Unit 10: Gases - tinamhall.weebly.com

Unit 10: Gases

Unit Outline

I. Introduction

II. Gas Pressure

III. Gas Laws

IV. Gas Law Problems

V. Kinetic-Molecular Theory of Gases

VI. Real Gases

I. Opening thoughts…

Have you ever:

Seen a hot air balloon?

Had a soda bottle spray all over you?

Baked (or eaten) a nice, fluffy cake?

These are all examples of gases at work!

4

Characteristics of Gases• Gases are highly compressible and occupy

the full volume of their containers.

• Gases exert pressure, P = F/A (force/area).

• Gases always form homogeneous mixtures

with other gases.

Properties of Gases

You can predict the behavior of gases

based on the following properties:

Pressure

Volume

Amount (moles)

Temperature

Lets review each of these briefly…

Pressure

Pressure is defined as the force the gas

exerts on a given area of the container in

which it is contained. The SI unit for

pressure is the Pascal, Pa.

• If you’ve ever inflated a tire,

you’ve probably made a

pressure measurement in

pounds (force) per square inch

(area).

Volume

Volume is the three-dimensional space inside

the container holding the gas. The SI unit for

volume is the cubic meter, m3. A more common

and convenient unit is the liter, L.

Think of a 2-liter bottle of soda to get

an idea of how big a liter is.

(OK, how big two of them are…)

Amount (moles)

As we’ve already learned, the SI unit for amount of

substance is the mole, mol. Since we can’t count

molecules, we can convert measured mass to the

number of moles, n, using the molecular or formula

weight of the gas.

By definition, one mole of a substance contains

approximately 6.022 x 1023 particles of the

substance.

Temperature

Temperature is the measurement of heat…or how

fast the particles are moving. Gases, at room

temperature, have a lower boiling point than things

that are liquid or solid at the same temperature.

Remember: Not all substance freeze, melt or

evaporate at the same temperature.

Water will freeze at zero degrees

Celsius. However Alcohol will not

freeze at this temperature.

II. Pressure

• Pressure is simply a force exerted over a

surface area.

760 mm

Hg

Pressure

If a tube is inserted into a container of mercury

open to the atmosphere, the mercury will rise 760

mm up the tube (at sea level).

(at sea level)

Atmospheric Pressure and the Barometer

. Standard atmospheric pressure is the pressure

required to support 760 mm of Hg in a column.

Units:

1 atm = 760 mmHg = 760 torr = 1.01325 105 Pa = 101.325 kPa.

II. Atmospheric Pressure

• Patm is simply the weight

of the earth’s

atmosphere pulled

down by gravity.

• Barometers are used to

monitor daily changes

in Patm.

• Torricelli barometer was

invented in 1643.

II. Units of Pressure

• The derived SI unit for pressure is N/m2,

known as the pascal (Pa).

• Standard conditions for gases (STP)

occurs at 1 atm and 0 °C. Under these

conditions, 1 mole of gas occupies 22.4 L.

How do they all relate?

Some relationships of gases may be easy to predict. Some are more subtle.Now that we understand the factors that affect the behavior of gases, we will study how those factors interact.

16

III. Gas Laws

Robert Boyle

1627-1691.

Boyle’s Law.

Jacques Charles

1746-1823.

Charles’ Law.

J. Charles 1783.

First ascent in

hydrogen balloon.

III. Gas Laws

• A sample of gas can be physically

described by its pressure (P),

temperature (T), volume (V), and

amount of moles (n).

• If you know any 3 of these variables,

you know the 4th.

• We look at the history of how the ideal

gas law was formulated.

Boyle’s Law• This law is named for Charles Boyle, who

studied the relationship between pressure, p, and volume, V, in the mid-1600s.

• Boyle determined that for the same amountof a gas at constant temperature, results in an inverse relationship:

•when one goes up, the othercomes down.

pre

ssu

re

volu

me

19

The Pressures-Volume Relationship: Boyle’s Law

(P vs. V at constant T)

What does Boyle’s Law

mean?

Suppose you have a cylinder with a piston in the

top so you can change the volume. The cylinder

has a gauge to measure pressure, is contained so

the amount of gas is constant, and can be

maintained at a constant temperature.

A decrease in volume will result in increased

pressure.

Boyle’s Law at Work…

Doubling the pressure reduces the volume by half. Conversely, when the volume doubles, the pressure

decreases by half.

NEXTPREVIOUSMAIN

MENU

Breathe Deeply!

It’s Boyle’s Law!

• When the diaphragm contracts,

the volume of the thoracic cavity

increases

• The lungs expand and pressure

decreases. Since Pair>Plungs, air

enters.

• When the diaphragm relaxes,

the volume of the thoracic cavity

decreases.

• The lungs contract and the

pressure in the lungs increases.

Plungs>Pair, so air is exhaled.

III. Volume and Temperature –

Charles’s Law

• The volume of a gas is directly related to its

temperature, i.e. if T is increased, V will

increase.

Charles’ Law• This law is named for Jacques Charles, who

studied the relationship volume, V, and temperature, T, around the turn of the 19th

century.

• This defines a direct relationship: With the same amount of gas he found that as the volume increases the temperature also increases. If the temperature decreases than the volume also decreases.

volu

me

tem

pe

ratu

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Charles’s Law

26

The Temperature-Volume Relationship: Charles’

Law

The Absolute Temperature

Scale• Temperature measures average

Kinetic Energy of particles:

• When motion stops, particles have

no kinetic energy.

• This means there must be an

absolute “zero” temperature!

• The Kelvin temperature scale starts

at Absolute Zero:

3 KE = RT

2

K = C + 273.15

What does Charles’ Law

mean?Suppose you have that same cylinder with a piston

in the top allowing volume to change, and a

heating/cooling element allowing for changing

temperature. The force on the piston head is

constant to maintain pressure, and the cylinder is

contained so the amount of gas is constant.

An increase in temperature results in increased

volume.

Charles’ Law at Work…

As the temperature increases, the volume increases. Conversely, when the temperature decreases, volume

decreases.

III. The Combined Gas Law

• Boyle’s and

Charles’s Laws can

be combined into a

convenient form.

III. Volume and Moles –

Avogadro’s Law

• The pressure of a

gas is directly

related to the

number of moles of

gas, i.e. if n

increases, V will

increase.

equal volumes of any gas at the same temperature and

pressure will contain the same number of molecules.

The Quantity-Volume Relationship: Avogadro’s Law

Same number

of particles

(same T and P)

V = constant n

at a constant P and T

33

V = constant n

at a constant P and T

22.4 L = constant 1 mole

at a 1 atm and 273 K

Avogadro’s Law (n, V)

Gay-Lussac’s Law (T, P)• Avogadro showed that the

volume of a gas varies

directly with the amount

of gas (# of moles)

• Thus, a similar

relationship exists as in

Charles’s Law:

• Gay-Lussac studied how

temperature affects the

pressure of a gas.

• He discovered a direct

relationship!

1 2

1 2

V V V = k or =

n n n

1 2

1 2

P P P = k or = T T T

Pressure & Temperature

are held constant here!

Moles & Volume

are held constant here!

Gas Laws

Summary

BOYLE

CHARLES

AVOGADRO

GAY-LUSSAC

The Combined Gas Law • The gas laws can be combined into one equation.

• Volume and pressure vary inversely, while volume varies directly with moles and temperature:

• When variables are held constant, they can be deleted from the combined law – this produces all four gas laws we studied earlier.

1 1 2 2

1 1 2 2

PV P V = n T n T

Ideal Gas Law, cont’d

• We can rewrite the combined law in a form that is known

as the Ideal Gas Law:

PV = nRT• The value of the Ideal Gas Law over the previous laws is

that only ONE set of conditions is required – if 3 of the

variables are known, the 4th can be calculated.

• Use substitution and some algebra to derive the related

equations from the Ideal Gas Law:

mRT PM M = and d =

PV RT

Standard Molar Volume: 22.4 L @ STP

39

The Ideal Gas Equation

• Summarizing the Gas Laws

P

nTV

P

nTRV

Boyle: V 1/P (constant n, T)

Charles: V T (constant n, P)

Avogadro: V n (constant P, T).

Combined:

Ideal gas equation

R = ideal gas constant

III. The Ideal Gas Law

• The ideal gas law is

a combination of the

combined gas law

and Avogadro’s Law.

R = 0.082058 L atm/K mole

41

The Ideal Gas Equation

• Ideal gas equation: PV = nRT

R = gas constant = 0.08206 L•atm/mol-K.

We define STP (Standard Temperature and Pressure)

= 0C (273.15 K)

= 1 atm.

Volume of 1 mol of gas at STP is 22.4 L.

IV. Gas Law Problems

• There are many variations on gas law

problems.

• A few things to keep in mind:

1) Temperature must be in Kelvin

2) If problem involves a set of initial and final

conditions, use combined gas law.

3) If problem only gives information for one

set of conditions, use ideal gas law.

IV. Sample Problem

• What’s the final pressure of a sample of

N2 with a volume of 952 m3 at 745 torr

and 25 °C if it’s heated to 62 °C with a

final volume of 1150 m3?

IV. Sample Problem

• What volume, in mL, does a 0.245 g

sample of N2 occupy at 21 °C and 750

torr?

IV. Sample Problem

• A sample of N2 has a volume of 880 mL

and a pressure of 740 torr. What

pressure will change the volume to 870

mL at the same temperature?

IV. Other Uses of Ideal Gas Law

• The ideal gas law can be used to find

other physical values of a gas that are

not as obvious.

gas density, d = mass/volume

gas molar mass, MW = mass/mole

stoichiometry, via moles and a balanced

equation

IV. Sample Problem

• Find the density of CO2(g) at 0 °C and

380 torr.

IV. Sample Problem

• How many mL of HCl(g) forms at STP

when 0.117 kg of NaCl reacts with

excess H2SO4?

H2SO4(aq) + 2NaCl(s) Na2SO4(aq) + 2HCl(g)

Dalton’s Law• The total pressure of a mixture

of gases equals the sum of the

partial pressures of the

individual gases.

Ptotal = P1 + P2 + ...When a H2 gas is

collected by water

displacement, the gas in

the collection bottle is

actually a mixture of H2

and water vapor.

Dalton’s Law of Partial

Pressures• In a mixture of gases, the TOTAL pressure of gas is the

sum of the pressures caused by each gas (the partial

pressures):

PTotal = P1 + P2 + P3 + …

• The MOLE FRACTION ()of a gas in a mixture can be

calculated in two different ways, then:

A AA A A Total

Total Total

n P= = so we get: P = P

n P

Dalton’s Law Illustrated

GIVEN:

PH2 = ?

Ptotal = 94.4 kPa

PH2O = 2.72 kPa

WORK:

Ptotal = PH2 + PH2O

94.4 kPa = PH2 + 2.72 kPa

PH2 = 91.7 kPa

B. Dalton’s Law• Hydrogen gas is collected over water at

22.5°C. Find the pressure of the dry gas if the atmospheric pressure is 94.4 kPa.

Look up water-vapor pressure

for 22.5°C.

Sig Figs: Round to least number

of decimal places.

The total pressure in the collection bottle is equal to atmospheric

pressure and is a mixture of H2 and water vapor.

GIVEN:

Pgas = ?

Ptotal = 742.0 torr

PH2O = 42.2 torr

WORK:

Ptotal = Pgas + PH2O

742.0 torr = PH2 + 42.2 torr

Pgas = 699.8 torr

• A gas is collected over water at a temp of 35.0°C

when the barometric pressure is 742.0 torr.

What is the partial pressure of the dry gas?

Look up water-vapor pressure

for 35.0°C.

Sig Figs: Round to least number

of decimal places.

B. Dalton’s Law

The total pressure in the collection bottle is equal to barometric

pressure and is a mixture of the “gas” and water vapor.

V. Kinetic-Molecular Theory (KMT)

Kinetic Molecular Theory

• Particles in an ideal gas…

– have no volume.

– have elastic collisions.

– are in constant, random, straight-line

motion.

– don’t attract or repel each other.

– have an avg. KE directly related to Kelvin

temperature.

Imagining a Sample of Gas

• We imagine a sample of gas –

chaos, molecules bumping into

each other constantly.

• After a collision, 2 molecules

may stop completely until

another collision makes them

move again.

• Some molecules moving really

fast, others really slow.

• But, there is an average speed.

57

Kinetic-Molecular Theory

0oC

100oC

N2

For gases, there is a range of velocities and energies at

each temperature.

Gas Molecular Speeds

• As temp increases,

avg. speed increases.

• i.e. avg. KE is related

to temp!!

• Any 2 gases at same

temp will have same

avg. KE!

Molecular Speeds The average kinetic energy per mole of

gas can be calculated in two different

ways:

We can rearrange and solve for “v”, the

velocity of a gas particle:

In order to get a velocity in “ms-1”, we

must use SI units for molar mass, kg

mol-1.

The gas constant (R) must be the SI

value, 8.314.

3RT v =

M

21 3 KE = Mv = RT

2 2

Don’t forget:

Molar Mass in Kg

for this equation!

Graham’s Law

• Diffusion

– Spreading of gas molecules throughout a

container until evenly distributed.

• Effusion

– Passing of gas molecules through a tiny

opening in a container

Graham’s Law

KE = ½mv2

• Speed of diffusion/effusion

– Kinetic energy is determined by the

temperature of the gas.

– At the same temp & KE, heavier molecules

move more slowly.

• Larger m smaller v

Graham’s Law• Graham’s Law

– Rate of diffusion of a gas is inversely related to the square root of its molar mass.

– The equation shows the ratio of Gas A’s speed to Gas B’s speed.

A

B

B

A

m

m

v

v

• Determine the relative rate of diffusion

for krypton and bromine.

1.381

Kr diffuses 1.381 times faster than Br2.

Kr

Br

Br

Kr

m

m

v

v2

2

A

B

B

A

m

m

v

v

g/mol83.80

g/mol159.80

Graham’s Law

The first gas is “Gas A” and the second gas is “Gas B”.

Relative rate mean find the ratio “vA/vB”.

• A molecule of oxygen gas has an average speed of 12.3 m/s at a given temp and pressure. What is the average speed of hydrogen molecules at the same conditions?

A

B

B

A

m

m

v

v

2

2

2

2

H

O

O

H

m

m

v

v

g/mol 2.02

g/mol32.00

m/s 12.3

vH2

Graham’s Law

3.980m/s 12.3

vH2

m/s49.0 vH 2

Put the gas with

the unknown

speed as

“Gas A”.

• An unknown gas diffuses 4.0 times faster than

O2. Find its molar mass.

Am

g/mol32.00 16

A

B

B

A

m

m

v

v

A

O

O

A

m

m

v

v2

2

Am

g/mol32.00 4.0

16

g/mol32.00 mA

2

Am

g/mol32.00 4.0

g/mol2.0

Graham’s Law

The first gas is “Gas A” and the second gas is “Gas B”.

The ratio “vA/vB” is 4.0.

Square both

sides to get rid

of the square

root sign.

Why is Diffusion so Slow??

• If molecular speeds are so incredibly fast,

why does a gas take so long to diffuse?

• The answer is in the completely random

path a gas particle takes as it diffuses.

• The gas particle constantly changes

direction when it collides with another

particle

• This slows down its outward diffusion

immensely!

• The MEAN FREE PATH is the distance a

particle travels before colliding with

another particle.

Effusion• Gas moving through

a pin-hole into a

vacuum

• The rate of effusion:

• Temp in Kelvin

• Molar mass in “kg

mol-1”

• Rate in “ms-1”

3RT v =

M

Graham’s Law of Effusion

• Graham compared the rates of effusion for two gases at the

same temperature.

• He derived the equation:

• Graham’s law is important because it can be used to

determine the Molar Mass of an unknown gas – if you

compare its rate of effusion with the rate of a known gas

under the same conditions!

1 2

2 1

v M =

v M

Here, Molar Mass can be

left in “grams”

Can you explain why??

Why Do Gas Laws Work So Well?

• Recall that the gas laws apply to any

gas – the chemical identity is not

important.

• Gas particles only interact when they

collide. Since this interaction is so

short, chemical properties don’t have

time to take effect!!

VI. Deviations from PV=nRT

• Under extreme

conditions (high P or

low T), gases

deviate from ideal

gas law predictions.

• Why? What’s so

different about these

conditions?

Real Gases

• Particles in a REAL gas…

– have their own volume

– attract each other

• Gas behavior is most ideal…

– at low pressures

– at high temperatures

– in nonpolar atoms/molecules

72

Real Gases: Deviations from Ideal Behavior(Temperature and Pressure Effects)

• As temperature increases, the gas molecules move faster and

further apart.

• Also, higher temperatures mean more energy available to

break intermolecular forces.

• Therefore, the higher the temperature, the more ideal the gas.

• As pressure increases, gas molecules are closer together making

the gas less ideal.

• Therefore, the lower the pressure, the more ideal the gas.

Gas Particle Volume

• Gas molecules do take up space! When very close

to one another, entire volume of container is not

available for travel, so actual volume of gas is larger.

Intermolecular Forces

• Gas molecules interact if they are very close

to one another…

VI. van der Waals Equation

• Under extreme conditions, ideal gas law

cannot be used.

correction terms for P and V

76

Real Gases: Deviations from Ideal

Behavior

The van der Waals Equation

• We add two terms to the ideal gas equation, one to

correct for volume of molecules, and the other to

correct for intermolecular attractions

• The correction terms generate the van der Waals

equation:

2

2

V

an

nb

V

nRTP

where a and b are empirical constants.