the ideal gas laws chapter 14. expectations after this chapter, students will: know what a mole is...

The Ideal Gas Laws

Chapter 14

Expectations

After this chapter, students will: Know what a “mole” is Understand and apply atomic mass, the atomic

mass unit, and Avogadro’s number Understand how an ideal gas differs from real

ones Use the ideal gas equation, Boyle’s Law, and

Charles’ Law, to solve problems

Expectations

After this chapter, students will: understand the connection between the

macroscopic properties of gases and the microscopic mechanics of gas molecules

Preliminaries: the Mole

A mole is a very large number of discrete objects, such as atoms, molecules, or sand grains.

Specifically, it is Avogadro’s Number (NA) of such things: 6.022×1023 of them.

The mole (“mol”) is not a dimensional unit; it is a label.

Amadeo Avogadro

1776 – 1856

Native of Turin, Italy

Hypothesized that equal volumes

of gases at the same temperature

and pressure contained equal

numbers of molecules.

(He was correct, too.)

The Mole and Atomic Mass

Mathematical definition: 12 g of C12 contains one mole of carbon-12 atoms.

Mass of one C12 atom:

The mass of one C12 atom is also 12 atomic mass units (amu), so:

kg 10993.1106.022

kg .012g 12 26-23

AN

kg 101.661 12

kg 10993.1amu 1 27-

-26

The Mole and Atomic Mass

Atomic masses for the elements may be found in the periodic table of the elements, located inside the back cover of your textbook.

These are often erroneously called “atomic weights.”

Atomic masses may be added to calculate molecular masses for chemical compounds (or diatomic elements).

The Mole: Calculations

If we have N particles, how many moles is that?

If we have a given mass of something, how many moles do we have?

AN

Nn number of moles

massmolecular or atomic

sample of mass

moleper mass

sample of massn

The Ideal Gas

The notion of an “ideal” gas developed from the efforts of scientists in the 18th and 19th centuries to link the macroscopic behavior of gases (volume, temperature, and pressure) to the Newtonian mechanics of the tiny particles that were increasingly seen as the microscopic constituents of gases.

The Ideal Gas

An ideal gas was one whose particles are well-behaved, in terms of the Newtonian theory of collisions: elastic collisions and the impulse-momentum theorem.

An ideal gas is one in which the particles have no interaction, except for perfectly-elastic collisions with each other, and with the walls of their container.

The Ideal Gas

An ideal gas has no chemistry. That is, the particles (atoms or molecules) have no tendency to “stick” to other particles through chemical bonds.

Inert gases (He, Ne, Ar, Kr, Xe, Rn) at low densities are very good approximations to the ideal gas.

Our analytic model of the ideal gas gives us insights into the properties of many real gases, inert or not.

The Ideal Gas Equation

Observations from experience

The pressure of a gas is directly proportional to the number of moles of particles in a given space. Example: blow up a balloon, and you’re adding to n, the number of moles of molecules.

Conclusion: nP

The Ideal Gas Equation

Observations from experience

The pressure of a gas is directly proportional to its temperature. Example: toss a spray can into a fire (no, wait, really, don’t do it, just think about it). Increasing pressure will cause the can to fail catastrophically.

Conclusion: TP

The Ideal Gas Equation

Observations from experience

The pressure of a gas is inversely proportional to its volume. Example: squeeze the air in a half-filled balloon down to one end and squeeze it tighter. Increased pressure makes the balloon’s skin tight.

Conclusion: V

P1

The Ideal Gas Equation

Combine the observations

A constant of proportionality, R, makes this an equation:

V

nTP

nRTPVV

nTRP or

The Ideal Gas Equation

The constant of proportionality, R, is called the universal gas constant. Its value and units depend on the units used for P, V, and T.

Value and SI units of R: 8.31 J / (mol K)

nRTPV pressure volume

number of moles

absolute temperature

universal gas constant

The Ideal Gas Equation

We can also write the ideal gas equation in terms of the number of particles, N, instead of the number of moles, n.

Since N = n·NA, we can both multiply and divide the right-hand side by NA:

J/K 101.38 where 23-

A

AA

N

RkNkTPV

TN

RnNPV Boltzmann’s constant

Ludwig Boltzmann

Austrian physicist

1844 – 1906

Boyle’s Law

Suppose we hold both n and T constant: how are P and V related?

This is called Boyle’s Law.

2211

constant

VPVP

PVnRTPV

Robert Boyle

Irish mathematician

1627 – 1691

Charles’ Law

Suppose we hold both n and P constant: how are T and V related?

This is called Charles’ Law.2

2

1

1

constant

T

V

T

V

P

nR

T

VnRTPV

Jacques Alexandre Cesar Charles

French scientist

1746 – 1823

Built and flew the first

large hydrogen-filled

balloon.

Kinetic Theory of the Ideal Gas

Macroscopic properties of a gas: temperature, pressure, volume, density

Microscopic properties of the particles making up the gas: mass, velocity, momentum, kinetic energy

How are they related?

Kinetic Theory of the Ideal GasConsider a gas molecule contained in a cube having

edge length L.

The molecule’s mass is m, and

its velocity (in the X direction

only) is v.

Time between collisions with the

right-hand wall:

v

Lt

2

Kinetic Theory of the Ideal GasThe time between collisions with the right-hand wall is

just the round-trip time:

From the impulse-momentum

theorem, we can calculate the

average force exerted on the

particle by the wall:

v

Lt

2

mvvmppptFJ f 0

Kinetic Theory of the Ideal GasSubstitute for the time and simplify:

By Newton’s third law, the average

force exerted on the wall is

L

mvF

mvv

LF

mvvmppptFJ f

2

0

22

L

vmF

2

Kinetic Theory of the Ideal GasThe average force on the wall from one particle is

If there are N particles, andtheir directions are random, wecould expect 1/3 of them to bemoving in the X direction.

Total force on the wall:

L

vmF

2

L

vmNF

2

3

Kinetic Theory of the Ideal GasAverage pressure on the wall:

But So:

3

2

2

2

33

L

vmN

LLvmN

A

FP

3LV

NkTvmN

PVV

vmNP 2

2

3

3

Kinetic Theory of the Ideal GasSubstituting kinetic energy:

So, we see that for an ideal gas,the average molecular kinetic energyis directly proportional to theabsolute temperature.

kTKE

KEvmvmkT

2

3

3

2

2

12

3

1

3

1 22

Kinetic Theory of the Ideal Gas

This result is true for any ideal gas.

By a similar argument, if an ideal gas is monatomic (the gas particles are single atoms), the internal energy of n moles of the gas at an absolute temperature T is

kTKE2

3

nRTU2

3