Three important paramters describe the properties of a dry gas, pressure, temperature, and specific volume (or density).  Pressure has units of force per unit mass (the SI unit is a pascal).  These three variables are related through the ideal gas law:

pv = R T

where R = 287 Joules per (degree kilogram).  Basically this law says that if the temperature of a gas increases, then either the volume must increase or the pressure must increase.  This law follows from the assumption that the gas is made up of molecules that bounce against each other occasionally.  Since the molecules are only expected to bounce occasionally there follows another law called Dalton's Law.  Dalton's Law says that the ideal gas law actually applies for each constituent of the gas (nitrogen, oxygen, etc.) where the pressure is replaced by the partial pressure of gthat constituent and the specific volume is replaced by the specific volume of that constituent.  The total pressure is the sum of the individual partial pressures.

Example:  Calculate the density of water vapor which exerts a pressure of 9 mb at 20C.

Solution: We have pv = R T where p is now  the partial pressure of water vapor and R is now the gas constant for water vapor.  R in this case is 461 J deg ^-1 kg ^-1.  9mb is equal to 900Pa, so the specific volume of water vapor is 461*293/900 = 150 m^3 kg^-1.  Density, which is the inverse of specific volume,  is 6.7*10^-3 kg m^-3.  (this is problem 2.1 in WH).

2. Hydrostatic equation

The hydrostatic equation says, simply, that the weight exerted by the atmosphere above a certain point is simply the mass of the air times the gravitational constant.  Effectively, we assume that vertical accelerations are small.  Thus,

dp/dz = - g*rho,

That is, any change of pressure in the vertical is the result of the intervening mass.  This is quite a useful approximation because it allows us to calculate pressure from density.  Integrating this  equation tells us that surface pressure reflects the total mass of the atmosphere (Fig. 1)

fig. 1 (WH Fig. 2.1)

Example: Calculate the mass of the atmosphere

Solution: Mean surface air pressure is 1013mb or 1.013*10^5Pa, so the mass of the atmosphere per square meter is:1.0*10^5/10 = 1*10^4 kg.
 

 3. First law of thermodynamics

The first law of thermodynamics expresses the idea of conservation of energy.  It states that if an incremental amount of heat, dq,  is added to a body, the sum of the internal energy and the work done by the fluid ( CPdT - vdp).  Here CP is the heat capacity at constant volume. Alternatively, the later expression may be written as (CVdT + pdv) where CV is the heat capacity at constant volume. CP = CV + R

4. Dry adiabatic lapse rate

Suppose a parcel of gas is raised without gaining or losing heat.  Then,

CVdT=vdp

using the ideal gas law gives us:

CV dT/T = R dp/p

or

theta = T (po/p)^(R/CP)

R/CP has a value of 0.286.  Here theta is the potential temperature, that is, the temperature that an air parcel would have if moved vertically without gaining or losing energy (adiabatically).  If all air parcels could be moved up or down without gaining or losing energy then the potential temperature would be constant in the vertical. The dependence of temperature on height in this case is given as:

d(CP*T + gz) = 0

dT/dz = -g/CP

This latter formula is known as the dry adiabatic lapse rate.  With g=9.8 m s^-2 and CP = 1004 J kg^-1 we get a dry adiabatic lapse rate of 9.8 degrees per kilometer.