Chapter 11: Properties of Gases

• Gases have a number of properties that are very different from liquids and solids:

1) Gases are compressible

2) Gases exert a pressure

3) Gas pressure depends on the amount of confined gas

4) Gases fill their container

5) Gases mix freely with each other

6) Gas pressure increases with temperature

• This simple model of gases is the basis of the kinetic-molecular theory (discussed in Section 7.2)

The qualitative observations of the properties of gases leads to the conclusion that a gas is comprised of widely spaced molecules in rapid motion. Collisions of molecules with the walls are responsible for the gas pressure.

• Recall the pressure is a force per unit area

• The earth exerts a gravitational force on everything with mass near it

• What we call weight is the gravitational force acting on an object

• The pressure due to air molecules colliding with an object is called the atmospheric pressure

area

forcepressure

Atmospheric pressure is measured with a barometer. A Torricelli barometer consists of a glass tube sealed at one end, about 80 cm in length. The tube is filled with mercury, capped, inverted, and the capped end immersed in a pool of mercury. When the cap is removed the atmosphere supports a the column of mercury about 760 mm high.

• The height of the mercury column varies with altitude

• The average pressure at sea level or the standard atmosphere (atm) was defined as the pressure needed to support a column of mercury 760 mm high measures at 0oC

• The SI unit of pressure is the pascal (Pa)

(exactly) Pa 101,325 atm 1

m N 1 m 1

N 1 Pa 1 2

2

• You may encounter a number of pressure units

• The standard atmosphere is

• Chemical reactions often involve gases

2

o

in lb 14.7

mb 1013 bar 1.013

kPa 101.325 Pa 101,325

torr760

C)0at (measured Hg mm 760

• A manometer is used to measure the pressure inside closed containers

Open-end manometer. (a) The pressure of the trapped gas, Pgas equals the atmospheric pressure, Patm. Trapped gas pressure (b) higher and (c) lower than atmospheric pressure.

A closed-end manometer for measuring gas pressures less than 1 atm. When constructed (a) the tube is fully evacuated and mercury is allowed to enter and fill the closed arm completely. (a) Mercury flows out of the closed arm when the bulb contains gas at low pressure. The difference in mercury levels, PHg, is the pressure of the confined gas, Pgas.

• Mercury is so dense (13.6 g mL-1) that small pressure changes are difficult to measure

• Other liquids can be used to make manometers

Columns of mercury and water that exert the same pressure. Mercury is 13.6 times more dense than water. Both columns have the same weight and diameter, so they exert the same pressure.

• Thus for a given difference in pressure, the difference in heights between the two levels is inversely proportional to the density of the liquid used in the manometer

• There are four variables that affect the properties of a gas: pressure, volume, temperature, and the amount of the gas

• Simple experiments can be conducted that relate how these variables change

• The gas laws summarize these experiments

Compressing a gas increases its pressure. A molecular view of what happens when a gas is squeezed into a smaller volume. The number of collisions with a given area of the walls increases which causes the pressure to rise.

Robert Boyle studied how the volume of a fixed amount of gas varies with pressure at constant temperature. (a) Air trapped in a J-tube by mercury. (b) As more mercury is added, the pressure of the trapped gas increases and the volume decreases.

(a) A typical graph of volume versus pressure showing volume decreasing as pressure increases. (b) A straight line is obtained when volume is plotted against (1/P), which shows that

PV

1

• The relationship between volume and pressure is called Boyle’s law or the pressure-volume law – The volume of a given amount of gas held at

constant temperature varies inversely with the applied pressure

• The proportionality can be removed by introducing a proportionally constant, C

re) temperatuandamount (constant or 1

CPVCP

V

• Boyle’s law is remarkably successful, especially for common laboratory conditions

• However, no real gas obeys Boyle’s law exactly over a wide range of temperatures and pressures

• The hypothetical gas that does exactly obey Boyle’s law is called an ideal gas

• Real gases act more like ideal gases as their pressures decrease and temperatures increase

• Jacques Alexander Charles studied how the volume of a gas sample varied with temperature Charles’ law plots.

Each line shows how the gas volume changes with temperature for different sized samples of the same gas.

• When extrapolated to zero volume all the samples have the same temperature:

• This temperature is called absolute zero and is the basis of the Kelvin temperature scale:

• Charles’ law or the temperature-volume law can be expressed mathematically

C15.273 o

15.273CK o TT

• Joseph Louis Gay-Lussac studied how the pressure and temperature of a fixed amount of gas at constant volume are related

• Gay-Lussacs’ law or the pressure-temperature law states:– The pressure of a fixed amount of gas held at

constant volume is directly proportional to the Kelvin temperature

• Mathematically this is

pressure) andamount (constant 'or TCVTV

• The three gas laws are often used in a single equation called the combined gas law

• When using this equation the temperature must always be in kelvins

• Alternate forms of the previous gas laws result when certain variables cancel

volume)andamount (constant "or TCPTP

amount) (fixed PP

or constant 2

22

1

11

T

V

T

V

T

PV

• Problems involving the gas laws are important

)(when //

:Law sLussac'-Gay

)(when //V

:Law Charles'

)(when

:Law sBoyle'

212211

212211

212211

VVTPTP

PPTVT

TTVPVP

Example: What will be the the final pressure of a sample of oxygen with a volume of 850 m3 at 655 torr and 25.0oC if it is heated to 80.0oC and given a final volume of 1066 m3?

ANALYSIS: Use the combined gas law with temperature in kelvins.

SOLUTION:

torr619

273.2)K(25.0

K)2.2730.80(

m 1066

m 850 torr655

3

3

1

2

2

112

T

T

V

VPP

• The law of combining volume states: – When gases react at the same temperature and

pressure, their combining volumes are in ratios of simple, whole numbers

• Example:

• Amedeo Avogadro studied this and devised Avogadro’s principle: – When measured at the same temperature and

pressure, equals volumes of gases contain equal number of moles

volumes2 volume1 volume1

chloridehydrogen chlorinehydrogen

• A corollary to Avogadro’s principle is:– The volume of a gas is directly proportional to

its number of moles, n

• Thus, the volume of one mole of any gas at standard temperature and pressure (STP) or 0oC and 1 atm is 22.4 L (a constant for all ideal gases)

• This is called the standard molar volume of a gas

) and constant (at PTnV

• The combined gas law can be generalized to include changes in the number of moles of sample

• The ideal gas law is

K mol

L atm 0.0821

constant gas universal

R

nRTPV

• The molecular mass is obtained by taking the ratio of mass to moles, which could be determined using the ideal gas law

• Gas densities (d), a ratio of gas mass to volume, can be calculated by taking the ratio of the molar mass to molar volume– Example: The molar mass of oxygen is 32.0

g/mol. What is the density of oxygen at STP?

1O L g 42.1

L/mol 22.4

g/mol 32.02

d

• We now need to consider mixtures of gases• One useful way to describe a composition

of a mixture is in terms of its mole fractions • The mole fraction is the ratio of the

number of moles of a given component to the total moles of all components

One mole of each gas occupies 22.4 at STP. Carbon dioxide is more dense that oxygen due to molar mass differences.

• For a mixture of A, B, … substances, the mole fraction of substance i (Xi) is

• This provides a convenient way to `partition’ the total pressure of a mixture of gases

• Dalton’s law of partial pressures states: the total pressure of a mixture of gases is the sum of their individual partial pressures

innnn

nX i

ZBA

AA of moles ,

...

• For a system of only gases, mole fractions and partial pressure partition the total pressure in the same fashion

• Gases are often collected over water in the laboratory

• These (collected) gases are saturated with water

...1

....

....

BA

BTotalATotal

BATotal

XX

XPXP

PPP

• The space above any liquid contains some of the liquid’s vapor

• The pressure this vapor exerts is called the vapor pressure

As the gas bubbles through the water, water vapor gets into the gas so the total pressure inside the bottle includes the partial pressure of the water vapor.

• The total pressure is the pressure of the gas plus the vapor pressure of water

760.0 100

55.32 40

31.82 30

17.54 20

4.579 0

(torr) PressureVapor C)( eTemperatur o

Vapor pressure of water at various temperatures.

aporwatertotalgas

aporwatergastotal

PPP

PPP

v

v or

Example: A sample of oxygen is collected over water at 20oC and a pressure of 738 torr. What is the partial pressure of oxygen?

ANALYSIS: The partial pressure of oxygen is less than the total pressure. Get the vapor pressure of water from table 11.2 (page 478).

SOLUTION:

• Partial pressures can be used to calculate mole fractions

torr720. torr )54.17738(

11.2) Table (from torr 54.17

gas

vaporwater

P

P

• This is possible because the number of moles of each gas is directly proportional to its partial pressure

• Using the ideal gas equation for each gas

• For a given mixture of gases, the volume and temperature is the same for all gases

• Using C=V/RT gives

RT

VPn A

A

• The partial pressure of a gas can be calculated using the total pressure and mole fraction

total

A

ZBA

A

ZBA

AA

P

P

PPP

P

CPCPCP

CPX

totalAA PXP

• Gas volumes can be used in stoichiometry problems

)(OH moles 2)(O mole 1

)(OH moles 2)(H moles 2

)(O mole 1)(H moles 2

asjust

)(OH volumes2)(O volumes1

)(OH volumes2)(H volumes2

)(O volume1)(H volumes2

O(g)H2)(O)(H2

22

22

22

22

22

22

pressure) and re temperatu(same volumes2 volume1 volumes2

222

gg

gg

gg

gg

gg

gg

gg

• Diffusion is the spontaneous intermingling of the molecules of one gas with another

• Effusion is the movement of gas molecules through a tiny hole into a vacuum

• The rates of both diffusion and effusion depend on the speed of the gas molecules

• The faster the molecules, the faster diffusion and effusion occur

• Thomas Graham studied effusion

• He found that the effusion rate of a gas was inversely proportional to the square root of the density (d)

• This is known as Graham’s law

• Where Mi is the molar mass of species i

A

B

A

B

M

M

d

d

B

A

TP

)( rateeffusion

)( rateeffusion

) and (constant d

1 rateeffusion

• The behavior of ideals gases can be explained

(a) Diffusion (b) Effusion

Postulates of the Kinetic Theory of Gases1) A gas consists of a large number of tiny

particles that are in constant, random motion.

2) The gas particles themselves occupy a net volume so small in relation to the volume of their container that their contribution to the total volume can be ignored.

3) The collisions between particle and with the walls of the container are perfectly elastic.

• Calculations show that PV is proportional to the average kinetic energy and the Kelvin temperature, thus

• The kinetic theory also explains the gas laws

KEmolecular average

KEmolecular average

T

TPV

PV

The kinetic theory and the pressure volume law (Boyle’s law). When the gas volume is made smaller going from (a) to (b), the frequency of collisions per unit area of the containers’ wall increases. Thus the pressure increases.

The kinetic theory and the pressure-temperature law (Gay-Lussac’s law). The pressure increases from (a) to (b) as measured by the amount of mercury that must be added to maintain a constant volume.

The kinetic theory and the temperature-volume law (Charles’ law). The pressure is the same in both (a) and (b). At higher temperatures the volume increases because the gas molecules have higher velocities.

• Graham’s law requires that the rate of effusion for the different gases be compared at the same temperature and pressure

• When different gases have the same temperature, they have the same average kinetic energy

• The average kinetic energy can be expressed in terms of the average of the velocities squared or root mean square

• For the two gases labeled 1 and 2

• Note that heavier gases move slower than lighter gases

or with

grearrangin 2

1

2

1

1

2

1

2

2

1

2

1

2

22

21

222

211

21

M

M

m

m

v

v

vvm

m

v

v

vmvm

KEKE

rms

rms

irmsi

• The rate of effusion is proportional to the average molecular speed, thus

• The kinetic theory predicts that absolute zero is the temperature at which the average kinetic energy of an ideal gas is zero

• Real gases exhibit non-ideal behavior

1

2

2

1

2) (gas rateeffusion

1) (gas rateeffusion

M

M

v

v

rms

rms

• Deviations from ideal behavior occur because:

1) Gas molecules interact, and

2) Gas molecules occupy a finite volume.

Deviation from the ideal gas law. A plot of PV/T versus P for an ideal gas is a straight line. The same plot for oxygen is not a straight line

• J. D. van der Waals corrected the ideal gas equation in a simple, but useful, way

(a) In an ideal gas, molecules would travel in straight lines. (b) In a real gas, the paths would curve due to the attractions between molecules.

• He did this by modifying the measured pressure and volume of a real gas so it fits the ideal gas equation

• The constants a and b are called the van der Waals constants

valuegas ideal to measured reduces :

valuegas ideal toup measured brings : 2

2

2

2

Vnb

PV

an

nRTnbVV

anP

measured

measuredmeasured

measured

• Table 11.3 (page 493) has a more complete set of van der Waals constants

0.03049 5.464 OH Water,

0.03707 4.170 NH Ammonia,

0.02661 0.02444 H Hydrogen,

0.01709 0.2107 Ne Neon,

0.02370 0.03421 He Helium,mol L

mol atmL

Substance

2

3

2

122

ba