## 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...

## Scalar and vector fields

A scalar field is a function defined on the three-dimensional space coordinates and possibly along the time axis. An example is the temperature field T(x, y, z t) T (r, t), where the position vector r is defined by and i, j, k are unit vectors pointing along the x,y and z axes (see Figure 9.5). A small increment in r is denoted as 1 dr dx i + dy j + dz k. (9.18) Figure 9.5 Schematic diagram of a position vector r whose components are x, y and z. Figure 9.5 Schematic diagram of a position vector...

## 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...

## Earth weight and mass

The Earth is an oblate spheroid, with slightly larger diameter in the equatorial plane than in a meridional (pole-to-pole) plane. The distance from the center to the poles Table 1.2 Selected physical quantities and their units Table 1.2 Selected physical quantities and their units Table 1.3 Greek prefixes applied to SI units Prefix Numerical meaning Example Abbreviation Table 1.3 Greek prefixes applied to SI units Prefix Numerical meaning Example Abbreviation Table 1.4 Selected conversions to...

## Cp QL dT

We can use the Clausius-Clapeyron equation to show that the first term in parentheses is much larger than the second term (recall that for every 10 K of increase in temperature there is a doubling of vapor pressure then dws ws 1, and dT T 10 300. Compare 1 > > 10 300). We may then substitute d(ws T) for dws T to a good approximation. where 0(T,p) T (p0 p)K. This last relationship (6.80) forms an implicit functional relationship that defines a curve in the T-P plane. The relationship can...

## Multiple phase systems

We proceed with the case of water in both its liquid and vapor forms in equilibrium in a container. This is a one-component (only one chemical species is present) system with two phases (liquid and gas) in equilibrium. Experience tells us that the two phases can coexist in equilibrium in this configuration. In fact, we have seen that for a given mass of the substance there is a range of values of volume for which the equilibrium exists with transfers of mass from one phase to the other as the...

## 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...

## 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...

## 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...

## 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...