Scheffler

Gas Laws

Borrowed from:

L. Scheffler

Lincoln High School

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Properties of Gases Variable volume and shape Expand to occupy volume available Volume, Pressure, Temperature, and

the number of moles present are interrelated

Can be easily compressed Exert pressure on whatever

surrounds them Easily diffuse into one another

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Kinetic Molecular Theory Matter consists of particles (atoms or molecules)

that are in continuous, random, rapid motion The Volume occupied by the particles has a

negligibly small effect on their behavior Collisions between particles are elastic (no

Energy is lost) Attractive forces between particles have a

negligible effect on their behavior Gases have no fixed volume or shape, but take

the volume and shape of the container The average kinetic energy of the particles is

proportional to their Kelvin temperature3

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Maxwell-Boltzman Distribution Molecules are in

constant motion Not all particles

have the same energy

The average kinetic energy is related to the temperature

An increase in temperature spreads out the distribution and the mean speed is shifted upward

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The distribution of speedsfor nitrogen gas moleculesat three different temperatures

The distribution of speedsof three different gases

at the same temperature

Velocity of a Gas

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Mercury Barometer Used to define and

measure atmospheric pressure

On the average at sea level the column of mercury rises to a height of about 760 mm.

This quantity is equal to 1 atmosphere

It is also known as standard atmospheric pressure

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Barometer The mercury barometer was

the basis for defining pressure, but it is difficult to use or to transport

Furthermore Mercury is very toxic and seldom used anymore

Most barometers are now aneroid barometers or electronic pressure sensors

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Pressure Units & Conversions

The above represent some of the more common units for measuring pressure. The standard SI unit is the Pascal or kilopascal. (kPa)

The US Weather Bureaus commonly report atmospheric pressures in inches of mercury.

Pounds per square inch or PSI is widely used in the United States.

Most other countries use only the metric system. Two other older units for pressure are still used

millimeters of mercury (mm Hg) atmospheres (atm) 8

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Standard Temperature and Pressure (STP) The volume of a gas varies with temperature

and pressure. Therefore it is helpful to have a convenient reference point at which to compare gases.

For this purpose, standard temperature and pressure are defined as:

Temperature = 0oC 273 K

Pressure = 1 atmosphere = 760 torr

= 101.3 kPa

This point is often called STP9

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Dalton’s Law of Partial Pressures The total pressure of a mixture of gases is

equal to the sum of the pressures of the individual gases (partial pressures).

PT = P1 + P2 + P3 + P4 + . . . .

where PT = total pressure

P1 = partial pressure of gas 1

P2 = partial pressure of gas 2

P3 = partial pressure of gas 3

P4 = partial pressure of gas 410

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Dalton’s Law of Partial Pressures

Determine the partial pressure of each gas in a vessel that holds 2.50 mole of O2, 1.00 mole of N2 and 0.50 mole of CO2. The total pressure in the vessel is 96.0 kPa.

2.50 mol + 1.00 mol + 0.50 mol = 4.00 mol

2.50/4.00 = .625 x 96.0 kPa = 60.0 kPa O2

1.00/4.00 = .25 x 96.0 kPa = 24.0 kPa N2

0.50/4.00 = .125 x 96.0 kPa = 12.0 kPa CO2

1.00 96.0 kPa Total

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Dalton’s Law of Partial Pressures

Applies to a mixture of gases

Very useful correction when collecting gases over water since they inevitably contain some water vapor.

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Sample Problem 7 Henrietta Minkelspurg

generates Hydrogen gas and collected it over water.

If the volume of the gas is 250 cm3 and the barometric pressure is 765.0 torr at 25oC, what is the pressure of the “dry” hydrogen gas?

(PH2O = 23.8 torr at 25oC)

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Boyle’s Law

According to Boyle’s Law the pressure and volume of a gas are inversely proportional at constant temperature.

PV = constant. P1V1 = P2V2

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Boyle’s Law

A graph of pressure and volume gives an inverse function

A graph of pressure and the reciprocal of volume gives a straight line

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= 340 kPa

If the pressure of helium gas in a balloon has a volume of 4.00 dm3 at 210 kPa, what will the pressure be at 2.50 dm3?

P1 V1 (conditions 1) = P2 V2 (new conditions)

(210 kPa) (4.00 dm3) = P2(2.50 dm3)

P2 = (210 kPa) (4.00 dm3) (2.50 dm3)

Sample Problem 1:

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Charles’ Law According to Charles’ Law the volume of a

gas is proportional to the Kelvin temperature as long as the pressure is constant

V = kT

V1

=

T1

V2

T2

Note: The temperature for gas laws must always be expressed in Kelvin where Kelvin = oC +273.15 (or 273 to 3 significant digits)

How do you remember? (Use Zero, get a Zero)

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Pressure, temperature and volume of gases also have a relationship to each other.

That relationships is summarized with:

PTV

Pressure, Temperature, Volume

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Charles’ Law

A graph of temperature and volume yields a straight line. Where this line crosses the x axis (x intercept) is defined

as absolute zero19

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Sample Problem 2 A gas sample at 40 oC occupies a volume of 2.32 dm3. If the temperature is increased to 75 oC, what will be the final volume?

2.58 dm3

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V1 = V2

T1 T2Convert temperatures to Kelvin. 40oC = 313K 75oC = 348K

2.32 dm3 = V2

313 K 348K

(313K)( V2) = (2.32 dm3) (348K)

V2 =

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Gay-Lussac’s Law

Gay-Lussac’s Law defines the relationship between pressure and temperature of a gas.

The pressure and temperature of a gas are directly proportional

P1 = P2

T1T2

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Sample Problem 3:The pressure of a gas in a tank is 3.20 atm at 22 oC. If the temperature rises to 60oC, what will be the pressure in the tank?

3.6 atm22

P1 = P2

T1 T2

Convert temperatures to Kelvin. 22oC = 295K 60oC = 333K

3.20 atm = P2

295 K 333K

(295K)( P2) = (3.20 atm)(333K)

P2 =

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The Combined Gas Law1. If the amount of the gas is constant, then

Boyle’s Charles’ and Gay-Lussac’s Laws can be combined into one relationship

2. P1 V1 = P2 V2

T2T1

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Sample Problem 4: A gas at 110 kPa and 30 oC fills a container at 2.0 dm3. If the temperature rises to 80oC and the pressure increases to 440 kPa, what is the new volume?

V2 = 0.58 dm3

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P1V1 = P2V2

T1 T2

Convert temperatures to Kelvin. 30oC = 303K 80oC = 353K

V2 = V1 P1 T2 P2 T1

= (2.0 dm3) (110 kPa ) (353K) (440 kPa ) (303 K)

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Advogadro’s Law Equal volumes of a gas under the same temperature

and pressure contain the same number of particles. If the temperature and pressure are constant the

volume of a gas is proportional to the number of moles of gas present

V = constant * n

where n is the number of moles of gas

V/n = constant

V1/n1 = constant = V2 /n2

V1/n1 = V2 /n2

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Universal (Ideal) Gas Equation

Based on the previous laws there are four factors that define the quantity of gas: Volume, Pressure, Kelvin Temperature, and the number of moles of gas present (n).

Putting these all together:

PVnT

= Constant = R

The proportionality constant R is known as the universal (Ideal) gas constant

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Universal (Ideal) Gas Equation

The Universal (Ideal) gas equation is usually written as

PV = nRTWhere P = pressure

V = volumeT = Kelvin Temperaturen = number of moles

The numerical value of R depends on the pressure unit (and perhaps the energy unit) Some common values of R include: R = 62.36 dm3 torr mol-1 K-1

= 0.0821 dm3 atm mol-1 K-1

= 8.314 dm3 kPa mol-1 K-1

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Sample Problem 5 Example: What volume will 25.0 g O2 occupy

at 20oC and a pressure of 0.880 atmospheres? :

V = (0.781 mol)(0.08205 dm3 atm mol-1 K-1)(293K)0.880 atm

V = 21.3 dm3

(25.0 g)n = ----------------- = 0.781 mol (32.0 g mol-1)

V =? P = 0.880 atm; T = (20 + 273)K = 293K R = 0.08205 dm3 atm mol-1 K-1

PV = nRT so V = nRT/P

Data

Formula

Calculation

Answer

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Density (d) Calculations PV=nRT

d = mass/Volume

n= moles …which can be equal to: mass (g) Molar Mass (g/mol)

Substituting:

d = mV

=PMRT

m is the mass of the gas in gM 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

Universal Gas Equation –Alternate Forms

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A 2.10 dm3 vessel contains 4.65 g of a gas at 1.00 atmospheres and 27.0oC. What is the molar mass of the gas?

dRTP

M = d = mV

4.65 g2.10 dm3

= = 2.21 g

dm3

M =2.21

g

dm3

1 atm

x 0.0821 x 300.15 Kdm3•atmmol•K

M = 54.6 g/mol

Sample Problem 6

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“Stupid” Gas Laws…

How did all these gas laws “come to be”???? All are based on the Ideal Gas Law for 2 sets of conditions. Remember: PV = nRT

we can rearrange things to solve for a constant (R)R=PV/nT n=# moles. (Not going to change

when conditions change) Therefore, if moles don’t change: R=PV/T

If we look at a gas that changes from one condition P1V1 = nRT1 to a new condition P2V2 = nRT2, we can set up a relationship (moles don’t change and R is constant)

P1V1/T1 = R = V2P2/T2

P1V1/T1 = V2P2/T2 (Combined Gas Law) 31

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“Stupid” Gas Laws (cont.)

P1V1 = P2V2 If Temperature is constant P1V1 = P2V2

T1 T2 (Boyle’s Law) T1 T2

P1V1 = P2V2 If Volume is constant P1V1 = P2V2

T1 T2 (Gay-Lassac’s Law) T1 T2

P1V1 = P2V2 If Pressure is constant P1V1 = P2V2

T1 T2 (Charles’ Law) T1 T2

P1V1 = P2V2 If Pressure & T are constant P1V1 = P2V2

n1T1 n2T2 n1T1 n2T2

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Gas diffusion is the gradual mixing of molecules of one gas with molecules of another by virtue of their kinetic properties.

NH3

17.0 g/mol

HCl36.5 g/mol

NH4Cl

Diffusion

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DIFFUSION AND EFFUSION

Diffusion is the gradual mixing of molecules of different gases.

Effusion is the movement of molecules through a small hole into an empty container.

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Graham’s Law (For Information only: do not need to know how to do this)

Graham’s law governs effusion and diffusion of gas molecules.

KE=1/2 mv2

(don’t need to know this)

Thomas Graham, 1805-1869. Professor in Glasgow and London.

The rate of effusion is inversely proportional to its molar mass.

The rate of effusion is inversely proportional to its molar mass.

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Kinetic Molecular Theory Matter consists of particles (atoms or molecules) that are

in continuous, random, rapid motion

The Volume occupied by the particles has a negligibly small effect on their behavior

Collisions between particles are elastic (no Energy is lost)

Attractive forces between particles have a negligible effect on their behavior

Gases have no fixed volume or shape, but take the volume and shape of the container

The average kinetic energy of the particles is proportional to their Kelvin temperature

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Ideal Gases v Real Gases

Ideal gases are gases that obey the Kinetic Molecular Theory perfectly.

The gas laws apply to ideal gases, but in reality there is no perfectly ideal gas.

Under normal conditions of temperature and pressure many real gases approximate ideal gases.

Under more extreme conditions more polar gases show deviations from ideal behavior.

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Real Gases

These deviations occur because Real gases do not actually have zero volume Polar gas particles do attract if compressed

For ideal gases the product of pressure and volume is constant. Real gases deviate somewhat as shown by the graph pressure vs. the ratio of observed volume to ideal volume below.

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In an Ideal Gas --- The particles (atoms or molecules) in continuous,

random, rapid motion. The particles collide with no loss of momentum The volume occupied by the particles is essentially zero

when compared to the volume of the container The particles are neither attracted to each other nor

repelled The average kinetic energy of the particles is proportional

to their Kelvin temperature

At normal temperatures and pressures gases closely approximate ideal behavior

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van der Waals Equation(For Information only: do not need to know how to do this)

(P + n2a/V2)(V - nb) = nRT

The van der Waals equation shown below includes corrections added to the universal gas law to account for these deviations from ideal behavior

where a => attractive forces between moleculesb => residual volume or molecules

The van der Waals constants for some elements are shown below

Substance a (dm6atm mol-2) b (dm3 mol-1)

He 0.0341 0.02370

CH4 2.25 0.0428

H2O 5.46 0.0305

CO2 3.59 0.043740