CHEM 331

Physical Chemistry

Fall 2014

The Historical Gas Laws

Some of the earliest scientific investigations concerning matter were performed by pneumaticists

trying to understand the physical and chemical properties of gases. It was these studies (~1650-

1800 AD) that helped establish chemistry as a scientific discipline and helped lay to rest the art of

alchemy. During this period a number of gases were discovered:

Gas Year Discoverer

CO2 1630 Jan Baptist van Helmont

H2 1766 Henry Cavendish

N2 1772 David Retherford

O2 1774 Joseph Priestley

In this lecture we wish to consider the development of the Equation of State for a gas which we

will call Ideal. This Ideal Gas will approximate the behavior of real gases only at moderate

temperatures and low pressures; roughly normal atmospheric conditions. Hence the

pneumaticists were able to unravel the form of the Equation of State for this "hypothetical gas"

using known gases. We will see that this Equation of State represents real gases only in the limit

of zero pressure.

Boyle's Law

Robert Boyle was one of the first to examine the physical properties of gases. He discovered the

law that bears his name while trying to disprove Franciscus Linus's argument that a column of

Mercury in a Toricellian tube was not held in place by the external force of the Air, but instead

was held in place by an invisible membrane called a funiculus. It was Linus's argument that this

membrane could only hold a column of Mercury 29 ½ " high. Linus's evidence for this claim

centered on the fact that if you open the end of the Toricellian tube and place your finger over it,

your finger is drawn into the tube.

If you take a tube open at both ends of a good length, suppose forty

inches long, and fill it with mercury, and place your finger on the top as

before, taking away your lower finger, you will find the mercury to

descend even to its wonted station [i.e., to a height of approximately

29 ½ inches], and your finger on the top to be strongly drawn within

the tube, and to stick close unto it. Whence again it is evidently

concluded that the mercury places in its own station is not there upheld

by the external air, but suspended by a certain cord [Linus's alleged

funiculus], whose upper end being fastened to the finger draws and

fastens it after this manner into the tube.

Robert Boyle's Experiments in Pneumatics

Ed. James Bryant Conant

Boyle's experiment designed to disprove this idea involved examining the pressure exerted on

the Air trapped in a bent tube filled with Mercury. One leg of the tube, the shorter, was closed

and contained the trapped Air. The other,

the longer, was open to the atmosphere.

Mercury could be added to the open

section of the tube, and thus made to

increase the pressure on the Air. The

"volume" of Air was determined by noting

the number of tick marks from the top end

of the tube to the level of the Mercury

containing the Air. The pressure on the gas

was determined by adding the pressure

head of the mercury to the atmosphere's

pressure. Boyle's data was published in

1662 in a book titled New Experiments

Physico-Mechanical Touching the Spring

of Air. Some of his data follows.

Boyle noted the P x V product in the final column of the Table. As can be seen, to within

experimental error, this product is constant. This is in essence Boyle's Law:

P x V = constant (at a given temperature)

A P-V plot of his data follows. The curve drawn is known as an Isotherm.

A plot of P vs. 1/V shows the appropriate linear relationship.

Modern, more precise data on O2, Ne and CO2 show that the PV product is not only not constant,

but depends on the pressure and the nature of the gas.

Oxygen Nitrogen Carbon Dioxide

P [atm] V [L/mol] PV P [atm] V [L/mol] PV P [atm] V [L/mol] PV

1.0 22.3939 22.3939 1.00 22.4280 22.4280 1.00 22.2643 22.2643

0.75 29.8649 22.3987 0.67 33.6360 22.4241 0.67 33.4722 22.3148

0.50 44.8090 22.4045 0.33 67.6567 22.4189 0.50 44.6794 22.3397

0.25 89.6384 22.4096 0.33 67.0962 22.3654

0.25 89.5100 22.3897

A plot of this data however shows that extrapolation to zero pressure for each gas converges on

the same intercept.

This suggests we can re-write Boyle's Law as:

That the Law becomes Universal in the limit of zero pressure is not unreasonable. If we view

our gas as being made up of discrete molecules, in the limit of zero pressure all the molecules are

infinitely far apart. So, in this limit, attractive or repulsive forces between the molecules will be

negligible, and the identity of the gas will be unimportant.

Note also that we have written the constant A(t) to indicate that it is dependent upon temperature.

It is this temperature dependence that was discovered by Charles and Gay-Lussac.

Charles' and Gay-Lussac's Laws

Jacques Charles Joseph Louis Gay-Lussac

In 1787, Charles observed that a volume of different gases, when heated to the same temperature,

would increase by the same amount. This result was extended and published by Gay-Lussac in

y = -0.0212x + 22.415

y = 0.0136x + 22.415

y = -0.1516x + 22.416

22.25

22.3

22.35

22.4

22.45

0 0.5 1 1.5

PV

[L

atm

/mo

l]

Pressure [atm]

PV Data for O2, Ne, CO2

1802. It was also re-discovered by John Dalton in 1801. According to these savants, the volume

is linearly dependent on the temperature:

V = Vo (1 + o)

where is the temperature when measured using the Celsius temperature scale and Vo is the gas'

volume at 0oC. is the Coefficient of Thermal Expansion for the gas. Gay-Lussac showed the

pressure is likewise linear in temperature:

P = Poo (1 + )

According to the Law of Charles, a plot of V vs. for any gas should extrapolate to the same

V = 0 point. And this is approximately, but not exactly, true.

We can see this by examining the data for gaseous Ethane:

Temp [oC] Vol. [cm

3] at 1 atm Vol. [cm

3] at 2 atm

26.85 24430 12120

36.85 25264 12544

46.85 26094 12967

56.85 26929 13389

66.85 27759 13810

76.85 28588 14229

Plotted we have:

At P = 1 atm

V = 84.001 + 22165

= 22165 (1 + 0.003790 )

So,

= 0.003790 oC

-1

At P = 2 atm

V = 42.607 + 10972

= 10972 (1 + 0.003883 )

So,

= 0.003883 oC

-1

We see from this data that the slope of

the P vs. plot, , is slightly dependent

on pressure (this is not entirely

unexpected) and so the temperature

when the volume of the gas is zero (V =

0) will also depend slightly on pressure.

We observe that for all gases (Physical

Chemistry by Walter J. Moore), when

extrapolated to zero pressure, the value

of o is indeed the same:

o(P=0) = 0.0036610 oC

-1

y = 84.001x + 22165

y = 42.607x + 10972

12000

14000

16000

18000

20000

22000

24000

26000

28000

30000

25 35 45 55 65 75 85

Vo

lum

e

cm3]

Temperature [oC]

Gaseous Ethane

This allows us to state that, universally, for all gases, when their volume is compressed to zero:

0 = Vo (1 + o(P=0) ) = Vo (1 + 0.0036610 oC

-1 To)

allowing us to determine for this coldest possible temperature:

To = - 1/o(P=0) = - 1/ 0.0036610 oC

-1 = - 273.15

oC

We can now define a new Temperature Scale, called the Ideal Gas Temperature Scale, as:

T = + To

where a "degree" on this new scale is referred to as the Kelvin and To is Absolute Zero.

Given this, Charles' Law reduces to:

V = Vo (1 + (T - To)/To) = VoT / To

More generally, we say that V is proportional to T, or:

V / T ~ Constant

when T is measured on the Ideal Gas Temperature Scale.

In 1954, the Tenth Conference of the International Committee on Weights and Measures defined

the Ideal Gas Temperature Scale such that Absolute Zero is taken as 0 Kelvin and the Triple

Point of Water is 273.16 Kelvin.

Triple Point Cell

Water Vapor

Liquid Water

Ice

William Thomson, Lord Kelvin

Avogadro's Hypothesis

Although the next piece of the puzzle concerning the inter-relationship of the State Variables of a

gas was put forth by the Italian chemist Amedeo Avogadro, it took a controversy concerning the

work of Gay-Lussac to bring his ideas to light.

Gay-Lussac noted that gases tend to combine in whole number ratios. For instance when

Hydrogen and Oxygen react to form Water vapor, they do so in a 2:1 ratio, forming 2 volumes of

Water.

Gay-Lussac and others tried explaining his Law of Combining Volumes, the ratio between the

volumes of the reactant gases and the products can be expressed in simple whole numbers, by

assuming equal volumes of gases contain equal numbers of particles. Assuming Water has a

chemical formula of H2O, this leads to a problem. By this account, only one volume of Water

vapor should be formed.

It was Avogadro that had noted the "equal volumes, equal particles"

hypothesis could explain the Law of Combining Volume data if it was

assumed Hydrogen and Oxygen gases contain particles that are

diatomic.

Stanislao Cannizzario discovered Avogadro's little noticed work of

1811 and published supporting data in 1860. It was then that the

importance of Avogadro's work was recognized and the "equal

volumes, equal particles" hypothesis was revived. Thus, it was shown

that:

V ~ N

Beyond reviving the hypothesis

that equal volumes of gases

contain equal numbers of

particles, Avogadro's

Hypothesis had other far

reaching consequences. It

helped to firmly establish the

atomic weights of the elements.

This in turn made possible the

assignment of unambiguous

chemical formulas for

compounds. Upon hearing of

Avogadro's ideas, the chemist

Julius Lothar von Meyer (right)

said " It was as though the scales had been lifted from my eyes.

Doubt vanished and was replaced by a feeling of peaceful clarity."

Equation of State

The State Variables P, V, T, and N can consequently all be related in a single equation of state:

where R is a Universal Gas Constant having the value 0.082057 L atm / K mol.

A molecular model for a gas that obeys this equation of state assumes:

The gas molecules are point particles; having no volume.

There are no attractive interactions between gas molecules.

The gas molecules only interact during brief elastic collisions.

These assumptions provide the basis for the Kinetic Molecular Theory of Gases and are capable

of accounting for all the historical gas laws.

A gas which follows the equation of state PV = NRT is said to be an Ideal Gas.

Consequences

1. Gas Density () is related to the Molar Mass (M).

P = NRT / V

Setting V/N = /M and we have:

P = RT / M

This rearranges to:

M = (/P) RT

Of course, this is only valid in the limit of zero pressure. So,

M =

This is illustrated using density data for Dimethyl Ether at 298.15K:

P [kPa] [kg/m

3] (/P)/10

-5 [kg/Pa m

3]

12.223 0.225 1.84079

25.20 0.456 1.80952

36.97 0.664 1.79605

60.37 1.062 1.75915

85.23 1.468 1.72240

101.3 1.734 1.71174

A graph of these data show that a quadratic fit can be extrapolated to give:

/P = (6.32857 x 10-11

) P2 + (-2.16784 x 10

-8) P + (1.86476 x 10

-5)

M =

= (1.86476 x 10

-5) (8.31447) (298.15)

= 0.04622 kg/mole

= 46.22 g/mole

If a plot of (P) vs. P shows significant curvature, extrapolation to P = 0 may not be

possible. In this case, a plot of (P/) vs. may be more appropriate. Inverting the data

frequently linearizes it enough to make the extrapolation to zero pressure possible.

2. Dalton's Law of Partial Pressures

Pi = Partial Pressure

PTot =

or

0.000017

0.0000172

0.0000174

0.0000176

0.0000178

0.000018

0.0000182

0.0000184

0.0000186

0 20 40 60 80 100 120

/P

[kg

/Pa

m3]

P [kPa]

Dimethyl Ether

Pi = xi PTot where xi is the mole fraction of species i in the gas

3. Constant Volume Gas Thermometry

We can now imagine measuring the temperature of a bath in the following manner. A

bulb of gas is placed in a bath at T1 and its pressure is read with a gauge; P1. The same

bulb is then transferred to a Water bath maintained at its Triple Point, Ttp. The pressure is

again read with the gauge; Ptp.

Some gas is then evacuated from the bulb and the process is repeated. If the process is

repeatedly performed with successively lower gas densities, the pressure data can then be

extrapolated to zero. Then,

If we take Ttp = 273.16K, as specified by the 10th

International Conference on Weights

and Measures, then:

T1 = 273.16 K x

and we have measured the temperature of the Bath at T1 on the Ideal Gas Temperature

Scale. This type of thermometer is a Constant Volume Gas Thermometer.