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Ch 10: Gases

Brown, LeMayAP Chemistry

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10.1: Characteristics of Gases

Particles in a gas are very far apart, and have almost no interaction. Ex: In a sample of air, only 0.1% of the

total volume actually consists of matter.

Gases expand spontaneously to fill their container (have indefinite volume and shape.)

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10.2: Pressure (P)

A force that acts on a given area A

FP

Atmospheric pressure: the result of the bombardment of air molecules upon all surfaces 1 atm = 760 mm Hg

= 760 torr= 101.3 kPa= 14.7 PSI

100 km

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Barometer: measures atmospheric P compared to a vacuum

* Invented by Torricelli in 1643 Liquid Hg is pushed up the closed glass tube by air

pressure

Measuring pressure (using Hg)

Evangelista Torricelli(1608-1647)

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1. Closed-end: difference in Hg levels (h) shows P of gas in container compared to a vacuum

closed

Manometers: measure P of a gas

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Difference in Hg levels (h) shows P of gas in container compared to Patm

2. Open-end:

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10.3: The Gas Laws

Amadeo Avogadro(1776 - 1856)

Robert Boyle(1627-1691)

Jacques Charles(1746-1823)

John Dalton(1766-1844)

Joseph Louis Gay-Lussac(1778-1850)

Thomas Graham(1805-1869)

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10.3: The Gas Laws Boyle’s law: the volume (V) of a fixed quantity (n)

of a gas is inversely proportional to the pressure at constant temperature (T).

2211 VPVP P

1constantV

P

V

1/P

V

Animation: http://www.grc.nasa.gov/WWW/K-12/airplane/aboyle.html

Ex: A sample of gas is sealed in a chamber with a movable piston. If the piston applies twice the pressure on the sample, the volume of the gas will be

. If the volume of the sample is tripled, the pressure of the gas will behalved

reduced to 1/3

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V of a fixed quantity of a gas is directly proportional to its absolute T at constant P.

T

V

Animation: http://www.grc.nasa.gov/WWW/K-12/airplane/aglussac.html

Charles’ law

2

2

1

1

T

V

T

V

TconstantV

Extrapolation to V = 0 is the basis for absolute zero. 

V = 11.5 L2730.1002730.50

0.10 2

VL

Ex: A 10.0 L sample of gas is sealed in a chamber with a movable piston. If the temperature of the gas increases from 50.0 ºC to 100.0 ºC, what will be the new volume of the sample?

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* Questionably named, seen as derivative of C’s and B’s laws

P of a fixed quantity of a gas is directly proportional to its absolute T at constant V.

T

P

“Gay-Lussac’s law”

2

2

1

1

T

P

T

P

TconstantP

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Equal volumes of gases at the same T & P contain equal numbers of molecules

n

V

Avogadro’s hypothesis

nconstantV

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Ex: A 10.0 L sample of gas at 100.0ºC and 2.0 atm is sealed in a chamber. If the temperature of the gas increases to 300.0ºC and the pressure decreases to 0.25 atm, what will be the new volume of the sample?

Combined gas law

2

22

1

11

T

VP

T

VPconstant

PV

T

V2 =120 L)2730.300(

)25.0(

)2730.100(

)00.10)(0.2( 2

VatmLatm

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Used for calculations for an ideal (hypothetical) gas whose P, V and T behavior are completely predictable.

R = 0.0821 L•atm/mol•K= 8.31 J/mol•K

Ex: How many moles of an ideal gas have a volume of 200.0 mL at 200.0ºC and 450 mm Hg?

10.4: The ideal gas law

nRTPV

n = 3.0 x 10-3 mol

)2730.200)(0821.0(1000

0.200

760

450

n

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What is the V of 1.000 mol of an ideal gas at standard temperature and pressure (STP, 0.00°C and 1.000 atm)

nRTPV

V = 22.4 L (called the molar volume)

22.4 L of an ideal gas at STP contains 6.022 x 1023 particles (Avogadro’s number)

)273)(0821.0)(000.1()000.1( KmolVatm

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10.5: More of the ideal gas law

Gas density (d):

Molar mass (M):

RT

PM

V

md

RTPV

mM

nRTPV M

massn

V

md RT

M

mPV

RTM

mPV

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10.6: Gas Mixtures & Partial Pressures

Partial pressure: P exerted by a particular component in a mixture of gases

Dalton’s law of partial pressures: the total P of a mixture of gases is the sum of the partial pressures of each gas

PTOTAL = PA + PB + PC + …

(also, nTOTAL = nA + nB + nC + …)

Ex: What are the partial pressures of a mixture of 0.60 mol H2 and 1.50 mol He in a 5.0 L container at 20ºC, and what is the total P?

=

nRTPV PH2 =(0.60)(0.0821)(293) / 5.0 = 2.9 atm

PHe =(1.50)(0.0821)(293) / 5.0 = 7.2 atm

PT = 2.9 + 7.2 = 10.1 atm

+

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Ratio of moles of one component to the total moles in the mixture (dimensionless, similar to a %)

RTVP

RTVP

T

A

Mole fraction (X):

Ex: What are the mole fractions of H2 and He in the previous example?

TOTAL

AA n

nX TA PP

T

A

n

n

T

A

P

P ∴

TAA PP X

29.02.10

0.60X

2H 714.02.10

1.50XHe

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Collecting Gases “over Water” When a gas is bubbled through water, the

vapor pressure of the water (partial pressure of the water) must be subtracted from the pressure of the collected gas:

PT = Pgas + PH2O

∴ Pgas = PT – PH2O

See Appendix B for vapor pressures of water at different temperatures.

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10.7: Kinetic-Molecular Theory* Formulated by Bernoulli in 1738

Assumptions:1. Gases consist of particles (atoms

or molecules) that are point masses. No volume - just a mass.

2. Gas particles travel linearly until colliding ‘elastically’ (do not stick together).

3. Gas particles do not experience intermolecular forces.

Daniel Bernoulli (1700-1782)

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10.7: Kinetic-Molecular Theory

4. Two gases at the same T have the same kinetic energy

KE is proportional to absolute T

2

2

1 rmsave muKE kTKEave 2

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urms = root-mean-square speedm = mass of gas particle

(NOTE: in kg)k = Boltzmann’s constant,

1.38 x 10-23 J/KLudwig Boltzmann

(1844-1906)

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Maxwell-Boltzmann distribution graph

James Clerk Maxwell(1831-1879)

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kTKEave 2

3

O2 at 273K

O2 at 1000K

H2 at 273K2

2

1 rmsave muKE

Nu

mb

er

at

sp

eed

, u

Speed, u

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10.8: Molecular Effusion & Diffusion Since the average KE of a gas has a specific

value at a given absolute T, then a gas composed of lighter particles will have a higher urms.

m

kTurms

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m = mass (kg)M = molar mass (kg/mol)R = ideal gas law constant, 8.31 J/mol·K

kTmuKE rmsave 2

3

2

1 2

kTmurms 32

M

RT3

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Effusion: escape of gas molecules through a tiny hole into an evacuated space

Diffusion: spread of one substance throughout a space or throughout a second substance

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Graham’s law

The effusion rate of a gas is inversely proportional to the square root of its molar mass

B

A

B

A

u

u

r

r

r = u = rate (speed) of effusiont = time of effusion

B

B

A

A

MRT

MRT

3

3

A

B

t

t

A

B

M

M

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10.9: Deviations from Ideal Behavior

a = correction for dec in P from intermolecular attractions (significant at high P, low T)

b = correction for available free space from V of atoms (significant at high concentrations)

nRTnbVV

anP

2

2

Particles of a real gas:1. Have measurable volumes2. Interact with each other

(experience intermolecular forces)

Van der Waal’s equation:

2

2

V

an

nbV

nRTP

or

Johannes van der Waals(1837-1923)

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10.9: Deviations from Ideal Behavior

A gas deviates from ideal: As the particles get larger (van der Waal’s “b”) As the e- become more widely spread out (van

der Waal’s “a”)

The most nearly ideal gas is He.