Kinetic Theory (Gas Laws)

Chapter 16

Atomic Mass Unit

• Uses Carbon-12 as standard • 1 atom of 12 C masses 12.000 u

1 u = 1.66 X 10 -27 kg

Review of Moles

• 1 mole = 6.022 X 10 23 atoms/molecules – GMA – GMMA • What is the molar mass of nitrogen, N 2 ?

• What is the molar mass of BaCl 2 ?

• How many moles are in 132 grams of CO 2 ?

• How many atoms are in a 200 gram sample of iron?

States of Matter

They wander in random patterns quite close to one another.

Can “wiggle” in place (these are the wiggle lines)

Plasma

• 4 th state of matter • Ionized gases – Electrons are removed from the atoms – Positive

ions

remain • Present in: – Stars – Lightning – Arc welding • Most common state of matter in the universe

Plasma

A hydrogen and helium plasma such as you would find in the sun

He + H + e He + e H + e e e H + e He e He + e H + +

Temperature

• Measure of the average molecular motion of a group of atoms/molecules

Conversion Formulas F = 1.8 (

o

C) + 32 K = C + 273 C = K – 273

Temperature

• • • • • Absolute Zero All atomic and molecular motion stops Coldest possible temperature Never reached absolute zero Liquid Nitrogen = 77 K (-196 Dry Ice = 216 K (-56.6 o C) o C)

102

o

F -10.0

o

C 25

o

C 177 K 310

o

C

    

o C o F K o C K

102

o

F -10.0

o

C 25

o

C 177 K 310

o

C

    

39 o C 14.0 o F 298 K -96 o C 583 K

Kinetic Molecular Theory

1. A gas is composed of small particles (molecules) that are spaced widely apart.

– Compressible – Low density - about a 1000 times less dense than a liquid 2. The molecules of a gas are in rapid, constant motion – Pressure – the force of the molecules hitting the side of a container

3. All collisions are elastic – Molecules don’t lose any energy when they collide.

4. Gas molecules have little/no attractive force on one another.

– Too far apart – Mix thoroughly – unlike oil and water (too far apart for polar/non-polar forces to matter)

5. The temperature of a gas is directly proportional to average kinetic energy of the molecules.

KE = 3kT 2 k = Boltzmann’s constant =

1.38 X 10 -23 J/K

Kinetic Molecular Theory: Ex 1

What is the average KE of molecules in a gas at 37 o C?

T = 273 + 37 = 310 K KE = 3kT 2 KE = (3/2)(1.38 X 10 -23 J/K)(310 K)=

6.42 X 10 -21 J

(this is

per

molecule)

Kinetic Molecular Theory: Ex 2

What is the average KE of molecules in a gas at 100 o C?

ANS:

7.72 X 10 -21 J

P 1 V 1 = n 1 RT 1 P 2 V 2 Solve both equations for R = n 2 RT 2 R = P 1 V 1 n 1 T 1 R = P 2 V 2 n 2 T 2 P 1 V 1 n 1 T 1 = P 2 V 2 n 2 T 2

• See what you can cross out (what you are not told) • Remember to convert to Kelvin and moles if needed.

Boyle’s Law

• Boyle’s Law – The pressure and volume of a gas are

inversely

related • Bicycle pump example – Piston down – low volume, high pressure – Piston up – high volume, low pressure

Example: The volume of a car’s cylinder is 475 mL at 1.05 atm. What is the volume when the cylinder is compressed and the pressure is 5.65 atm?

P 1 V 1 = P 2 V 2 n 1 T 1 n 2 T 2

Collapses to: P 1 V 1 = P 2 V 2 (Answer: 88.3 mL)

Example: 2. A weather balloon has a volume of 40.0 liters on the surface of the earth at 1.00 atm. What will be the volume at 0.400 atm as it rises?

P 1 V 1 = P 2 V 2 n 1 T 1 n 2 T 2

Barometer

• Torricelli (1643) • Height of column stayed about 760 mm (760 torr) • The higher the elevation, the lower the mercury • Weather – Rising pressure – calm weather – Dropping pressure – storm (fast moving air)

Charles Law

• Charles Law – The temperature and volume of a gas are

directly

related – “HOTTER = BIGGER” – Can be used to find absolute zero – Temperature

must

be in Kelvin

-300

Volume as a function of Temperature at Constant pressure

-200 45 40 35 30 25 20 15 10 5 0 -100 0 100

Temperature (Celsius)

200 300

45 40 35 30 25 20 15 10 5 0 0

Volume as a function of Temperature at Constant pressure

100 200 300

Temperature (K)

400 500 600

1. A basketball has a volume of 12.0 L when blown up at 25.00 o C. What will be the volume if it is taken outside on a day when it is only 5.00 o C?

P 1 V 1 = P 2 V 2 n 1 T 1 n 2 T 2

Collapses to: V 1 T 1 = V 2 T 2

2. If a tire contains 30.0 L of air at 10.0 o C, what volume will it occupy when it is driven and warms up to 50.0 o C?

Guy-Lussac’s Law

Gay-Lussac’s Law = The temperature and pressure of a gas are directly related.

– Temperature

must

be in Kelvin 1. Gas in a spray can has a pressure of 5.00 atm at 25.0 o C. What will be the pressure at 400.0 o C?

P 1 V 1 = P 2 V 2 n 1 T 1 n 2 T 2

Avagadro’s Law

Avagadro’s Law = The volume of a gas is directly proportional to the moles present – “MORE = BIGGER” 1. A balloon has a volume of 1.00 L when 50.0 grams of N 2 are in the balloon. What is the volume if an additional 25.0 grams of N 2 are added?

Putting it all together

• Often you change more than one thing at a time.

• Ex: In a car, volume, temperature, and pressure may change.

1. The volume of 0.0400 mol of a gas is 500.0 mL at 1.00 atm and 20.0 o C. What is the volume at 2.00 atm and 30.0

o C?

2. The gauge pressure in a tire is 200 kPa at 10 o C. After driving, the temperature rises to 40 o C. What will be the new gauge pressure? (Remember to add 101.3 kPa to the gauge pressure to get absolute pressure)

The Ideal Gas Law

• Works very well in situations close to Earth’s pressures and temperatures • Does not work for “extreme” situations (Jupiter’s atmosphere is too cold and too dense)

PV = nRT – P = pressure in atmosphere – V = volume in Liters – n = number of moles – T = Temperature in Kelvin – R = gas constant

R = 8.31 J/ mol-K

STP

Standard Temperature & Pressure – Standard Temperature = 0 o C (273 K) – Standard Pressure = 1.013 X 10 5 N/m 2 atm) (101.3 kPa, 1

The Ideal Gas Law

Examples: 1. What is the volume of 1.00 mole of a gas at STP?

2. What is the

mass

of oxygen in a container at STP that has a volume of 10.0 m 3 ?

3. A helium balloon has a radius of 18.0 cm. How many moles and grams of helium are needed to fill the balloon at 20 o C and 1.05 atm? (V = 4/3 p r 3)

The Ideal Gas Law

4. Estimate the number of molecules you exhale in one breath at STP.

Three Processes

1. Constant Volume (isochoric) 1. Vertical Line on PV diagram 2. No work done 3. Pressure cooker 2. Constant Pressure (isobaric) 1. Horizontal line on PV graph 2. Work done 3. Constant Temperature (isothermal) 1. Hyperbola curve on PV graph

Stop here in Knight’s Text

Graham’s Law of Diffusion

• Gases mix to fill their volume evenly • Graham’s Law of Diffusion – the speed of a gas is inversely proportional to its molar mass • The larger the molar mass, the slower the gas molecule

Graham’s Law Example

At the same temperature, which moves faster, an He atom or an N 2 molecule?

Calculating Average Speed

• Root-mean-square velocity v rms = 3kT m • Heavier molecules are slower • Temperature increases speed Molar mass

Average Speed: Example 1

What is the rms speed of one O 2 molecule at 20 o C?

First we need the mass of one O 2 in kilograms (32 u)(1.67 X 10 -27 kg) = 5.3 X 10 -26 kg v rms = (3)(1.38 X 10 -23 J/K)(293 K) (5.3 X 10 -26 kg) ½ v rms = 480 m/s (about 1000 mph)

Average Speed: Example 2

What is the rms speed of one N 2 molecule at 20 o C?

ANS: 510 m/s (about 1100 mph)

Relative Humidity

•Vapor exists above all liquids •Even solids have a vapor pressure •Saturated vapor pressure depends on temperature •When saturated vapor pressure exceeds atmospheric pressure, boiling occurs

Vapor pressure at 100 o C is now the same as atmospheric pressure

Our Atmosphere

99% N 2 and O 2 – 78% N 2 – 21% O 2 – 1% CO 2 Gases and the Noble

80 70 60 50 40 30 20 10 0 Gas Nitrogen Oxygen Carbon dioxide and Noble Gases

Relative Humidity

• Partial Pressure - Pressure caused by one component of the atmosphere

Rel Humidity = partial pressureH 2 O X 100 saturated vapor pressure H 2 O

Relative Humidity: Example 1

On a hot day, the temperature is 30 o C and the partial pressure of water vapor is 21.0 torr. What is the relative humidity?

21 torr X 100 = 66% 31.8 torr

Relative Humidity: Example 2

What is the air pressure at a place where the water boils at 95 o C?

ANS: About 643 torr

Relative Humidity: Example 3

On a given day, the relative humidity is 73%. What is the temperature if you assume that the air pressure is normal?

ANS: Above 90 o C

Thermal Expansion

• Most substances expand when heated • Do not all expand to the same degree • Bimetallic strips often used in thermometers

D L = a L o D T D L a L o D T - Change in length - Coefficient of linear expansion - Original Length - Change in temperature ( o C or K)

Thermal Expansion: Example 1

Why does running hot water sometimes help you open a glass jar with a metal lid?

Thermal Expansion: Example 2

A circular hole is cut in a cookie sheet. When the sheet is baked, will the hole expand or contract?

(ANS: Expands)

Thermal Expansion: Example 3

A steel ( a =12 X 10 -6 ) bridge is 200 m long at 20 o C. How long will it be at 40 o C? At -30 o C?

(ANS: 4.8 cm longer, 12 cm shorter)

Thermal Expansion: Example 4

An iron ring must fit snugly on an iron rod. The ring starts with a diameter of 6.420 cm at 20 o C, and must expand to 6.453 cm. To what temperature should the rod be heated?

(ANS: 450 o C)