5 Gases Contents 5-1 Gases and Pressure 5-2 Relation Between Pressure and Volume of a Gas 5-3...

5 Gases

Contents

5-1 Gases and Pressure

5-2 Relation Between Pressure and Volume of a Gas

5-3 Relations Between Volume and Temperature

5-4 Standard Temperature and Pressure

5-5 Gay-Lussac’s Law of Combining Volumes and

Avogadro’s Law

5-6 The Ideal Gas Equation and Its Uses

5-7 Dalton’s Law of Partial Pressures

5-8 The Kinetic-Molecular Theory

5-9 Real Gases

Why Study Gases?

1. Many elements and compounds are gases under everyday c

onditions. N2, H2, O2, O3, F2, Cl2, He, Ne, Ar, Kr, Xe, Rn, CO2, C

O, NO2, SO2, NH4, etc.

2. Many chemical reactions involve gases as reactants or

products or both.

3. Earth’s weather is largely the result of changes in the

properties of the mixture of gases called air.

Elements that exist as gases at 250C and 1 atmosphere

5.1

Characteristics of Gases

• Gases occupy containers uniformly and

completely. Gases always form homogeneous

mixtures with other gases

• Gases are highly compressible and occupy the full

volume of their containers.

• Gases can be expanded infinitely. When a gas is

subjected to pressure, its volume decreases.

Gas properties can be modeled using math. Model depends on:

V = volume of the gas (L)

T = temperature (K)

n = amount (moles)

P = pressure (atmospheres)

Characteristics of Gases

•Gas experiments revealed that these four variables will

affect the state of a gas. These variables are related through

equations know as the gas laws.

Units of Pressure

1 pascal (Pa) = 1 N/m2=1kg/(m∙s2)

1 atm = 760 mmHg = 760 torr

1 atm = 101,325 Pa (~105)

Barometer

Pressure = ForceArea

1.0 N is the force required to accelerate 1.0 kg 1.0 m/s2

5-1 Gases and Pressure

Pressure is the force acting on an object per unit area.

The air in Earth’s atmosphere

is attracted to earth by gravity

and pushes against every

surface it touches.

Atmosphere Pressure and The BarometerAtmosphere Pressure and The Barometer

• If a tube is inserted into a container of mercury open to t

he atmosphere, the mercury will rise 760 mm up the tu

be.

• Atmospheric pressure is measured with a 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 P

a = 101.325 kPa.

Atmosphere Pressure and The

Manometer

• The pressures of gases not open t

o the atmosphere are measured in

manometers.

• A manometer consists of a bulb o

f gas attached to a U-tube contain

ing Hg:

– If Pgas < Patm then Pgas + Ph2 = Patm.

– If Pgas > Patm then Pgas = Patm + Ph2.

The Gas Laws: Boyle’s LawThe Gas Laws: Boyle’s Law

The Pressure-Volume Relationship:

• Weather balloons are used as a practical consequence to

the relationship between pressure and volume of a gas.

• As the weather balloon ascends, the volume increases.

• As the weather balloon gets further from the earth’s

surface, the atmospheric pressure decreases.

• Boyle’s Law: At constant temperature, the volume of a

sample is inversely proportional to the pressure of the gas.

• Boyle used a manometer to carry out the experiment.

5-2 Relation Between Pressure and Volume of a Gas

Boyle’s LawBoyle’s Law

• Mathematically:

• A plot of V versus P is a hyperbola.

• Similarly, a plot of V versus 1/P must be a straight line passing through the origin.

• The Value of the constant depends on the temperature and quantity of gas in the sample.

The Pressure-Volume RelationshipThe Pressure-Volume Relationship

PV

1constant constantPV

A sample of chlorine gas occupies a volume of 946 mL a

t a pressure of 726 mmHg. What is the pressure of the

gas (in mmHg) if the volume is reduced at constant temp

erature to 154 mL?

Solution: at constant temperature: P1 x V1 = P2 x V2

P1 = 726 mmHg

V1 = 946 mL

P2 = ?

V2 = 154 mL

P2 = P1 x V1

V2

726mmHg×946mL

154ml= = 4.46×103 mmHg

5.3

• We know that hot air balloons expand when they are

heated.

• Charles’s Law: At constant pressure, the volume of a

sample of a gas is directly proportional to the Kelvin or

absolute temperature.

• Mathematically:

Charles’s LawCharles’s Law

TV constant constantT

V

5-3 Relations Between Volume and Temperature

Plotting Charles’s LawPlotting Charles’s Law

• A plot of V versus T is a straight line.

• When T is measured in C, the intercept on the

temperature axis is -273.15C.

• We define absolute zero, 0 K = -273.15C.

• Note the value of the constant reflects the assumptions:

amount of gas and pressure.

All gases will solidify or liquefy before reaching zero volume.

Variation of gas volume with __________________at constant ________________.

5.3

V T

V = constant x T

V1/T1 = V2/T2T (K) = t (0C) + 273.15

Charles’ Law

Temperature must bein _Kevin._

A sample of carbon monoxide gas occupies 3.20 L at

125 0C. At what temperature will the gas occupy a

volume of 1.54 L if the pressure remains constant?

V1 = 3.20 L

T1 = 398.15 K

V2 = 1.54 L

T2 = ?

T2 = V2 x T1

V1

= =

5.3

V1/T1 = V2/T2

5-4 Standard Temperature and Pressure

Standard Temperature: 0℃ or 273.15K

Standard Pressure: 760mmHg or 1 atm

or 1.01325×105 Pa

STP: standard temperature and pressure

The Quantity-Volume Relationship:• Gay-Lussac’s Law of combining volumes: at constant

temperature and pressure, the volumes of gases involved in

chemical reactions are ratios of small whole numbers.

5-5 Gay-Lussac’s Law of Combining Volumes and

Avogadro’s Law

Avogadro’s LawAvogadro’s Law

• Avogadro’s Hypothesis: equal volumes of gas at the

same temperature and pressure will contain the same

number of molecules.

• Avogadro’s Law: the volume of a gas at constant

temperature and pressure is directly proportional to the

number of molecules of the gas, n.

• Mathematically:

• We can show that 22.4 L of any gas at STP contain 6.02 1023 gas molecules.

nV constant

__________________ Law__________________ Law

V number of moles (n)

V = constant x n

V1/n1 = V2/n2

5.3

Constant _________

Constant _________

Ammonia burns in oxygen to form nitric oxide (NO)

and water vapor. How many volumes of NO are

obtained from one volume of ammonia at the same

temperature and pressure?

4NH3 + 5O2 4NO + 6H2O

__ mole NH3 __ mole NO

At constant T and P

__ volume NH3 __ volume NO

5.3

• Consider the three gas laws.

• We can combine these into a general gas law:

• Boyle’s Law:

• Charles’s Law:

• Avogadro’s Law:

The Ideal Gas EquationThe Ideal Gas Equation

), (constant 1

TnP

V

), (constant PnTV

),(constant TPnV

P

nTV

• If R is the constant of proportionality (called the gas constant), then

• The ideal gas equation is:

The Ideal Gas ConstantThe Ideal Gas Constant

P

nTRV

nRTPV

Kmol

J

Kmol

atmLR

314.808206.0

5-6 The Ideal Gas Equation and Its Uses

• We define STP (standard temperature and pressure) =

0C, 273.15 K, 1 atm.

• Volume of 1 mol of gas at STP is:

Applying The Ideal Gas EquationApplying The Ideal Gas Equation

L 41.22

atm 000.1K 15.273))/(08206.0(mol 1 KmolatmL

PnRT

V

Argon is an inert gas used in light bulbs to retard the va

porization of the filament. A certain light bulb containing

argon at 1.20 atm and 180C is heated to 850C at consta

nt volume. What is the final pressure of argon in the lig

ht bulb (in atm)?

PV = nRT n, V and R are _________________

nRV

= PT

= constant

P1

T1

P2

T2

=

P1 = 1.20 atm

T1 = 291 K

P2 = ?

T2 = 358 K

P2 = P1 x T2

T1

= 1.20 atm x 358 K291 K

= _________ atm

5.4

• For an ideal gas, calculate the following quantities:

• (a) the pressure of the gas if 1.04 mol occupies 21.8 L at 2

5 oC;

• (b) the volume occupied by 6.72 x 10-3 mol at 265 oC and p

ressure of 23.0 torr;

• (c) the number of moles in 1.50 L at 37 oC and 725 torr;

• (d) the temperature at which 0.270 mol occupies 15.0 L a

t 2.54 atm.

Relating the Ideal-Gas Equation and the Gas LawsRelating the Ideal-Gas Equation and the Gas Laws

• If PV = nRT and n and T are constant, then PV = constant and we h

ave Boyle’s law.

• Other laws can be generated similarly.

• In general, if we have a gas under two sets of conditions, then

22

22

11

11

Tn

VP

Tn

VP

• A sample of argon gas is confined to a 1.00-L tank at 27.

0 oC and 1 atm. The gas is allowed to expand into a large

r vessel. Upon expansion, the temperature of the gas drop

s to 15.0 oC, and the pressure drops to 655 torr. What is

the final volume of the gas?

• Density has units of mass over volume.

• Rearranging the ideal-gas equation with M as molar mass

we get

Molar MassMolar Mass

RT

Pd

V

nRT

P

V

n

nRTPV

MM

• The molar mass of a gas can be determined as follows:

.

Gas DensitiesGas Densities

P

dRTM

• What is the density of carbon tetrachloride vapor at 714 t

orr and 125 oC?

Class Guided Practice ProblemClass Guided Practice Problem

Volumes of Gases in Chemical ReactionsVolumes of Gases in Chemical Reactions

• The ideal-gas equation relates P, V, and T to number of

moles of gas.

• The n can then be used in stoichiometric calculations

• The safety air bags in automobiles are inflated by nitroge

n gas generated by the rapid decomposition of sodium azi

de, NaN3:

2 NaN3(s) 2 Na(s) + 3N2(g)

If an air bag has a volume of 36 L and is filled with nitro

gen gas at a pressure of 1.15 atm at a temperature of 26 o

C, how many grams of NaN3 must be decomposed?

Class Guided Practice ProblemClass Guided Practice Problem

Density (d) Calculations

d = mV =

PMRT

Molar Mass (M ) of a Gaseous Substance

dRTP

M =

5.4

How do we arrive at these equations??

Let’s see…

PV = nRT(1)

(2)

(3)

(4)

(5)

P = nRTV

Divide by V

Divide by RT

PRT

nV

=

mM

=

=nn = number of molesm = mass in gramsM = molar mass

sub. (4) into (3)

mMV

PRT Multiply by M

(7) = mV

(8)

(9) =MdRT

P

Multiply by RT anddivide by P in order toSolve for M

(6) =mV

PMRT

Since d and =mV

PMRT

d = PMRT

Density (d) Calculations

d = mV =

PMRT

m is the mass of the gas in g

M is the molar mass of the gas

Molar Mass (M ) of a Gaseous Substance

dRTP

M = d is the density of the gas in g/L

5.4

Gas Mixtures and Partial PressuresGas Mixtures and Partial Pressures

• Since gas molecules are so far apart, we can assume they behave

independently.

• Dalton’s Law: in a gas mixture the total pressure is given by the

sum of partial pressures of each component:

• Each gas obeys the ideal gas equation:

• Combining the equations we get:

321total PPPP

V

RTnP ii

V

RTnnnP 321total

5-7 Dalton’s Law of Partial Pressures

Dalton’s Law of ___________________

V and T are

constant

P1 P2 Ptotal = P1 + P2

5.6

Consider a case in which two gases, A and B, are in a container of volume V.

PA = nART

V

PB = nBRT

V

nA is the number of moles of A

nB is the number of moles of B

PT = PA + PB XA = nA

nA + nB

XB = nB

nA + nB

PA = XA PT PB = XB PT

Pi = Xi PT

5.6

X is the mole fraction

A sample of natural gas contains 8.24 moles of CH4, 0.

421 moles of C2H6, and 0.116 moles of C3H8. If the tot

al pressure of the gases is 1.37 atm, what is the partial

pressure of propane (C3H8)?

Pi = Xi PT

Xpropane = 0.116

8.24 + 0.421 + 0.116

PT = 1.37 atm

= __________

Ppropane = 0.0132 x 1.37 atm = __________ atm

5.6

Collecting Gases over Water

• It is common to synthesize gases and collect them by

displacing a volume of water.

• To calculate the amount of gas produced, we need to

correct for the partial pressure of the water:

Collecting Gases over Water

watergastotal PPP

• Theory developed to explain gas behavior.

• Theory of moving molecules.

• Assumptions:

– Gases consist of a large number of molecules in constant

random motion.

– Volume of individual molecules negligible compared to

volume of container.

– Intermolecular forces (forces between gas molecules)

negligible.

5-8 The Kinetic-Molecular Theory

• Assumptions:

– Energy can be transferred between molecules, but total kinetic

energy is constant at constant temperature.

– Average kinetic energy of molecules is proportional to

temperature.

• Kinetic molecular theory gives us an understanding of

pressure and temperature on the molecular level.

• Pressure of a gas results from the number of collisions per

unit time on the walls of container.

Kinetic Molecular Theory

• Magnitude of pressure

given by how often and

how hard the molecules

strike.

• Gas molecules have an

average kinetic energy.

• Each molecule has a

different energy.


• As kinetic energy increases, the velocity of the gas

molecules increases.

• Root mean square speed, v, is the speed of a gas molecule

having average kinetic energy.

• Average kinetic energy, E, is related to root mean square

speed:


2

21 mvE

• As volume increases at constant temperature, the average

kinetic of the gas remains constant. Therefore, u is

constant. However, volume increases so the gas

molecules have to travel further to hit the walls of the

container. Therefore, pressure decreases.

• If temperature increases at constant volume, the average

kinetic energy of the gas molecules increases. Therefore,

there are more collisions with the container walls and the

pressure increases.

Application to Gas Laws

• From the ideal gas equation, we have

• For 1 mol of gas, PV/RT = 1 for all pressures.

• In a real gas, PV/RT varies from 1 significantly.

• The higher the pressure the more the deviation from ideal

behavior.

Real Gases: Deviations from Ideal Behavior

nRT

PV

5-9 Real Gases

• From the ideal gas equation, we have

• For 1 mol of gas, PV/RT = 1 for all temperatures.• As temperature increases, the gases behave more ideally.• The assumptions in kinetic molecular theory show where

ideal gas behavior breaks down:– the molecules of a gas have finite volume;

– molecules of a gas do attract each other


nRT

PV

• As the pressure on a gas increases, the molecules are

forced closer together.

• As the molecules get closer together, the volume of the

container gets smaller.

• The smaller the container, the more space the gas

molecules begin to occupy.

• Therefore, the higher the pressure, the less the gas

resembles an ideal gas.


• As the gas molecules get closer together, the smaller the intermolecular distance.


• The smaller the distance between gas molecules, the

more likely attractive forces will develop between the

molecules.

• Therefore, the less the gas resembles and ideal gas.

• 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.


nRTnbVVan

P ))((2

2

Correction for attractive force

between molecules

Correction for volume of molecules

5 Gases Contents 5-1 Gases and Pressure 5-2 Relation Between Pressure and Volume of a Gas 5-3...

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