Gas Lecture NOtes UNSW

7/30/2019 Gas Lecture NOtes UNSW

1/21

PHYS 2060

Thermal Physics

Macroscopic observations about gases

Boyles law

Charles law

The missing gas law

The combined gas law

Joules law

The ideal gas law

The adiabatic gas relations

PVT surfaces and diagrams

The PV diagram

Isotherms and adiabats

Real gases Van der Waals equation

PHYS2060 Lecture 4 Gas Laws

Updated: 2/08/2007 10:46 AM

Lecture 4: Gas lawsLecture 4: Gas laws


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PHYS 2060

Thermal Physics

Macroscopic observations about gasesMacroscopic observations about gases

We are now going to switch back to thinking about the macroscopic properties of

gases, and look at how the three key measurable parameters that we discussed in the

previous two lectures Pressure, Volume and Temperature are related in gases.

These outcomes emerge from two of the most important early experiments performedin thermodynamics.

The first is Boyles law, which relates pressure and volume at constant

temperature.

The second is Charles law, which relates volume and temperature at constant

pressure.

By combining these two observations, you can end up (quite simply) with the idealgas law

We will also see where the adiabatic gas relationship PV = constant comes from.


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PHYS 2060

Thermal Physics

Boyles law was discovered by an Irishman, Robert Boyle, in 1662.

His basic observation was that:

Boyles lawBoyles law

till further trial hath more clearly informed me, I shall not venture to determine,whether or no the intimated theory will hold universally and precisely, either incondensation of air, or rarefaction: all that I shall now urge being, thatthe trial

already made sufficiently proves the main thing, for which I here allege it; since by it, itis evident, that as common air, when reduced to half its wonted extent, obtainednear about twice as forcible a spring as it had before, so this thus comprest airbeing further thrust into half this narrow room, obtained thereby a springabout as strong again as that it last had, and consequently four times as strongas that of the common air(BW, 3:60, Birch 1772, I:159). 1

1. Sourced from http://plato.stanford.edu/entries/boyle/#5 as are both pictures shown.

In other words, all he suggests is that air, as long as it stays a

gas, will have a pressure inversely proportional to its volume, or

PV= k, where kis a constant (not the Boltzmann constant). We

can also write it as:

2211VPVP = (4.1)

Boyles law is pretty easy to demonstrate, and it is commonly

observed in real life, but in the 1600s, it was state of the art!


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PHYS 2060

Thermal Physics

Charles law has a more complex history. It was discovered by

Joseph Gay-Lussac in 1802, but since he referenced unpublished

work by Charles from 1787, it ended up bearing his name instead.

Charles lawCharles law

(4.2)

In addition to hot-air ballooning, Charles law is central to convection, which appears

in many places from heat transport in liquids to thunderstorm cells.

Incidentally, Jacques Charles made the first flight in a hydrogen

balloon in 1783 (only a week or two after Montpelliers first flight),and while Boyles law was about learning the properties of air, this

discovery was all about flying balloons, which were all the rage in

France at the time.

Their finding was that the volume of a gas is directly proportional to its temperature atconstant pressure, or more simply V= kT, where kis a constant (n.b., not the

Boltzmann constant, which I will always write as kB). We can also write this as:

2

2

1

1

T

V

T

V=


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PHYS 2060

Thermal Physics

Given Boyles law P1V1 = P2V2 (eqn. 4-1), which is at constant T, and Charles LawV1/T1 = V2/T2 (eqn 4-2), which is at constant P, it is probably clear that there is a third

equivalent that we should get for constant V.

The missing gas lawThe missing gas law

V/T = k

PV = k

P

TV

?

Charles

Boyles

(4.3)

which is the remaining gas law relating Pand Tat constant V. Unfortunately, it doesnt

have a name like the other two do.

If we take Boyles law at constant V(which now makes it constant Vand T) we get P1= P2. And if we take Charles law at constant V(which now makes it constant Vand P)

we get 1/T1 = 1/T2. If you compare these, it should be clear that the general case is:

2

2

1

1

T

P

T

P

=

P/T = k


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PHYS 2060Thermal Physics

If you now look at these three relations Eqns 4-1, 4-2 and 4-3 it becomes pretty

clear that there should be one equation that covers all three cases. Intuition tells us it

should look like:

The combined gas lawThe combined gas law

(4.4)

This is called the combined gas law, and youll see that for constant T(i.e., T1 = T2)

you get eqn 4-1, for constant Pyou get eqn 4-2, and for constant Vyou get eqn 4-3,

all starting from eqn 4-4.

2

22

1

11

T

VP

T

VP=

V/T = k

PV = k

P

TV

?

Charles

Boyles

P/T = k

kT

PV=Or, more importantly:

The ideal gas law is the specific case ofk= nR= NkB, but we need to return to our

microscopic view to get this constant out (or measure it experimentally).


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Remembering back to Lecture 2, our analysis of pressure in the kinetic theory gave usthe result:

Joules lawJoules law

(2.11)UPV3

2=

n.b., holds for amonatomic gas

only!

2

2

1

2

3mvTk

B= (3.7)

while in Lecture 3, we looked at temperature and found:

We can take this result one step further for an ideal monatomic gas, where the

internal energy is due to the kinetic energy of the atoms alone, and write:

TNkmvNUB

2

3

2

1 2 == (4.5)

This brings us directly to Joules law, which states:

The internal energy of an ideal gas depends only on its temperature.

Note that this is the other Joules law, the more famous one is that the

heat Qproduced by a current in a wire is Q= VIdt, where Vis the voltage

across the wire, Iis the current and dtis the time the current flows for.


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If we now combine these results, we get:

Ideal gas lawIdeal gas law

(4.6)TNkTNkUPVBB

===2

3

3

2

3

2

We often dont use N, because in most cases its such a large number that its painfulto use, so instead we use the number of moles n = N/NA, where NA = 6.022 10

23 mol-

1 is Avogadros number. Which gives:

(4.7)( ) nRTTkNN

NTNk

N

NTNkPV

BA

A

B

A

A

B====

Youll notice that weve now defined a new constant R= NAkB = 8.316 J/mol.K, usually

called the ideal gas constant. I never bother remembering the exact value because

its just one moles worth of Boltzmanns constant so I can always get it from NAkB.

Hence we have two forms of the ideal gas law:

(4.6)TNkPVB

= (4.7)nRTPV =


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An interesting thing to realise here is that we can rewrite eqn 4-6 as:

Features of an ideal gasFeatures of an ideal gas

(4.7)Tk

PVN

B

=

What this tells us is that equal volumes at the same pressure and temperature have

the same number of molecules N, even if they are different gases! This is reallyuseful if youre a chemist because it provides an easy way of measuring out an

amount of gas.

We can also see eqn. 4.6 as: (4.8)T

Pv

nT

PVR ==

In an ideal gas Pv/Tis constant

and equal to R. By plotting Pv/T

vs P for different T, we candetermine how ideal a gas is.


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Weve now got a number of important relationships between P,Vand Tfor gases, buttheres one last important set to go the adiabatic gas relations.

The adiabatic gas relationsThe adiabatic gas relations

An adiabatic process is one where no heat is allowed to enter or leave a system. They

are given a special place in thermal physics because they allow us to avoid some

complications in many problems. They can be achieved in two ways:

Fast: A process can be made adiabatic by making it happen very quickly

compared to other characteristic times in the system (I.e., heat leak rates, etc). In

this sense, we aim to beat the heat.

Well insulated: A process can be made adiabatic by insulating it very well from itssurroundings, in a sense, we aim to keep the heat in.

The most important aspect of an adiabatic process relates to the first law of

thermodynamics. If the heat Qentering or leaving the system is zero, then any change

in the internal energy Uof the system will be due to work Wdone by or on the system.

In other words:

(4.9)dWdU =n.b., holds only for

adiabatic processeswhere dQ= 0.


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If we go back to Newtonian mechanics, the work done by or on an object is just the

force applied by or to the object times its displacement. If we apply this to our piston

and cylinder earlier on:


OK, next Im going to generalise the result in Eqn. 2.11 a little, to:

(4.10)( ) PdVPAdxdxFdW ===

(4.11)( )UPV 1= such that for a monatomic gas = 5/3, giving back our (5/3 1) = 2/3 we had in Eqn.

2.11. Well see the reason for this next week, but its probably clear that wontalways equal 5/3.

So we can rewrite Eqn 4.11 as U = PV/( 1) and so a small change in U could be

considered to be the combination of a small change in Vat constant Pand a small

change in Pat constant V, so we can write:

(4.12)1

+=

VdPPdVdU


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We now have two expressions fordU (the other is dU= dW= PdV) so lets equatethem:


(4.13)1

+=

VdPPdVPdV

(4.14)VdPPdVPdVPdV +=+

multiplying both sides by 1 gives:

we can then subtract PdVon each side:

(4.15)VdPPdV =

divide both sides by PVand pull to one side:

(4.16)0=+

P

dP

V

dV

You might spot that Eqn. 4.16 is something we can solve easily, because the integral

ofdx/xis just ln(x). This relies on being a constant, which it is for a simple gases at

least. So the solution to Eqn. 4-16 is:

(4.17)CPV lnlnln =+

where Cis a constant.


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Finally, we can take the exponential of both sides of Eqn 4.17 to get our final result:


(4.18)CPV =

There are equivalent relations between V& Tand P& T, which can be obtained quite

simply by using the combined gas relation to swap PorVout forT.

Ill let you work these out for yourself (because you willneed to know how to do it in

the exam), but the results are:

V T= C

PV = CP

TV

P1 T = CAdiabaticrelations

A final reminder: These relations are only used when a process (i.e., some change in

the system from an initial to a final state) is adiabatic. Otherwise, dQ 0, dU dWand

you can only use the combined or ideal gas laws.


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PVT surfaces and diagramsPVT surfaces and diagrams

An essential skill is to be able to read/draw the various PVT relationships.

It might be evident that this PVTstuff can get rather complicated, we have a lot of

different relationships going on PV= nRT(ideal), PV = const, etc (there are others).

One way to cope with these is to present the information visually in the form either of

a 3D graph ofPvs Vvs T, or one of the 2D sections Pvs V, orPvs T, orVvs T. All

four of these are shown below for an ideal gas.


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The PV diagramThe PV diagram

The most common section is the PV-diagram, which you will see very frequentlythrough this course. It is probably wise to get used to the four key processes

isothermal (const. T), isobaric (const. P), isochoric (const. V) and adiabatic (const.

heat Q), as shown below.


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Isotherms vs AdiabatsIsotherms vs Adiabats

On a PV diagram, isotherms and adiabats look quite similar (curved lines), but there is

a crucial difference.

An Isotherm is a hyperbola (PV= k) and an adiabat is not (because PV = kand 1).

This, combined with the fact that Tis not constant in an adiabatic process (Qis

instead), means that adiabats span between two isotherms, one corresponding to Tiand one corresponding to Tfas shown below.


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Real gases Van der Waals equationReal gases Van der Waals equation

Finally, I want to mention that you can add to the simple ideal gas equation to better

account for the real behaviour of gases (few gases actually behave as ideal gases,

particularly ifPlarge orTis small).

One of the more famous examples is the Van der Waals model, where two

corrections are added to the ideal gas law to make its predictions more realistic.These are:

Intermolecular forces: The first is a correction to the pressure to account for the

existence of intermolecular forces, replacing Pwith P+ a/v2, where vis the

volume per mole v= V/n.

Finite particle size: The second is a correction accounting for the volume

occupied by the molecules themselves, replacing vwith v b.

This results in a real gas law of the form:

( ) RTbvv

aP =

+

2( ) TNkNbv

V

aNP

B=

+

2

2

(4.4)or


18/21


Real gases Van der Waals equationReal gases Van der Waals equation

The 3D PVT-surface and the PV-diagram for the Van der Waals gas are shown below.The constants a and b vary from gas to gas and have units of Jm3/mol2 and m3/mol.

I will leave it at that for now I very strongly recommend reading pages 180-185 of

Schroeder for more on this topic, so much so that I will scan this section of the book

and put it on the website for you.


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Some useful websitesSome useful websites

There are some really nice websites relating to lectures 3 & 4. They have some javaapplets that you can play with to get a better idea of how some of the things weve

discussed work.

Thermal equilibrium: http://jersey.uoregon.edu/Thermodynamics/therm1a.html

Gas laws: http://phet-web.colorado.edu/new/simulations/sims.php?sim=Gas_Properties


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AssignmentAssignment

The assignment will be to do some calculations related to the feasibility of filming a

scene like this in real-life.

The assignment will be due at the Thursday lecture in week 6 (Thurs. 30th August).

Although there will be right and wrong ways to approach the questions, the correct

answers will not be precise. There are things you will need to estimate, reason, etc.

in the assignment, and part of the marks will relate to how you handle these. Thequality of your reasoning and explanations will also be very important.

Do NOT leave this to the last minute, it will be hard to do in a rush. Also, I am happy to

discuss things if they are unclear, but may decline to answer some questions.


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Boyles law states that pressure Pis inversely proportional to volume V, that is, PV=const .

Charles law states that volume Vis directly proportional to temperature T, that is, V=

kTwhere kis a constant. I always remember which is Charles law as he was French

and into flying balloons the hot air balloon is all about increasing a gas volume by

heating it.

If you combine these two laws, you get the combined gas law PV/T= const. For an

ideal gas, this constant is equal to NkB or nR. R is just a moles worth of (or

Avogadros number times) kB.

In the next lecture we will take a break from P,V and T and talk instead about the

specific heat of gases and how this led to the first inklings that there was something

beyond classical physics that something we now know as quantum mechanics.

Joules law says that the internal energy of an ideal gas depends only on its

temperature.

SummarySummary

Van der Waals came up with a real gas law by adding corrections for intermolecular

forces and the volume of the molecules to the pressure and volume respectively.

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Gas Lecture NOtes UNSW