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Gases & the Kinetic Molecular Theory

The particles of a gas are considered to be small, hard spheres with an insignificant volume

Gas particles are spaced relatively far apart compared to the particles of solids and liquids
Between the gas particles there is empty space
No attractive or repulsive forces exist between gas particles

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Gases & the Kinetic Molecular Theory

The motion of gas particles is rapid, constant, and random

Gases fill their containers, regardless of the shape and volume of the container
Uncontained gases will spread out into space without limit
Gas particles travel in straight paths until they collide with another particle, or object (i.e., the walls of their container)

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Gases & the Kinetic Molecular Theory

All collisions between gas particles are perfectly elastic

During elastic collisions, kinetic energy is transferred without loss from one particle to another
The total kinetic energy of the particles remains constant, even after collision

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Gas Pressure

Gas pressure results from the force exerted by a gas per unit surface area of an object

An individual gas particle exerts an extremely small force as it moves around
Gas pressure is the result of billions of rapidly moving particles in a gas simultaneously colliding with an object

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Gas Pressure - Units

The SI unit for pressure is the Pascal (Pa)

SI = International System of Units which is a globally agreed system of measurements that has been in place for more than two centuries

Other units for pressure include:

mmHg = millimeters of mercury
atm = atmospheres (equal to the average air pressure at sea level at a temperature of 15^oC)
PSI = pounds per square inch
kPa = kiloPascals

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Atmospheric Pressure

Atmospheric pressure is the pressure exerted by gas particles in Earth's atmosphere as those particles collide with objects

Air exerts pressure on Earth because gravity holds the particles in air within Earth’s atmosphere
Atmospheric pressure decreases as elevation increases because the density of Earth’s atmosphere decreases at higher altitudes

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Compressibility

Compressibility is a measure of how much the volume of matter decreases under pressure

Gases are easily compressed or squeezed into a smaller volume
Under increased pressure, the gas particles are forced closer together

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Factors Affecting Gas Pressure

Four variables are generally used to describe a gas’s behavior:

Pressure (P) usually in kilopascals
Volume (V) in liters
Temperature (T) in Kelvin
Number of moles (n)

The amount of gas, the volume, and the temperature are factors that affect gas pressure

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STP

The volume of a gas varies with a change in temperature and pressure so volume for gases is usually measured at standard temperature and pressure (STP)

Temperature at STP = 0^oC or 273 K
Pressure at STP = 1 atm or 101.3 kPa

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The Gas Laws

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The Gas Laws

The gas laws help predict the behavior of gases under certain conditions of pressure, volume, and temperature

The gas laws we will study are:

2.1 - Avogadro’s Law
2.2 - Boyle’s Law
2.3 - Charles’s Law
2.4 - Gay-Lussac’s Law

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Avogadro’s Law

2.1

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Avogadro’s Gas Law

Avogadro’s law states that the volume of any gas is proportional to the number of molecules of gas

If the amount of gas increases, then so does its volume and vice versa
Moles of gas and volume of gas are directly related

V₁

V₂

n₁

n₂

Volume can be in any unit of measure for this law

V₁ = initial volume

n₁ = initial moles

V₂ = final volume

n₂ = final moles

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Boyle’s Law

2.2

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Boyle’s Gas Law

Boyle’s law describes the relationship between the pressure and the volume of a gas

Boyle determined that with temperature held constant, the pressure and volume of a gas are inversely related

P₁V₁ = P₂V₂

P₁ = initial pressure P₂ = final pressure

V₁ = initial volume V₂ = final volume

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Boyle’s Gas Law

Units for pressure include:

mmHg
atm
PSI
kPa or Pa
torr

P₁V₁ = P₂V₂

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Boyle’s Gas Law - Example

A balloon contains 30.0 L of helium gas at 103 kPa. What is the volume of the helium gas when the balloon rises to an altitude where the pressure is only 25.0 kPa? Assume temperature remains constant.

P₁V₁ = P₂V₂

P₁ = P₂ =

V₁ = V₂ =

103 kPa

30.0 L

25.0 kPa

( 103 kPa ) ( 30.0 L ) = ( 25.0 kPa ) ( V₂ )

3090 = 25.0 V₂

V₂ = 123.6

V₂ = 124 L

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Charles’s Law

2.3

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Charles’s Gas Law

Charles’s law describes the relationship between the volume and the temperature of a gas

Charles determined with pressure held constant, the volume and temperature of a gas are directly related

V₁ = initial volume V₂ = final volume

T₁ = initial temperature T₂ = final temperature

T₁

V₁

T₂

V₂

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Charles’s Gas Law

🔥Temperature🔥 must be in Kelvin!

To convert from Celsius to Kelvin:

^oC + 273 = K

T₁

V₁

T₂

V₂

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Charles’s Gas Law - Example

A balloon inflated in a room at 24^oC has a volume of 4.00 L. The balloon is then heated to a temperature of 58^oC. What is the new volume if the pressure remains constant?

V₁ = V₂ =

T₁ = T₂ =

4.00 L

24^oC

58^oC

( 4.00 L) ( 331 K ) = ( V₂ ) ( 297 K )

1324 = 297 V₂

V₂ = 4.4579

V₂ = 4.5 L or 4.46 L

T₁

V₁

T₂

V₂

+ 273 =

297 K

331 K

4.00 L

297 K

V₂

331 K

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Gay-Lussac’s Law

2.3

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Gay-Lussac’s Gas Law

Gay-Lussac’s law describes the relationship between the pressure and the temperature of a gas

Gay-Lussac determined that with volume held constant, the pressure and temperature of a gas are directly related

P₁ = initial pressure P₂ = final pressure

T₁ = initial temperature T₂ = final temperature

T₁

P₁

T₂

P₂

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Gay-Lussac’s Gas Law

🔥Temperature🔥 must be in Kelvin!

To convert from Celsius to Kelvin:

^oC + 273 = K

T₁

P₁

T₂

P₂

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Gay-Lussac’s Gas Law - Example

Aerosol cans carry label warnings not to incinerate (burn) the cans or store them above a certain temperature. The gas in a used aerosol can is at a pressure of 103 kPa at 25^oC. If the can is thrown onto a fire, what will the pressure be when the temperature reaches 928^oC?

P₁ = P₂ =

T₁ = T₂ =

103 kPa

25^oC

928^oC

( 103 kPa) ( 1201 K ) = ( P₂ ) ( 298 K )

123703 = 298 P₂

P₂ = 415.1107

P₂ = 415 kPa or 420 kPa

T₁

P₁

T₂

P₂

+ 273 =

298 K

1201 K

103 kPa

298 K

P₂

1201 K

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The Combined Gas Law

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The Combined Gas Law

There is a single expression, called the combined gas law, that combines Boyle’s, Charles’s, and Gay-Lussac’s Laws:

The amount of gas is held constant

P₁ = initial pressure P₂ = final pressure

V₁ = initial volume V₂ = final volume

T₁ = initial temperature T₂ = final temperature

T₁

P₁V₁

T₂

P₂V₂

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Combined Gas Law - Example

The volume of a gas filled balloon is 30.0 L at 313 K and 153 kPa. What would the volume be at standard temperature and pressure (STP)?

P₁ = P₂ =

V₁ = V₂ =

T₁ = T₂ =

153 kPa

30.0 L

101.3 kPa

( 153 kPa) ( 30.0 L) ( 273 K ) = (101.3 kPa) ( V₂ ) ( 313 K )

1253070 = 31706.9 V₂

V₂ = 39.52041

V₂ = 39.5 L

T₁

P₁V₁

T₂

P₂V₂

( 153 kPa )

313 K

( 101.3 kPa )

273 K

313 K

273 K

( 30.0 L )

( V₂ )

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The Ideal Gas Law

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Ideal Gases

An ideal gas is a theoretical model of a gas where particles are assumed to have no volume and no intermolecular forces, meaning they don't attract or repel each other

In reality, no gases are perfectly ideal because no gas totally obeys all of the gas laws perfectly
The concept of an ideal gas helps us understand and predict the behavior of real gases under many conditions

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Real Gases

A real gas is most like an ideal gas when the real gas is at low pressure and a high temperature

This usually only applies to small mass gases such as hydrogen or helium
At high temperatures, the kinetic energy of the gas molecules is high, making intermolecular forces less significant
At low pressures, the gas particles are far apart, and the volume they occupy is much larger than their actual volume, allowing them to act more independently

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The Ideal Gas Law

The ideal gas law brings together all of the simple gas laws (Boyle’s, Charles’s, Gay-Lussac’s, and Avogadro’s Laws)

PV = nRT

P = pressure

V = volume (in liters)

n = moles

R = the ideal gas law constant

T = temperature (in Kelvin)

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Values for the Ideal Gas Constant - R

0.0821

atm • L
mol • K

8.314

kPa • L
mol • K

62.4

torr • L
mol • K

62.4

mmHg • L
mol • K

How do I know which R value to use?

Look at the unit for the pressure value in the problem.

That pressure unit will match one of the R values above.

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Example Problem 1:

How many moles of hydrogen gas are in a 3.10 L sample of H₂ gas at 300.0 kPa and 20.0°C?

300.0 kPa

3.10 L

8.314 ^{kPa ● L}/_{mol ● K}

P =

V =

n =

R =

T =

20.0^oC

= 293 K

PV = nRT

( 300.0 )( 3.10 ) = ( n )( 8.314 )( 293 )

930 = ( 2436.002 )( n )

n = 0.38177

n = 0.382 mol

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Example Problem 2:

What is the volume that 500.5 g of iodine gas will occupy at 320°C and 745 mmHg?

745 mmHg

1.970 mol

62.4 ^{mmHg ● L}/_{mol ● K}

P =

V =

n =

R =

320^oC

= 593 K

PV = nRT

( 745 )( V ) = ( 1.970 )( 62.4 )( 593 )

745 V = 72896.304

V = 97.84738

V = 98 L or 97.8 L

500.5 g I₂ =

1.970 mol I₂