignore gravity). There are gas...

## Pz Pol if 628

Note that the formula fails when z > 6 cp g 30 km. The isentropic (6 constant) profile is often observed in the daytime boundary layer and up to the LCL where the air is well mixed vertically. Above this level the lapse rate is smaller because of warming due to condensation of water vapor in the parcel into droplets. When does the hydrostatic approximation not work This can happen in unusual circumstances, but first consider typical conditions. If the force...

## Kz Kz0 fZ Paz Pezdz Jz0

Substituting the hydrostatic equation, dp dz - p g, and the ideal gas equation of state,p pRT, yields K(z) - K(z0) -R fP(Ta - Te)d(lnp). (7.6) The result is that the kinetic energy of a parcel is proportional to the area in the closed loop defined by doubly intersecting environmental and adiabatic curves in a T-lnp diagram It is worth remembering that the above derivation is valid for both dry and moist adiabatic processes. As the parcel rises adiabatically, its kinetic energy goes up, if there...

## Photochemistry

Further examples of endothermic reactions include the photochemical reactions. In this case the additional source of energy necessary for the endothermic reaction to proceed is solar radiation which can break the chemical bonds of atmospheric species. In this book we will consider only one photochemical process photodissociation.1 Physics refresher Solar radiation consists of electromagnetic waves. Electromagnetic radiation has a dual wave-particle nature. This means that electromagnetic...

## Convective available potential energy CAPE

In previous sections we analyzed the stability of the displacement of a small parcel in terms of temperature lapse rate. In this section we will continue to analyze stability, but in terms of energy. We have already shown that when there is a positive area in the closed loop between environmental and adiabatic curves on a T-ln p diagram or, in other words, if a parcel (after a nudge) is positively buoyant, the parcel's kinetic energy increases. Consider a parcel being initially unsaturated in a...

## Problems

4.1 A parcel is lifted adiabatically from z 0 to z H, what is its change in entropy 4.2 Compute the change in entropy for an ideal dry gas of mass M which is heated at constant volume from T1 to T2. Take M 1 kg, T1 300 K and T2 310 K. 4.3 A parcel is lifted isothermally from pressure p0 to p1. Find its change in potential temperature. Take po 1000 hPa and pi 500 hPa, To 300 K. 4.4 A 1 kg parcel at 500 hPa and 250 K is heated with 500 J of radiation heating. What is the change in its enthalpy...

## Reversible and irreversible work

In the preceding we assumed that the work done by the system was along a well-defined path p(V). Actually this is a rather strong assumption - that at each infinitesimal adjustment the curve p(V) exists. We are implicitly assuming that we are in a state of thermodynamic equilibrium at each step - in other words the system has time to come to equilibrium (i.e., uniform temperature throughout, etc.) before the next infinitesimal step. In real processes such as the compression of a piston in an...

## Van der Waals equation

As we learned earlier, the approximation of an ideal gas works well if we can neglect the intermolecular forces. This is virtually always the case for the major constituents of air at Earth-like conditions. But as a gas nears its critical temperature and the liquid or solid state can coexist with the gas phase, the departure from ideality is important. As we see from Figure 5.2 the ideal gas equation of state describes the behavior of real gases in limiting cases of high temperatures and low...

## Integration of the Clausius Clapeyron equation

To proceed we divide each side of the equation by es and multiply through by dT. The left-hand side will be a function only of es and the right-hand side will be only a function of T. This allows us to integrate Next we choose the lower limit to be 273.2 K so that es(0) 6.11 hPa. The value 6.11 hPahas to come from observations -thermodynamics cannottell us the value of such a constant. After all, this integration constant should be different for different substances (e.g., compare this value to...

## Systems and equilibrium

Thermodynamics is the study of macroscopic or bulk systems of masses and their interrelations under conditions of steady state (no dependence on time). By macroscopic we mean the system contains large numbers of individual molecules (within a few orders of magnitude of a mole1 which contains 6.02 x 1023 molecules). We call these states equilibrium states if they are not only time independent but also stable under small perturbations. Thermodynamic states are describable by a set of dimensional...

## The thermodynamic equation

In this chapter we derive two of the fundamental equations of atmospheric science, the equation of continuity and the thermodynamic equation. The equation of continuity expresses the conservation of mass in the form of a partial differential equation, the form needed to implement it in numerical simulations or forecasts. The thermodynamic equation expresses the combined First and Second Laws of Thermodynamics into a similar form. But before we come to these important formulas we need some...

## Ideal gas basics

Gases are a form of matter in which the individual molecules are free to move independently of one another except for occasional collisions. Most of the time the individual molecules are in free flight out of the range of influence of their neighbors. Gases differ from liquids and solids in that the force between neighbors (on the average over time) is very weak, since the intermolecular force is of short range compared to the typical intermolecular distances for the individual gas molecules....

## Distribution of velocities

Obviously the molecules in a box are not moving parallel to the x, y and z directions exclusively. Instead molecules will have instantaneous velocity components vx, vy, and vz. Consider the vx component for an individual molecule at a given time. The value of vx will take on a range of values from one time to the next because of collisions with other molecules (it can be thought of as a random variable). Computer simulations suggest that after only a hundred or so collisions per molecule the...

## Constraints

An important concept in the study of thermodynamic systems is that of constraints. This notion is best illustrated by example. Consider the gas in a cylinder whose volume is determined by the position of a piston as in Figure 1.3. Several constraints are operative in this case. Most obvious is the position of the piston. It constrains the volume to have a certain value. If the piston is removed by a small amount the constraint is said to be relaxed. Note that a force must be applied Figure 1.3...

## Mean free path

The average distance a molecule travels in the gas before collision is called the mean free path. To obtain an estimate of the mean free path imagine the background gas particles to be stationary. Take our test molecule of radius r0 to be moving through the lattice of fixed points used in the last subsection. A collision between our prototype molecule and a background molecule will occur when their centers are within 2r0 of each other (Figure 2.2). We can think of the test molecule having...