Atmospheric_thermodynamics_Lecture

8/7/2019 Atmospheric_thermodynamics_Lecture

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Atmospheric Thermodynamics

The Empirical Gas Laws and the

Ideal Gas Law


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Empirical Gas Laws

The empirical gas laws form the basis for

the analysis and forecasting of

thermodynamics processes in theatmosphere.

Empirical means that the laws were

developed through experimentation and

observation of gases.


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Empirical Gas Laws (Cont.)

According to Boyles law at a constant

temperature the pressure of a gas is

inversely proportional to its volume.

p w 1/V


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According to Gay-Lussacs law at a

constant pressure the temperature of a

gas is directly proportional to its volume.

T w V


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Ideal Gas

An ideal gas is any gas in which

thermodynamic processes can be

described exactly by Boyles Law andGay-Lussacs Law.


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Ideal Gas Law

Boyles Law and Gay Lussacs Law can be

combined into a single equation called the

ideal gas law (or the equation of state).

pV/T = constant

This form of the ideal gas law applies to a

gas composed a single type of molecule.


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Ideal Gas Law (Cont.)

We need to modify this form of the ideal gas

law to account for the different gases that

make up the composition of a mixture ofgases like we find in the Earths

atmosphere.


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Avagadros Law states that at a specifictemperature and pressure one mole of agas will occupy the same volume as one

mole of any other gas.

A mole of a gas contains as many

elementary units (molecules, atoms, etc.)as there are C atoms in exact 0.012 kg ofC12.


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Based on this observation the ideal gas lawis sometimes written as

pV = nR*T

where

n is the number of moles of a gas, and

R* is the universal gas constant

R* = 8.314 J mol-1 K-1


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Since

n = m/Mwhere

m is the mass of the gas, and

M is the molecular mass (weight) of the gas,


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we can rewrite the ideal gas law as

pV = mR*T/M

If we divide both sides by the volume, we

getp = mR*T/(MV)


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p = mR*T/(MV)

Since density, , is mass divided by volume,

= m/V

we can write the ideal gas law as

p = R*T/M


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p = R*T/M

The specific gas constant, R, for any gasis defined as

R = R*/M

Substitution producesp = RT


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This form of the ideal gas law is valid for a

single gas, because it contains this

specific gas constant, R, for a single gas.

We need to modify R, if we want to create

an ideal gas law that we can use for the

mixture of gases in the Earths

atmosphere.


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According to Daltons Law the total

pressure exerted by a mixture of gases is

equal to the sum of the partial pressures

exerted by the individual gases that

comprise the mixture.

p = pi = p1 + p2 + p3 + + pn


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p = pi

Using a form of the ideal gas law we hadearlier we can substitute forpi and write

Daltons Law as

p = ((miR*Ti)/(ViMi)


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Rule for Summations

Anything that is a constant can be pulled

outside the summation symbol and

multiplied after summing the remaining

terms.


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Assumptions About a Mixture of

GasesIf the mixture of gases is well mixed by the

winds, then all of the individual gases in

the mixture will occupy the same volume

(i.e. V1 = V2 = V3, etc.)


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Assumptions About a Mixture of

Gases (Cont.)If the mixture of gases is well mixed by the

winds, then molecular collisions will

transfer internal energy and all of the

gases in the mixture will eventually have

the same temperature (i.e. T1=T2=T3, etc.)

By definition the universal gas constant, R*,

is a constant.


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p = ((miR*Ti)/(ViMi)

Thus, we can pull R*, T, and V out of thesummation and write

p = (R*

T/V) (mi/Mi)


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p = (R*T/V) (mi/Mi)

Multiply the right hand side by

(mi/mi)

to getp = (R*T)(mi/V)((mi/Mi)/(mi)


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Since mi = m, total mass of all of the

gases in the mixture, then

(mi/V) =


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and

p = (R*T)(mi/V)((mi/Mi)/(mi))

becomesp = (R*T)(mi/Mi)/(mi)


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Mean Molecular Mass (Weight)

Meteorologists define a mean molecular

mass (weight), M, such that

1/M = ( (mi/Mi))/(mi)


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Dry Air

In thermodynamics the term dry airrefers to

the normal mixture of gases observed in

the Earths atmosphere, except for the fact

that it contains no watervapor.


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Major Components of Dry Air

Gas M (kg/mol) Mass Fraction R (J kg-1 K-1)

Nitrogen (N2) 0.028 0.755 296.8

Oxygen (O2) 0.032 0.232 259.8

Argon (Ar) 0.040 0.013 208.1Carbon

dioxide (CO2) 0.044 0.0005 188.9


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Mean Molecular Mass of Dry Air

Meteorologists define the mean molecular

mass of dry air, Md based on the mass

fractions and molecular masses of the

individual gases in the table.

If we assume we have one kilogram of dry

air and plug the appropriate numbers into

the equation for 1/M, we get


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Mean Molecular Mass (Weight)

Meteorologists define a mean molecular

mass (weight), M, such that

1/M = ( (mi/Mi))/(mi)


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Mean Molecular Mass for Dry Air

(Cont.)

0.755 kg + 0.232 kg + 0.013 kg + 0.0005 kg1/Md = 0.028 kg mol

-1 0.032 kg mol-1 0.040 kg mol-1 0.044 kg mol-1

0.755 kg + 0.232 kg + 0.013 kg + 0.0005 kg

Md = 0.029 kg mol-1


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Md = 0.029 kg mol-1

Gas M (kg/mol) Mass Fraction R (J kg-1 K-1)

Nitrogen (N2) 0.028 0.755 296.8

Oxygen (O2) 0.032 0.232 259.8

Argon (Ar) 0.040 0.013 208.1Carbon

dioxide (CO2) 0.044 0.0005 188.9

Since approximately 75% of the mass of dry air is N2

, itshould make sense that our Md is just a little larger thanM for N2.


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Ideal Gas Law for Dry Air

p = (R*T)(mi/Mi)/(mi)

If we let

1/Md = ( (mi/Mi))/(mi)

then we can substitute to get

p = (R*T)/Md


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Gas Constant for Dry Air

Meteorologists often define as gas constantfor dry air, Rd, as

Rd = R*/Md

Rd = 8.314 J mol-1 K-1 / 0.029 kg mol-1

Rd = 287 J kg-1 K-1


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Ideal Gas Law for Dry Air (Cont.)

p = (R*T)/Md

and

Rd = R*

/Mdso substitution produces

p = RdT

which is the most common form of the idealgas law for dry air.

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