chapter 3 the properties of gases liquids and solids

Another aspect of gas behaviorAvogadro’s PrincipleUnder the same conditions of temperature and pressure, a given number of gas molecules occupy the same volume regardless of their chemic

Chapter 3

THE PROPERTIES OF GASES, LIQUIDS AND SOLIDS

THE NATURE OF GASES

3.1 Observing Gases3.2 Pressure

3.3 Alternative Units of PressureTHE GAS LAWS

3.4 The Experimental Observations3.5 Applications of the Ideal Gas Law3.6 Gas Density

3.7 The Stoichiometry of Reaction Gases3.8 Mixtures of Gases

MOLECULAR MOTION

3.9 Diffusion and Effusion3.10 The Kinetic Model of Gases3.11 The Maxwell Distribution of SpeedsREAL GASES

3.12 Deviations from Ideality3.13 The Liquefaction of Gases3.14 Equations of State of Real Gases

LIQUID STRUCTURE

3.15 Order in Liquids3.16 Viscosity and Surface TensionSOLID STRUCTURES

3.17 Classification of Solids3.18 Molecular Solids

3.19 Network Solids3.20 Metallic Solids3.21 Unit Cells

3.22 Ionic Structures

States of Matter

The fundamental difference between states of matter is the

distance between particles

In the solid and liquid states particles are closer together, we

States of Matter

The state a substance is in at a particular temperature and

pressure depends on two antagonistic entities:

– the kinetic energy of the particles

– the strength of the attractions between the particles

Observing Gases

Many of physical properties of gases are very similar, regardless

of the identity of the gas Therefore, they can all be described

simultaneously

Samples of gases large enough to study are examples of bulk

matter – forms of matter that consist of large numbers of

molecules

Two major properties of gases:

Compressibility – the act of reducing the volume of a sample of a gas

Expansivity - the ability of a gas to fill the space available to it rapidly

Measurement of Pressure

Barometer – A glass tube, sealed at one

end, filled with liquid mercury, and

inverted into a beaker also containing

liquid mercury (Torricelli)

where h = the height of a column, d =

density of liquid, and g = acceleration of

gravity (9.80665 ms-2)

Measurement of Pressure

Ex Suppose the height of the column of mercury in abarometer is 760 mm (written 760 mmHg, and read “760millimeters of mercury”) at 15°C What is the atmosphericpressure in pascals? At 15°C the density of mercury is13.595 g.cm-3 (corresponding to 13 595 kg.m-3) and thestandard acceleration of free fall at the surface of the Earth

is 9.806 65 m.s-2

Measurement of Pressure

Manometer – a U-shaped tube filled with

liquid and connected to an experimental

system, whose pressure is being

monitored

Two types of Hg manometer:

(a) open-tube (b) Closed tube system

vacuum

Measurement of Pressure

The height of the mercury in the system-side column of anopen-tube mercury manometer was 10 mm above that ofthe open side when the atmospheric pressurecorresponded to 756 mm of mercury and the temperaturewas 15°C What is the pressure inside the apparatus inmillimeters of mercury and in pascals?

1 bar = 105 Pa = 100 kPa

1 atm = 760 Torr = 1.01325×105 Pa (101.325 kPa)

1 Torr ~ 1 mmHg = 133.322 Pa

Ex The US National Hurricane Center reported that the eye

of Hurricane Katrina (2005) fell as low as 902 mbar What isthe pressure in atmospheres?

The Experimental Observations

Boyle’s law: For a fixed amount of gas at constanttemperature, volume is inversely proportional to pressure

This applies to an isothermal system (constant T) with afixed amount of gas (constant n)

THE GAS LAWS

The Experimental Observations

For isothermal changes between two states

THE GAS LAWS

Ex In a petroleum refinery a 750 L container containingethylene gas at 1.00 bar was compressed isothermally to5.00 bar What was the final volume of the container?

The Experimental Observations

Charles’s law: For a fixed amount of gas under constant

pressure, the volume varies linearly with the temperature

THE GAS LAWS

This applies to an isobaric system

(constant P) with a fixed amount of

gas (constant n)

If a Charles’ plot of V versus T (at constant P and n) isextrapolated to V = 0, the intercept on the T axis is ~-273

oC

The Kelvin Scale of Temperature

- Kelvin temperature scale

Another aspect of gas behaviorGay-Lussac’s Law

This applies to an isochoric

system (constant V) with a fixed

amount of gas (constant n)

Another aspect of gas behavior

Ex A rigid oxygen tank stored outside a building has apressure of 20.00 atm at 6:00 am when the temperature is

10 0C What will be the pressure in the tank at 6:00 pm,when the temperature is 30.oC?

Solution

Volume is constant, hence Gay-Lussac’s law can be used

P1/T1 = P2/T2 => (20.00 atm)/(283.15 K) = P2/(303.15 K)

=> P2 = 21.41 atm

Another aspect of gas behavior

Avogadro’s Principle

Under the same conditions of temperature and pressure, a given number of gas molecules occupy the same volume regardless of their chemical identity

- This defines molar volume

Ex A helium weather balloon was filled at -20.oC and a

certain pressure to a volume of 2.5 x 104 L with 1.2 x 103

mol He What is the molar volume of helium under these

conditions?

Molar volume = Volume/No.moles

Another aspect of gas behavior

The Ideal Gas Law

This is formed by combining the laws of Boyle, Charles,

Gay-Lussac and Avogadro

Gas constant, R = PV/nT It is sometimes called a “universal constant” and has the value 8.314 J K-1 mol-1 in SI units,

although other units are often used

Applications of the Ideal Gas Law

- Standard ambient temperature and pressure (SATP)

298.15 K and 1 bar, molar volume at SATP = 24.79 L·mol-1

- Standard temperature and pressure (STP)

0 oC and 1 atm (273.15 K and 1.01325 bar)

- Molar volume at STP

- For conditions 1 and 2,

- Molar volume

Applications of the Ideal Gas Law

Ex In an investigation of the properties of the coolant gasused in an air-conditioning system, a sample of volume 500

mL at 28.0 oC was found to exert a pressure of 92.0 kPa.What pressure will the sample exert when it is compressed

to 300 mL and cooled to -5.0 oC?

Applications of the Ideal Gas Law

Ex An idling, badly tuned automobile engine can release asmuch as 1.00 mol of CO per minute into the atmosphere At

27 oC, what volume of CO, adjusted to 1.00 atm, is emittedper minute?

Applications of the Ideal Gas Law

Ex A sample of argon gas of volume 10.0 mL at 200 Torr isallowed to expand isothermally into an evacuated tube with

a volume of 0.200 L What is the final pressure of the argon

in the tube?

Solution

The volume is increased by a factor of 20, so we expect adecrease in pressure by the same factor, under isothermalconditions

P1V1/n1T1 = P2V2/n2T2 , where T1 = T2 , n1 = n2 (reduces toBoyle’s law)

(200 Torr)(10.0 mL) = P2(Torr)(200 mL)

P2 = 10.0 Torr

Applications of the Ideal Gas Law

Ex Calculate the volume occupied by 1.0 kg of hydrogen at

25 oC and 1.0 atm

Solution

We can use the ideal gas equation PV = nRT, after firstfinding the number of moles of H2 in 1.0 kg

n = Mass/Molar mass = 1.0 x 103 g/2.016 g/mol = 496 mol

(1.0 atm)V(L) = (496 mol)(8.206.10-2 L.atm.K-1.mol

-1)(298.15K)

V = 1.2 x 104 L

Gas Density

Molar concentration of a gas is the number moles divided

by the volume occupied by the gas

Molar concentration of a gas at STP

(where molar volume is 22.4141 L):

Density, however, does depend on the identity of the gas

This value is the same for all gases, assuming ideal behavior

• the density of a gas increases with pressure

When a gas is compressed, its density increasesbecause the same number o f molecules are confined

in a smaller volume

Similarly, heating a gas that is free to expand at constantpressure increases the volume occupied by the gasand therefore reduces its density

M = 154 g mol-1

The Stoichiometry of Reacting Gases

CO2 generated by the personnel in the artificial

atmosphere of submarines and spacecraft must

be removed form the air and the oxygen

recovered Submarine design teams have

investigated the use of potassium superoxide,

KO2, as an air purifier because this compound

reacts with carbon dioxide and releases oxygen:

The Stoichiometry of Reacting Gases

Ex Calculate the volume of carbon dioxide, adjusted to 25

oC and 1.0 atm that plants need to make 1.00 g of glucose,

C6H12O6, by photosynthesis in the reaction

6CO2(g) + 6H2O(l) → C6H12O6(s) + 6O2(g)Solution

From the equation, the stoichiometry of CO2: glucose is 6:1The molar mass of glucose is 180 g/mol The molar volume

of CO2 at 25 oC and 1 atm is 24.47 L/mol

there are no interactions— neither attractions nor repulsions— between the two kinds

of molecules (is a characteristic feature of an ideal gas)

The partial pressures to describe the composition of a humid gas For example, the total pressure of the damp air in our lungs is

P = Pdry air + Pwater vapor

the total pressure is 0.87 atm.

Mixtures of Gases

A baby with a severe bronchial infection is in respiratory

distress The anesthetist administers heliox, a mixture of

helium and oxygen with 92.3% by mass O2 What is the

partial pressure of oxygen being administered to the baby of the atmosphere pressure is 730 Torr?

Solution

n(He) = 0.077 g/4.00 g mol-1 = 0.0193 mol

n(O2) = 0.923 g/32.0 g mol-1 = 0.0288 mol

x(He) = 0.401 x(O2) = 0.599P(O ) = x(O )P = 0.599 x 730 Torr = 437 Torr

Molecular Motion

The empirical results summarized by the gas laws suggest

a model of an ideal gas in which widely spaced (most of thetime), noninteracting molecules undergo ceaseless motion,with average speeds that increase with temperature In thenext three sections, we refine our model in two steps

• use experimental measurements of the rate at whichgases spread from one region to another to discoverthe average speeds of molecules

• use these average speeds to express our model of anideal gas quantitatively, check that it is in agreementwith the gas laws, and use it to derive detailedinformation about the proportion of molecules havingany specified speed

Diffusion and Effusion

Diffusion: gradual dispersal of one substance throughanother substance

Effusion: escape of a gas through a small hole into avacuum

Diffusion and Effusion

Effusion occurs whenever a gas is separated from a

vacuum by a porous barrier a barrier that contains

microscopic holes— or a single pinhole

A gas escapes through a pinhole because there are more

“collisions” with the hole on the high-pressure side than on the low-pressure side, and so more molecules pass from the high-pressure region into the low-pressure region than pass in the opposite direction

Effusion is easier to treat than diffusion, so we concentrate

on it; but similar remarks apply to diffusion too

Diffusion and Effusion

At constant termperature, the rate of effusion of a gas is

inversely proportional to the square root of its molar mass:

𝑅𝑎𝑡𝑒 𝑜𝑓 𝑒𝑓𝑓𝑢𝑠𝑖𝑜𝑛 ∝ 1

𝑚𝑜𝑙𝑎𝑟 𝑚𝑎𝑠𝑠 =

1𝑀𝑅𝑎𝑡𝑒 𝑜𝑓 𝑒𝑓𝑓𝑢𝑠𝑖𝑜𝑛 𝑜𝑓 𝐴 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠

𝐸𝑣𝑒𝑟𝑎𝑔𝑒 𝑠𝑝𝑒𝑒𝑑 ∝ 1

Diffusion and Effusion

Rate of effusion and average speed increase as the square root of the temperature is raised:

Combined relationship: The average speed of molecules in

a gas is directly proportional to the square root of the

temperature and inversely proportional to the square root of

the molar mass

Diffusion and Effusion

It takes 30 mL of argon 40 s to effuse through a porous

barrier The same volume of vapor of volatile compound

extracted from Caribbean sponges takes 120 s to effuse through the same barrier under the same conditions What

is the molar mass of the compound?

Solution

𝑇𝑖𝑚𝑒 𝑓𝑜𝑟 𝐴𝑟 𝑡𝑜 𝑒𝑓𝑓𝑢𝑠𝑒𝑇𝑖𝑚𝑒 𝑓𝑜𝑟 𝑢𝑛𝑘𝑛𝑜𝑤𝑛 𝑡𝑜 𝑒𝑓𝑓𝑢𝑠𝑒 =

𝑀(𝐴𝑟)𝑀(𝑢𝑛𝑘𝑛𝑜𝑤𝑛)

40 (𝑠)

120 (𝑠) =

39.95 𝑔/𝑚𝑜𝑙𝑀(𝑢𝑛𝑘𝑛𝑜𝑤𝑛) ⇒ 𝑀 = 360 𝑔/𝑚𝑜𝑙

The Kinetic Model of Gases

Kinetic molecular theory (KMT) of a gas makes four

assumptions:

1 A gas consists of a collection of molecules in

continuous random motion

2 Gas molecules are infinitesimally small points

3 The molecules move in straight lines until

they collide

4 The molecules do not influence one another

except during collisions

- Collision with walls: consider molecules

traveling only in one dimensional x with a

velocity of vx

The Kinetic Model of Gases

The change in momentum

(final – initial) of one

molecule: 2mvx

All the molecules within a distance vxDt

of the wall and traveling toward it willstrike the wall during the Interval Dt

If the wall has area A, all the particles

in a volume AvxDt will reach the wall if

they are traveling toward it

The Kinetic Model of Gases

The number of molecules in the volume

AvxDt is that fraction of the total volume V,

multiplied by the total number of molecules:

The average number of collisions with the

wall during the interval Dt is half the

number in the volume AvxDt:

The Kinetic Model of Gases

Force = rate of change of momentum =

(total momentum change)/Dt

Where <vx2 > is the average value of vx2 for all the molecules in the sample

Mean square speed: From the Pythagorean theorem,

because the particles are moving randomly, the average of

v 2 is the same as the average of v 2 and the average of v 2

The Kinetic Model of Gases

Pressure on wall:

or

- The temperature is proportional to the mean square speed of the molecules in a gas.

- This was the first acceptable physical interpretation of temperature: a measure of molecular motion.

v rms is the root mean square speed

The Kinetic Model of Gases

Ex What is the root mean square speed

of nitrogen molecules in air at 20 oC?

about 1140 miles per hour.

The Kinetic Model of Gases

Ex Estimate the root mean square of water molecules in the vapor above boiling water at 100 oC?

Solution

Molar mass of water is 18.01 g/mol or 0.01801 kg/mol

From vrms = (3RT/M)1/2 = 719 m s-1

Where R = 8.3145 J K-1 mol-1

Useful as it is

gives only the root mean square speed of gas molecules Like cars in traffic, individual molecules have speeds that vary over a wide range

Like a car in a head-on collision, a molecule might be brought almost to a standstill when it collides with another

In the next instant (but now unlike a colliding car), it might be struck by another molecule and move off at the speed of sound

An individual molecule undergoes several billion changes of

speed and direction each second

The Maxwell Distribution of Speeds

The Maxwell Distribution of Speeds

v = a particle’s speed

DN = the number of molecules with speeds in the narrow range

between v and v + Dv

N = total number of molecules; M = molar mass

f(v) = Maxwell distribution of speeds

For an infinitesimal range,

And average speed

For calculating the fraction of gas molecules having the speed v at any instant, from the kinetic model Maxwell derived equation,

with

The Maxwell Distribution of SpeedsHOW DO WE KNOW THE DISTRIBUTION OF MOLECULAR SPEEDS?

It can be determined experimentally;

The gas is heated to the required

temperature in an oven;

The molecules then stream out of the oven

through a small hole into an evacuated

region (a series of slits);

Each disk contains a slit that is offset by a

certain angle from its neighbor;

A molecule that passes through the first slit will pass through the slit in the next disk only if the time that it takes to pass between the disks is the same as the time required for the slit in the second disk to move into.

The Maxwell Distribution of SpeedsHOW DO WE KNOW THE DISTRIBUTION OF MOLECULAR SPEEDS?

The distribution of molecular

speeds is determined by measure

the intensity of the beam of

molecules arriving at the detector

for different rotational rates of

the disks

The points represent a typical result of a molecular speed distribution

measurement They are superimposed on the theoretical curve.