Chemistry: The Central Science

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Chemistry: The Central Science

Fourteenth Edition

Chapter 10

Gases

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Characteristics of Gases

There are 5 important properties of gases:

• Gases have an indefinite shape

• Gases have low densities

• Gases can compress

• Gases can expand

• Gases mix completely with other gases in the same container

Let’s take a closer look at these properties.

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Some Common Gases

Table 10.1 Some Common Compounds That Are Gases at Room Temperature

Formula Name Characteristics

HCN Hydrogen cyanide Very toxic, slight odor of bitter almonds

H sub 2 S Hydrogen sulfide Very toxic, odor of rotten eggs

CO Carbon monoxide Very Toxic, colorless, odorless

C O sub 2 Carbon dioxide Colorless, odorless

C H sub 4 Methane Colorless, odorless, flammable

C sub 2 H sub 4 Ethene (Ethylene) Colorless, ripens fruit

C sub 3 H sub 8 Propane Colorless, odorless, bottled gas

N sub 2 O Nitrous oxide Colorless, sweet odor, laughing gas

N O sub 2 Nitrogen dioxide Toxic, red-brown, irritating odor

N H sub 3 Ammonia Colorless, pungent odor

S O sub 2 Sulfur dioxide Colorless, irritating odor

H S2

CO2

CH4

C H2 4

C H3 8

N O2

NO2

NH3

SO2

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Properties That Define the State of a Gas Sample

1) Temperature 2) Pressure

3) Volume

4) Amount of gas, usually expressed as number of moles – Having already discussed three of these, we need to

define pressure.

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Chapter 11 5

Gas Pressure

Gas pressure is the result of constantly moving gas molecules striking the inside surface of their container.

–The more often the molecules collide with the sides of the container, the higher the pressure.

– The higher the temperature, the faster gas molecules

move.

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Atmospheric Pressure

• Atmospheric pressure is a result of the air molecules in the environment.

• Evangelista Torricelli

invented the barometer in

1643 to measure atmospheric pressure.

• Atmospheric pressure is 29.9

inches of mercury or 760 torr

at sea level.

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Pressure

• Pressure is the amount of force applied to an area:

P F

= A

• Atmospheric pressure is the weight of air per unit of area.

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Units of Pressure

• Pascals: 1 Pa 1 N/m= ² (SI unit of pressure)

• Bar: 1 bar 10= ⁵ Pa = 100 kPa

• mm Hg or torr: These units are

literally the difference in the heights measured in millimeters of two

connected columns of mercury, as in the barometer in the figure.

• Atmosphere:

1 atm = 760. torr = 760. mm

Hg = 101.325 kPa = 1.10325 bar

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Manometer

• The manometer is used to measure the difference in pressure between

atmospheric pressure and that of a gas in a vessel.

(The barometer seen on the last slide is used to measure the pressure in the atmosphere at any given time.)

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Standard Pressure

• Normal atmospheric pressure at sea level is referred to as standard atmospheric pressure.

• It is equal to – 1 atm

– 760 torr (760 mmHg) – 101.325 kPa

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11

Volume and Pressure

• When the volume decreases, the gas molecules collide with the container more often and the pressure

increases.

• When the volume increases, the gas molecules collide

with the container less often and the pressure decreases.

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Temperature and Pressure

• When the temperature decreases, the gas molecules move slower and collide with the container less often and the pressure decreases.

• When the temperature increases, the gas molecules move

faster and collide with the container more often and the

pressure increases.

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Chapter 11 13

Molecules and Pressure

• When the number of molecules decreases, there are fewer gas molecule collisions with the side of the container and the pressure decreases.

• When the number of molecules increases, there are

more gas molecule collisions with the side of the

container and the pressure increases.

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Boyle’s Law

• The volume of a fixed

quantity of gas at constant temperature is inversely proportional to the

pressure.

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15

Boyle’s Law

• Inversely proportional

means two variables have a reciprocal relationship.

• Mathematically, we write:

• Boyle’s Law states that the volume of a gas is inversely proportional to the pressure at constant temperature.

V a 1 .

P

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Boyle’s Law Continued

• If we introduce a proportionality constant, k, we can write Boyle’s law as follows:

• We can rearrange it to PV = k.

• Lets take a sample of gas at P

₁

and V

₁

and change the

conditions to P

₂

and V

₂

. Since the product of pressure and volume is constant, we can write:

P

₁

V

₁

= k = P

₂

V

₂

V = k × 1 .

P

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17

Applying Boyle’s Law

• To find the new pressure after a change in volume:

• To find the new volume after a change in pressure:

V

₁

× P

₁

P

₂

= V

₂

P

₁

× V

₁

V

₂

= P

₂

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Boyle’s Law Problem

• A 1.50 L sample of methane gas exerts a pressure of 1650 mm Hg. What is the final pressure if the volume changes to 7.00L?

• The volume increased and the pressure decreased as we expected.

P

₁

× V

₁

V

₂

= P

₂

1650 mm Hg × 1.50 L

7.00 L = 354 mm Hg

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Mathematical Relationships of Boyle’s Law

• PV = a constant.

• This means, if we compare two conditions: ^PV_{1 1} = ^{P V}_{2 2}^.

• Also, if we make a graph of V v^ersus P, it will not be linear.

However, a graph of V v^ersus ¹

P will result in a linear relationship!

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Charles’s Law

• The volume of a fixed

amount of gas at constant pressure is directly

proportional to its absolute temperature.

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Mathematical Relationships of Charles’s Law

• ^V ⁼ ^Constant ^´ ^T^.

• This means, if we

compare two conditions:

1 2

V V . T = T

• Also, if we make a graph of V v^ersus T, it will be linear.

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Charles’ Law

• In 1783, Jacques Charles discovered that the volume of a gas is directly proportional to the temperature in Kelvin.

• This is Charles’ Law.

• V a T at constant pressure.

• Notice that Charles’

Law gives a straight

line graph.

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23

Charles’ Law

• We can write Charles’ Law as an equation using a proportionality constant, k.

V = k T or = k

• Again, lets consider a sample of gas at V

₁

and T

₁

and change the volume and temperature to V

₂

and T

₂

. Since the ratio of volume to temperature is constant, we can write:

V T

V

₁

T

₁

V

₂

T

₂

= k =

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Illustration of Charles’ Law

• Below is an illustration of Charles’ Law.

• As a balloon is cooled from room temperature with liquid

nitrogen (–196°C), the volume decreases.

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25

Applying Charles’ Law

• To find the new volume after a change in temperature:

• To find the new temperature after a change in volume:

V

₁

× T

₂

T

₁

= V

₂

T

₁

× V

₂

V

₁

= T

₂

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Charles’ Law Problem

• A 275 L helium balloon is heated from 20°C to 40°C.

What is the final volume at constant P?

• We first have to convert the temp from °C to K:

20°C + 273 = 293 K

40°C + 273 = 313 K V

₁

× T

₂

T

₁

= V

₂

275 L × 313 K

293 K = 294 L

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Gay-Lussac’s Law of Combining Volumes

• At a given temperature and pressure, the volume of gases that react with each other is in small whole numbers.

• Example: two volumes of hydrogen gas and one volume of oxygen gas will make two volumes of water vapor.

• Stoichiometry based!

• Led to …

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Avogadro’s Law

• The volume of a gas at constant temperature and pressure is directly

proportional to the number of moles of the gas.

• So, at STP, one mole of Any gas occupies 22.4 L.

• V ⍺ n (number of moles)

• Mathematically:

1 2

= constant , or V = V .

V n

n n

´

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Ideal-Gas Equation

• So far we’ve seen that

µ 1 (Boyle s l’ aw).

V P

µ (Charles s l’ aw).

V T

µ (Avogadro s’ law .) V n

• Combining these, we get

V nΤ µ P

• Finally, to make it an equality, we use a constant of

proportionality (R) and reorganize; this gives the ideal-gas equation: PV = nRT.

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Calculation of Gas Constant, R

PV = nRT

R = P x V / n x T

R = 760 mm x 22400 mL / 1 mol x 273 K.

= 62,400 mL-mmHg/mol-K OR in other units;

R = 1 atm. X 22.4L / 1 mol x 273K

= 0.0821 L-atm/mol-K

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R

• The ideal gas constant makes the equation and equality, not only a proportion.

• PV = nRT

Table 10.2 Numerical Values of the Gas Constant R in Various Units

Unit Numerical Value

L - atm/mol - K 0.08206

Joule per mole kelvin asterisk

8.314

cal/mol - K 1.987

Pascal cubic meters per kelvin mole, asterisk 8.314

L - torr/mol - k 62.36 J / mol- K *

m - Pa/mol- K *3

*SI unit

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Ideal Gas Law Problem

• How many moles of hydrogen gas occupy 0.500 L at STP?

• At STP, T=273K and P = 1 atm. Rearrange the ideal gas equation to solve for moles:

n = PV . RT n = (1 atm)(0.500 L) .

(0.0821 atm×L/mol×K)(273K)

n = 0.0223 moles

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Absolute Zero

• The temperature where the pressure and volume of a gas theoretically reaches zero is absolute zero.

• If we extrapolate T vs. P or T

vs. V graphs to zero pressure

or volume, the temperature is

0 Kelvin, or -273°C.

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Density of Gases

If we divide both sides of the ideal-gas equation by V, we get; (PV = nRT or n = PV/RT)

n P . V = RT

Also: moles molecular mass ´ = mass M .

n ´ = m

If we multiply both sides by M, we get

m MP V = RT

and ^m

V is density, d; the result is: MP . d = RT

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Density and Molar Mass of a Gas

• To recap:

– One needs to know only the molecular mass, the

pressure, and the temperature to calculate the density of a gas.

▪ ^d ⁼ ^MP_RT

– Also, if we know the mass, volume, and temperature of a gas, we can find its molar mass (a gas can be

identified).

M mRT

= PV m MP

V = RT or MPV = mRT or

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Dalton’s Law of Partial Pressures

• If two gases that don’t react are combined in a container, they act as if they are alone in the container.

• The total pressure of a mixture of gases equals the sum of the pressures that each would exert if it were present alone.

• In other words,

t 1 2 3

P = P + P + P +!

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Mole Fraction

• Because each gas in a mixture acts as if it is alone, we can relate amount in a mixture to partial pressures:

1

1 1

t t t

n RT

P V n

P n RT n V

= =

• That ratio of moles of a substance to total moles is called the mole fraction,

c .

c₁ = Moles of compound 1 = ¹ Total moles _t

n n

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Pressure and Mole Fraction

• The end result is

1

1 t 1 t

t

P n P X P n

æ ö

= ç ÷ = è ø

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A sample of neon gas at 1.20 atm compresses from 0.250 L to 0.125 L.

If the temperature remains constant, what is the final pressure?

• P1 V1 = P2 V2

• Ans. 2.4 0 atm.

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Kinetic-Molecular Theory

• Laws tell us what happens in nature. Each of the gas laws we have discussed tells us what is observed under certain conditions.

• Why are these laws

observed? We will discuss a theory to explain our

observations.

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Main Tenets of Kinetic-Molecular Theory

^{(1 of 2)}

1) Gases consist of large numbers of molecules that are in continuous, random motion.

2) The combined volume of all the molecules of the gas is negligible relative to the total volume in which the gas is contained.

3) Attractive and repulsive forces between gas molecules are negligible.

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Main Tenets of Kinetic-Molecular Theory

^{(2 of 2)}

4) Energy can be transferred between molecules during collisions, but the average kinetic energy of the

molecules does not change with time, as long as the temperature of the gas remains constant.

5) The average kinetic energy of the molecules is proportional to the absolute temperature.

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How Fast Do Gas Molecules Move?

• Temperature is related to the average kinetic.

• Individual molecules can have different speeds.

• The figure shows three different speeds:

– u_mp is the most probable.

– u_av is the average speed of the molecules.

– u_rms, the root-mean-square speed, is the one associated with their average kinetic energy.

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u Sub r m s_rms

and Molecular Mass

u

• At any given temperature, the average kinetic energy of molecules is the same.

• So,

(

rms

)

²

1m

2 u is the same for two gases at the same temperature.

• If a gas has a low mass, its speed will be greater than for a heavier molecule.

rms

u 3RT

= M

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Effusion and Diffusion

• Effusion is the escape of gas molecules

through a tiny hole into an evacuated space.

• Diffusion is the spread of one substance throughout a space or a second

substance.

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Graham’s Law Describes Diffusion and Effusion

• Graham’s law relates the molar mass of two gases to their rate of speed of travel.

• The “lighter” gas always has a faster rate of speed.

1 2

2 1

r M

r = M

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Real Gases

• In the real world, the behavior of gases conforms to the ideal-gas equation only at relatively high temperature and low pressure.

• Even the same gas will show wildly different behavior under high pressure at different temperatures.

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Deviations from Ideal Behavior

The assumptions made in the kinetic-molecular model

(negligible volume of gas molecules themselves, no attractive forces between gas molecules, etc.) break down at high

pressure and/or low temperature.

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Corrections for Nonideal Behavior

• The ideal-gas equation can be adjusted to take these deviations from ideal behavior into account.

• One equation corrected for real gases is known as the van der Waals equation.

• The pressure adjustment is due to the fact that molecules attract and repel each other.

• The volume adjustment is due to the fact that molecules occupy some space on their own.

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The van der Waals Equation

( )

2 2

P n a V nb nRT V

æ ö

+ - =

ç ÷

è ø

Table 10.3 Van der Waals Constants for Gas Molecules

Substance a, atmosphere square liters per square mole b, liters per mole

He 0.0341 0.02370

Ne 0.211 0.0171

Ar 1.34 0.0322

Kr 2.32 0.0398

Xe 4.19 0.0510

H sub 2 0.244 0.0266

N sub 2 1.39 0.0391

O sub 2 1.36 0.0318

F sub 2 1.06 0.0290

C l sub 2 6.49 0.0562

H sub 2 O 5.46 0.0305

N H sub 3 4.17 0.0371

C H sub 4 2.25 0.0428

C O sub 2 3.59 0.0427

C C l sub 4 20.4 0.1383

(L - atm / mol² ²)

a b(L / mol)

H2

N2

O2

F2

Cl2

H O2

NH3

CH4

CO2

CCl4

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