Gas Laws and Equation of State
What are the “guts”
of air parcels within
thunderstorms?
Thermodynamics
M. D. Eastin
Gas Laws and Equation of State
Outline:
 Basic Definitions
 Kinetic Theory of Gases
 Early Gas Laws
 Avogadro’s Hypothesis
 Ideal Gas Law
Thermodynamics
M. D. Eastin
Basic Definitions
Thermodynamics:
The study of equilibrium states of a system
System: Any specific sample of matter
(Examples: an “air parcel” or the air within a balloon)
• Open system: Exchanges matter and energy with its surroundings.
(Example: a balloon with a lot of holes)
• Closed system: Does not exchange matter with its surroundings,
but can exchange energy.
(Example: a balloon without insulation)
• Isolated System: Does not exchange matter or energy with its
surroundings.
(Example: a balloon with insulation)
Note: In meteorology we assume most systems are closed
Thermodynamics
M. D. Eastin
Basic Definitions
Thermodynamics:
State:
The study of equilibrium states of a system
Defines the total volume (V), pressure (p), and temperature (T) of each
constituent composing a system at any given time
If the system is homogeneous (contains only one component), then
only the V, p, T values are needed.
If the system is heterogeneous (contains multiple components), then
the concentrations of the different components as well as the V, p, T
are needed.
Note: Our atmosphere is a
heterogeneous system:
Thermodynamics
M. D. Eastin
Basic Definitions
Thermodynamics:
Equillibrium:
The study of equilibrium states of a system
Defines as a state in which a system’s properties do not change
as long as the properties of the surroundings remain unchanged
Example:
Surroundings: V2, P2, T2
What would happen if the air
pressure in the surroundings
decreases?
Closed
System
V1, P1, T1
Equilibrium: P1 = P2 and T1 = T2
Thermodynamics
What would happen if the air
temperature in the surroundings
decreases?
M. D. Eastin
Basic Definitions
Transformation:
When a system changes from an initial state to a new final state
Symbolically:
i→f
Reversible transformation: Any transformation that can be reversed and
arrive back at the initial state ( i → f → i )
Occur when the external surroundings change
very slowly so the system has time to adjust to
new equilibrium states along the way
Irreversible transformation: Any transformation that can not be reversed
and arrive back at its original state.
Occur when surroundings or the system
experiences a rapid change
Thermodynamics
M. D. Eastin
Basic Definitions
Transformation:
When a system changes from an initial state to a new final state
Isothermal transformation: Any transformation that occurs without a
change in temperature
Isochoric transformation:
Any transformation that occurs without a
change in volume
Isobaric transformation:
Any transformation that occurs without a
change in pressure
Adiabatic transformation:
Any transformation that occurs without
exchanging heat with the environment
Note: Adiabatic transformations are not
isothermal
Thermodynamics
M. D. Eastin
Basic Definitions
Energy: The total internal energy of a system (U) is the sum of the kinetic energy
and potential energy of all the particle’s that compose the system.
Each particle is moving
in some random direction
(kinetic energy)
Each particle is being
pulled down by gravity
(potential energy)
Thermodynamics
M. D. Eastin
Kinetic Theory of Gases
Basic Idea:
We have a closed system containing N molecules moving in random
directions and random speeds:
The mean kinetic energy of all molecules is the system temperature (T)
The mean force exerted on the system’s edges due to particle collisions
with the edge is the system pressure (p).
The three-dimensional space occupied by the moving molecules is the
system volume (V)
Thermodynamics
M. D. Eastin
Kinetic Theory of Gases
Basic Idea:
By applying some simple laws of physics (see Chapter 2 in your text),
a simple equation that combines the three state variables into a
single equation can be obtained:
pV  NkT
where:
p = pressure
V = volume
N = Number of molecules
k = Boltzmann’s constant
T = temperature
Note: This is one form of the ideal gas law
This equation was obtain from the results of numerous laboratory experiments
during the 18th and 19th centuries. Lets examine some of those early results…
Thermodynamics
M. D. Eastin
Early Gas Laws
Gay-Lussac Law #1:
When pressure (p) is constant:
V1 T1

V2 T2
Gay-Lussac Law #2:
When volume (V) is held constant:
p1 T1

p2 T2
Boyles Law:
When temperature (T) is held constant:
p1V1  p2V2
Note: The subscripts refer to two different system states
Thermodynamics
M. D. Eastin
Early Gas Laws
Example:
Assume a hot air balloon is anchored to the ground and the air
within occupies V = 10 m3 at p = 1000 mb at T = 300 K.
The burners at the balloon based are turned on and the air heats
up to T = 320 K. Assume the balloon volume remains constant.
What is the new pressure inside the balloon?
Answer:
Apply the Gay-Lussac Law #2:
T1 = 300 K
p1 = 1000 mb
T2 = 320 K
Solve the equation for p2
p1 T1

p2 T2
p1T2
p2 
T1
Plug in the values → p2 = 937.5 mb
Thermodynamics
M. D. Eastin
Avogadro’s Hypothesis (Law)
Avogadro’s Number:
The number of atoms in 12 g of carbon
Called a mole → 1 mole = 6.022 ×1023
Avogadro’s Hypothesis:
One mole of any gas at constant temperature and
pressure occupies the same volume
If we increase the number of molecules (or mass)
of the gas at a constant temperature and pressure,
then the volume of the system increases.
Thermodynamics
M. D. Eastin
Ideal Gas Law
• If we combine Boyles Law and Gay-Lussac’s second law we obtain:
p1V1 p2V2

T1
T2
which implies that if we change either p, V, or T, the value of pV/T remains
constant. Thus we can re-write the equation above as:
pV  AT
where A is a constant that depends on the type and amount of the gas.
Note:
The dependence on the amount (or mass) of the gas is a result of
Avogadro’s Hypothesis
Thermodynamics
M. D. Eastin
Ideal Gas Law
• Since A is a function of both gas type and its mass, we can define A as:
A  mR
where:
m = total mass of the gas
R = constant independent of mass, but dependent on gas type
(will change for different gas types: oxygen vs. hydrogen)
and re-write our equation as:
pV  mRT
Thermodynamics
Ideal Gas Law
M. D. Eastin
Ideal Gas Law
• Since density (ρ) is defined as the mass (m) per unit volume (V):
m

V
we could re-write our equation as:
p  RT
pV  mRT →
Ideal Gas Law
• Also, since we can define a specific volume (α):

1

we could re-write our equation as:
pV  mRT →
Thermodynamics
p  RT
Ideal Gas Law
M. D. Eastin
Ideal Gas Law
• Also, if we let M equal the molecular weight of the gas, then the number of moles
of that gas is defined as:
m
n
M
where:
n = number of moles
m = total mass of the gas
M = molecular weight of the gas
We could also re-write our equation as:
pV  mRT →
Thermodynamics
pV  nMRT
Ideal Gas Law
M. D. Eastin
Ideal Gas Law
• Both M and R are constants and each are dependent on gas type
• However, their product results in a constant that is independent of gas type,
and is thus called the universal gas constant (R*):
R *  MR
where:
R* = 8.3143 J K-1 mol-1
We can then rewrite our equation as:
*
→
pV

nR
T
pV  nMRT
Thermodynamics
Ideal Gas Law
M. D. Eastin
Ideal Gas Law
Formulations of the Ideal Gas Law:
pV  NkT
pV  nMRT
pV  nR*T
pV  mRT
p  RT
→
where:
Thermodynamics
p  RT
←
p = pressure
V = volume
T = temperature
N = number or molecules
k = Boltzmann’s constant (1.38 ×10-23)
n = number of moles
M = molecular weight of a gas particle
R = gas constant dependent on gas type
m = total mass
ρ = density (m/V)
α = specific volume (1/ρ)
R* = universal gas constant (8.3143 J K-1 mol-1)
M. D. Eastin
Ideal Gas Law: Earth’s Atmosphere
Ideal Gas Law for Earth’s Dry Atmosphere:
• We need to find the gas constant (R) valid for our atmosphere, which contains
multiple gases in various percentages:
R*
R
M
→
R*
Rd 
M
• We will neglect water vapor, and find the gas constant for our dry atmosphere
Gas
Molecular Weight
Nitrogen (N2)
28.016
Oxygen (O2)
32.000
Argon (Ar)
39.940
Carbon Dioxide (CO2) 44.010
Thermodynamics
Percentage by mass
75.51%
23.14%
1.30%
0.05%
M. D. Eastin
Ideal Gas Law: Earth’s Atmosphere
Ideal Gas Law for Earth’s Dry Atmosphere:
• Thus, we need to find the mean molecular weight of our atmosphere:
Dalton’s Law:

M

K
mi
K mi
i 1
Mi
i 1
See Example 3.1
(on page 21-22)
of your text for details
M  28.97 g / mol
Rd  287 J / kgK
Thermodynamics
M. D. Eastin
Ideal Gas Law: Earth’s Atmosphere
Be careful about your interpretation:
• The ideal gas law relates three variables (p, ρ or V, and T)
• Thus, any change in one does not automatically produce a change in another.
• If we differentiate the ideal gas law:
p  Rd T
dp  Rd dT  Rd Td
• A change in pressure (dp) can lead to either a change in temperature (dT) or
a change in density (dρ), or both.
Let’s see a couple examples…
Thermodynamics
M. D. Eastin
Ideal Gas Law: Earth’s Atmosphere
Be careful about your interpretation:
Example #1:
Assume you are in Charlotte getting ready to board an airline
flight to Denver. You have a soft drink bottle that is filled with air
at p = 1000 mb, T = 300 K, and ρ = 1.00 kg/m3. Before you
boarded the plane, you sealed the bottle tightly. Once you arrived
at Denver you noticed the bottle was partially collapsed. You
know the temperature in the bottle remained unchanged, but the
pressure dropped to p = 850 mb.
What was the new air density?
dp  RdT  RTd
Can you solve this problem?
If “yes”, then do so… (pay attention to units)
Thermodynamics
M. D. Eastin
Ideal Gas Law: Earth’s Atmosphere
Be careful about your interpretation:
Example #2:
Assume you are in Charlotte getting ready to board an airline
flight to Denver. You have a soft drink bottle that is filled with air
at p = 1000 mb, T = 300 K, and ρ = 1.00 kg/m3. Before you
boarded the plane, you sealed the bottle tightly. Once you arrived
at Denver you noticed the bottle was partially collapsed. You did
not know the bottle temperature but the pressure dropped
to p = 850 mb.
What was the new air density?
dp  RdT  RTd
Can you solve this problem?
If “yes”, then do so… (pay attention to units)
Thermodynamics
M. D. Eastin
Gas Laws and Equation of State
Summary:
• Basic Definitions:
• Equilibrium, System, State
• Reversible, Irreversible, Isobaric, Isothermal, Isochoric, Adiabatic
• Kinetic Theory of Gases (know the basic idea)
• Early Gas Laws:
• Gay-Lussac Laws
• Boyles Law
• Avogadro’s Hypothesis (know the basic idea)
• Ideal Gas Law:
• Basic Derivation
• Various Forms
• Universal Gas Constant
• Dalton’s Law for (mean molecular weights)
Thermodynamics
M. D. Eastin
References
Petty, G. W., 2008: A First Course in Atmospheric Thermodynamics, Sundog Publishing, 336 pp.
Tsonis, A. A., 2007: An Introduction to Atmospheric Thermodynamics, Cambridge Press, 197 pp.
Wallace, J. M., and P. V. Hobbs, 1977: Atmospheric Science: An Introductory Survey, Academic Press, New York, 467 pp.
Thermodynamics
M. D. Eastin