Wednesday, November 11, 1998

Chapter 10: Zeroth Law, Temperature Ideal Gas Law Kinetic Theory of Gases

Thermal Physics, Part I

The material in this chapter is almost certainlyreview for any of you who have sat througha chemistry course, so we’ll move ratherquickly. If you have trouble with this material,come by and see me. I’m happy to work withyou the extra hours you may require to masterthis material.

Cola

Insulated “coolers”

I’ve got a can of cola at room temperature.I like my cola cool. If I put a block of icein one “cooler” and my can of cola in another,what happens to my cola?

Cola

Insulated “coolers”

This is an example of a system in which thetwo objects are thermally isolated from oneanother: that is, the cola and block of icecannot exchange energy (heat) with oneanother.

If I want to cool off the can of cola, I’m goingto put it in the same “cooler” as the ice block.In this situation, the two bodies are said tobe in thermal contact, which means they CANexchange energy (heat).

Cola

When the energyexchange betweenthe two objectsceases, the twoobjects are saidto have the sametemperature.

If objects A is in thermal equilibrium with B,

A B

And object B is in thermal equilibrium with C,

C

Then object A is in thermal equilibrium with C.

T T TA B C

We will avoid the Fahrenheit scale with whichwe are all most familiar from the eveningweather forecasts. In the Fahrenheit system,water freezes at 32o and boils at 212o.

The Celsius (or Centigrade) scale is morenatural physical scale with which to measuretemperature. In the Celsius system, waterfreezes at 0o and boils at 100o.

The best scale on which to measure temperaturesin physics problems, however, is the Kelvin scale.In the Kelvin system, water freezes at 273.15 andboils at 373.15. The Kelvin scale of temperaturenicely translates temperature to energy.

The scale is chosen so that at 0 K, the moleculeswithin a gas (assuming the substance couldremain a gas) would come to a complete stop.We’ll explore this more when we talk about thekinetic theory of gases.

Notice that 1 K = 1oC, so converting betweentemperature in the Celsius and Kelvin scalesis simply done:

T TC 27315.

Temperaturein Celsius

Temperaturein Kelvin

The response of materials to increases intemperatures is typically to expand, withone exception: water expands when cooled!

Section 10.3 of your book covers the responseof materials to changes in temperature (I.e.,thermal expansion). You will NOT be heldresponsible for this material on the test. Hereare the results, FYI:

Linear expansions L L T 0

L0 L = coefficient oflinear expansion

Area expansions

Volume expansions

A A T 0

V V T 0

A

A0

= coefficient ofarea expansion

V

V0

= coefficient ofvoulme expansion

A property related to the molecule itself, youcan use the atomic weights associated witheach given element to compute molar mass.

For example, oxygen has an atomic weight of16. This means that 1 mole of atomic oxygenwill have a mass of 16 grams.

1 mole of ANY substance contains Avagadro’snumber of molecules

N A 6 022 1023. molecules

Most gases withwhich we deal ona day-to-day basisbehave as idealgases.

Ideal gases can be identified by the fact thatthey obey the ideal gas law, a physical lawthat governs the relationship of the pressureof a gas, the number of molecules in the gas,the volume of the gas, and its temperature.

We can reason out this relationship fairlyeasily. Keeping everything else constant...

What happens to the temperature ofa gas if we increase its pressure?

Pressure is proportional to temperature.

P T

What happens to the pressure of agas if we increase the number ofmolecules for fixed V and T?

Pressure is proportional to thenumber of molecules. P NWhat happens to the volume of agas if we increase the number ofmolecules at fixed P and T?

Volume is proportional to N. V N

What happens to the pressure of a gasif we increase its volume?

So pressure is inversely related to volume.

PV

1

Summarizing, we have:

P T P N PV

1

Therefore, we can confidently state that

P kNT

VB

where kB is the proportionality constant

known as Boltzman’s constant.

kB 138 10 23. J / K

PV Nk TBMore typically, theideal gas law iswritten as:

It is also often convenient to write the idealgas law using the number of moles ratherthan the number of molecules. In this case,we have

PVN

NNk T

N

NN k T nRTA

AB

AA B ( )

PV nRTSo a common, butequivalent form ofthe ideal gas law, is:

Where n is the number of moles of gas (N/NA)and R is the ideal (universal) gas constant

R

8 314

0 08206

. J / mole / K

. L atm / mole / K

What volume is occupied by 1 moleof gas at temperature 0oC andpressure (1 atm)?

PV nRT( ) ( )( . )( )1 1 0 0821 273 atm mole KL atm

mole KV

V 22 4. L

How many molecules will you find in1 cm3 of gas at temperature 0oCand pressure of 1 atm?

PV Nk TB( )( ) ( . )( )101.3 10 138 10 2736 23 10 Pa m K3 3 J

KN

N 2 68 1019. molecules

d

I’ve got a boxfull of molecules.They’re bouncingaround in my boxin all differentdirections, at avariety of speeds.

d

What happenswhen a moleculeruns into a wallof my box?

It exerts a force onthat wall. Can wefigure out themagnitude of thisforce?

Let’s simplify our problem first by examininga 1-D gas instead of a 3-D gas. Let’s assumethat our molecules are only capable of movingin the x-direction, and that all collisions theymake with the walls are perfectly elastic.

vxi

m

d

BEFOREvxf

m

d

AFTER

vxi

m

d

BEFOREvxf

m

d

AFTER

What is the magnitude of the force that hasbeen applied by the right wall to our molecule?

F mv

t

mv

txi

2