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5.7/5.1 Kinetic Molecular Theory of Gases

1. Gases consist of large number of molecules (atoms) that are in continuous, random motion.

2. The volume of all molecules of the gas is negligible compared to the volume in which the gas is contained.

3. Attractive and repulsive forces between gas molecules are negligible.

4. Energy can be transferred between molecules during collisions, but the average KE of the molecules does not change with time as long as the temperature of the gas remains constant.

5. The average kinetic energy of a molecule is proportional to the Kelvin temperature. At any given temperature the molecules of all gases have the same average KE.

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5.7/5.1 Notes: Kinetic Molecular Theory of Gases

a. Gases are compressible: because the gas particles have a small volume compared to the container.

b. Elastic collisions: when gases are left alone in a container do not seem to lose energy and do not spontaneously convert to the liquid. Energy is not lost during collisions.

c. The assumptions have limitations. gases can be liquefied if cooled enough. real gas molecules do attract one another to

some extent otherwise the particles would condense to form a liquid.

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5.7 Gas Laws and Molecular Kinetic Theory

• Pressure is due to collisions between gas molecules and the walls of the container. Mgnitude determined by: force of collisions and frequency.

• Boyle’s Law: decreasing the volume of the gas at constant n and T increases the frequency of collisions with the container walls and therefore increases the pressure. (temp. constant – KE constant)

• Charles’ Law: Raising temperature increases the number of collisions and force of collisions (KE increases) with container wall. If the walls are flexible, they will be pushed back and the gas expands.

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5.7 Gas Laws and Molecular Kinetic Theory

• Gay Lussac’s Law: Raising the temperature increases the number of collisions and the kinetic energy of the molecules. More collisions with greater energy (force) means higher pressure.

• Dalton’s Law: Molecules of a gas do not attract or repel each other. The distances between particles are very large, therefore each particular gas occupies the entire container.

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Maxwell Distribution Curves

• Average Kinetic Energy at a given temperature is constant for a gas sample

• But, the speeds of the molecules vary (during to collisions with each other and with the walls of the container)

• Physics: momentum is conserved (palying pool)

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5.7 Maxwell’s Distribution Curves

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5.7 Maxwell’s Distribution Curves

Molecules in a gas move at many different speeds.

The Maxwell distribution shows the fraction of the molecules that are moving at a particular speed.

The distribution shifts to higher speeds at higher temperatures.

http://jersey.uoregon.edu/vlab/Balloon/index.htmlhttp://intro.chem.okstate.edu/1314F00/Laboratory/GLP.htm

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Maxwell- Boltzmann Velocity (energy) Distribution

KE = ½ m(urms)2

Plot of Probability (fraction of molecules with given speed) versus root mean square velocity of the molecules.

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Maxwell Distribution Curve

• Variation in particle speeds for hydrogen gas at 273K

The vertical line on the graph represents the root-mean-square-speed (urms).

urms

The root-mean-square-speed is the square root of the averages of the squares of the speeds of all the particles in a gas sample at a particular temperature.

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5.7 MKT of Gases: Equations

• Average KE = (3/2) RT

• Urms = M

RT3

The Urms is not the same as the mean (average ) speed. The difference is small.

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Average Molecular speed

Average molecular kinetic energy depends only on temperature for ideal gases.

Therefore: Higher temperature means higher root-mean- square

speed (RMS), rms

The higher the molecular weight (molar mass), the lower the urms speed (same temperature)

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Average Root Mean Square: Examples

18. Calculate the Urms speed, urms, of an N2 molecule at 25ºC.

(5.15 x 102 m/s)

19. Calculate the urms speed of helium atoms 25ºC. (1.36 x 103

m/s)

20. Calculate the Urms speed of chlorine atoms at 25ºC. (323 m/s)

M

RT3

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5.8 Average Speed of Some Molecules

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5.8 Properties and Urms

• Diffusion rates: – gas movement due to random molecular motion– faster molecules = faster diffusion rate

• Effusion rates:– Gas movement through pinhole in a container– faster molecules = faster diffusion rate

• Heat conduction: – faster molecules = faster transfer of heat energy– sound travels faster in hot air then in cold air

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5.8 Diffusion and Effusion

(a) Diffusion is the mixing of gas molecules by random motion under conditions where molecular collisions occur.

(Ib) Effusion is the escape of a gas through a pinhole without molecular collisions

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5.8 Graham’s Law of Diffusion

• Under the same conditions of temperature and pressure, the rate of diffusion of gas molecules are inversely proportional to the square root of their molecular masses.

1

2

2

1

M

M

r

r

Aof mass Molar

B of mass Molar

Aofdensity

B ofdensity

B of effusion of Rate

Aof effusion of Rate

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5.8 Graham’s law of Diffusion

21. It has taken 192 seconds for 1.4 L of an unknown gas to effuse through a porous wall and 84 seconds for the same volume of N2 gas to effuse at the same temperature and pressure. What is the molar mass of the unknown gas? (146 g/mol)

21. In a given period of time, 0.21 moles of a gas of RMM = 26 gmol-1 effuses. How many moles of HCN would effuse in the same period of time?

22. Calculate and compare the urms of Nitrogen gas at 35oC and 299K.

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Homework : section 5.7/5.8

Page 203, question 5.90, 100, 101, 102, 117

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5.9 Real Gases

Two problems with the Kinetic Molecular Theory of "Ideal" Gases:

1. Gas particles have volume (they are not point masses). The volume becomes important under certain conditions.

2. When gas particles are close to each other, they attract each other.

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1RT

PVFor 1 mole of gas we can rearrange the PV= nRT equation and get:

Plot of (PV)/(RT) for

1 mole of gas

The value for the equation is not always equal to 1

Corrections to the Ideal Gas Equation is needed

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Factors Affecting Ideality of Gases

• Distance between molecules is related to gas

concentration:

RT

P

V

n

• At high concentration (high P, low V):

molecules are closer,

therefore stronger interactions between molecules:

repulsions or attractions

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5.9 Real Gases: Effect of Volume

(a) At low pressure, the gas occupies the entire container and its volume is insignificant compared to the volume of the container.

(b) At high pressure, the volume of a real gas is somewhat larger than the ideal value for an ideal gas as gas molecules take up space.

Volume should go to zero, but it does not.

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5.9 Real Gases: Effect of Pressure

At high pressures• Intermolecular distances

between molecules decrease

• Attractive forces start to play a role

• Stickiness factor• Measured pressure is less

than expected

2

2

V

naPP realideal

2

2

V

naPP realideal

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5.9 Real Gases: Corrections

• Constant needed to correct intermolecular attractive forces (make it larger)

• Constant needed to correct for volume of individual gas molecules (make it smaller)

The constants are characteristic properties of the substances: depend on the make-up and geometry of the substance

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Van der Waals Constantsfor Common Gases

Compound     a (L2-atm/mol2)    b (L/mol)

He 0.03412 0.02370

Ne 0.2107 0.01709

H2 0.2444 0.02661

Ar 1.345 0.03219

O2 1.360 0.03803

N2 1.390 0.03913

CO 1.485 0.03985

CH4 2.253 0.04278

CO2 3.592 0.04267

NH3 4.170 0.03707

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Deviations are greater if : •Intermolecular attractive forces (IMF) of gas molecules are greater. •Mass (and subsequently volume) of gas molecules is greater.

Conditions are "Ideal" at: •high temperature •low pressure           

Conditions are "Real" at: •Low Temperature •High Pressure

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Real Gases: Deviations

Deviations are greater if : Intermolecular attractive forces (IMF) of gas molecules are greater. Mass (and subsequently volume) of gas molecules is greater.

Conditions are "Ideal" at: high temperature low pressure   

Conditions are "Real" at: Low Temperature High Pressure

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Ideal gas Real gas

Obey PV=nRT Always Only at low pressures

Molecular volume

Zero Small, but not zero

Molecular attraction

Zero Small

Molecular repulsion

Zero small

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Factors Affecting Ideality of Gases

– Repulsion make pressure higher than expected by decreasing the free volume in the container

– Attractions make pressure lower than expected by breaking molecular collisions (less collisions) against the walls of the container).

– Tug-of-war between these two effects causes the following:

• Repulsion win at very high pressure• Attractions win at moderate pressure• Neither attractions nor repulsions are important at

low pressure.

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PV

P (at constant T)

CO2

O2

22.4

1 L

atm

PV versus P at Constant T (1 mole of Gas)

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• V of a real gas > V of an ideal gas because V of gas molecules is significant when P is high.  Ideal Gas Equation assumes that the individual gas molecules have no volume.