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METR 2413
11 and 13 February 2004
Thermodynamics
I
Introduction
What is thermodynamics?
• Study of energy exchange between a system and its
surroundings
• In meteorology, how these changes relate to the state of the
atmosphere
• Important for understanding the temperature distribution in
the atmosphere, cloud processes and severe weather, and the
general circulation
Introduction
Thermodynamics considers a “system” and its
“surroundings”.
In meteorology
• System = parcel of air
• Surroundings = rest of the atmosphere, environment
Two types of exchange between system and surroundings
• Energy exchange (heat, friction, work, radiation, etc)
• Matter exchange (movement of molecules)
Introduction
Types of systems:
• Isolated system – no exchange of matter or energy with the
surroundings
• Closed system – exchange of energy, but no exchange of
matter with the surroundings
• Open system – exchange of both energy and matter
Introduction
“Laws” of thermodynamics
These are actually empirical relationships (based on
observations) that over time and repeated experiments by
scientists have been shown to hold.
• Zeroth law - equilibrium
• First law - conservation of energy
• Second law – entropy can never decrease
State variables
Variables of state: pressure, density and temperature
Pressure
Pressure = force/area, with units (MKS) of Newton/meter2
1 Nm-2 = 1 Pa (Pascal)
Mean sea level pressure 1001 mb = 100.1 kPa = 1001 hPa
Pressure is isotropic; at any point, it is the same in all directions
Dalton’s law: total pressure of a mixture of gases is equal to the
sum of the partial pressures exerted by each individual gas
State variables
Temperature
T = average kinetic energy of molecules in a gas
Depends on heat content
Heat = total kinetic energy of a gas
Density
ρ = mass/volume, with units (MKS) of kg/meter3
Average density of air at sea level ρ0 = 1.23 kgm-3
State variables
Temperature of an air parcel can be affected by:
• Radiation being absorbed or emitted
• Energy exchange with the surrounding air
• Energy exchange with the adjacent ground
• Evaporation or condensation of water
• Expansion or compression of the parcel due to vertical motion
Equation of State
Want to relate variables of state to each other.
Experiments have found that gases follow approximately the
same equation of state over a wide range of conditions.
Although the atmosphere is a mixture of gases, it behaves as
though it is a single “ideal” gas:
• made up of a large number of molecules in random motion
• separation of molecules is large compared with their size
• molecules experience perfect elastic collisions (momentum
and energy are conserved)
• forces between molecules are negligible (except during
collisions)
Equation of State
Empirical relationships
Boyle’s law: If a gas is kept at a constant temperature, then
pressure is inversely proportional to volume
1
p
V
Charles’ law: When pressure is kept constant, volume is directly
proportional to temperature
V T
Equation of State
Ideal gas law
Equation of state for dry air
p = ρ Rd T
where Rd is the gas constant for dry air = 287 Pa K-1 m3 kg-1
Gas constant depends on mean molecular weight of air.
Now, we shall derive the equation of state and gas constant from
kinetic theory of gases.
Kinetic theory of gases
We shall derive the ideal gas law from first principles.
You won’t need to reproduce this in a quiz or homework
Consider a Cartesian coordinate system (x,y,z), with z upward,
and a wall in the y-z plane with air on one side
At any time, many air molecules are colliding with the wall.
Let’s consider one molecule as it
air
z
bounces off the wall. It must have
wall
experienced a force, since it changed its
motion. From Newton’s Second Law,
x
the molecule must have exerted an
y
equal and opposite force on the wall.
Kinetic theory of gases
The momentum of the molecule Mx is altered by the force acting
on it, so that
dM x
Fx 
dt
Assuming that the collision is elastic, then the x-component of the
molecule’s motion must be exactly reversed. The x-component of
its momentum must be equal and opposite in sign to that prior to
the collision. So Mxf = -Mxi = mu and the change in momentum is
ΔMx = mu – (-mu) = 2mu, with m the mass of the molecule.
This gives dM x  Fx dt
 M x   Fx dt  2mu
(1)
Kinetic theory of gases
Now we need to extend this to an arbitrary number of molecules.
Assuming random isotropic motion, at any time half the
molecules are moving towards the wall. The flux of molecules
N
striking the wall is nu where n 
is the number of
V
2
molecules per unit volume of air (N the total number of molecules
and V the volume of the air).
Combining this with (1), the total force over time t is
1
0 Fx dt  ( 2 nuAt )  (2mu ) where ½nuAt is the total no. of
t
molecules striking the wall in time t, and 2mu is the change in
momentum of a single molecule striking the wall.
Kinetic theory of gases
Now divide this result by t to get the average instantaneous force
t
< Fx > on A gives
Fx dt nuAtmu

0
 Fx  

 nmAu 2  nmA  u 2 
t
t
where <u2> is the average square of the x-component of the
velocity over all the molecules.
Now, we define pressure as force per unit area, giving
 Fx 
p
 nm  u 2  (2)
A
Kinetic theory of gases
Since the molecular motion is random and isotropic,
<u2> = <v2> = <w2> and
2
 V    u 2    v2    w 2   3  u 2 
Hence
2
1N
p  nm  u  
m V 
3V
2
Putting this into the form of the ideal gas law pV = NkT gives a
definition of absolute temperature T as
2
2
1
1
T m  V 
 mV 
3k
3k
with k = Boltzmann constant = 1.38×10-23 J K-1
Kinetic theory of gases
Now, defining the gas constant
mN
and density  
V
k
R
m
gives the ideal gas law
NkT  m  N k
p

T  RT
V
V
m
For dry air, Rd = 287 J kg-1 K-1 = 287 Pa K-1 m3 kg-1
Kinetic theory of gases
From the definition of absolute temperature, T increases as <u2>
increases.
Does this mean that all molecular motion ceases at absolute zero?
In practice, as T → 0, the assumptions in the kinetic theory of
gases are no longer valid; collisions are not elastic, molecules are
not widely spaced and intermolecular forces cannot be neglected.
Another interpretation of T = 0 is, using pV = NkT, that it would
be the temperature that the volume of the gas would go to zero if
pressure remained constant.
Virtual temperature
Water vapor has a lower molecular weight than dry air
MH2O ~18, MO2 ~ 32, MN2 ~ 28
so water vapor in air makes the mean molecular weight of moist
air lower than that for dry air.
This means the gas constant for moist air is greater than for dry air
and depends on the amount of water vapor.
Rather than define a variable gas constant, we define a new
temperature, the virtual temperature Tv = T (1 + 0.61 r)
where r is the water vapor mixing ratio.
Hence, moist air with temperature T behaves like dry air with
temperature Tv with gas law p = ρ R Tv