bahan ajar gas ideal

8/8/2019 bahan ajar gas ideal

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[ Ideal Gas ] 1

IDEAL GAS

Hydrogen is most abundant matter in the

universe. This means our atmosphere

should have has hydrogen in great number.

But in fact, in our atmosphere number of

hydrogen is less than number of nitrogen

and oxygen. Why is this happen? How can

hydrogen release to the outer space?


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[

d

l

s ] 2INDICATOR

Cogniti e1. Name 3 ideal gas characters2. E lain 3 ideal gas characters 3. E lainlawthat accom lishideal gas character4. E lain 3 u ontool Boylelaw GayLussac5. E lainideal gas pressure according to kinetictheoryof gases 6. E plainequipartitiontheoryofenergy7. Sol e problemnatural philosophywhichis engaged ideal gas character,

kinetictheoryof gases, and equipartitiontheoryenergy

8. Analyze relationship variablethat available onideal gas laws , kinetictheoryof gases, and equipartitiontheoryofenergy

Psychomotor

1. Utilizing up ontool Boyle Gay-LussaclawAffective1. Pointing outconfidence2. Involve selfin discussion3. Onhands4. Pricing opinion5. Pointing out attention onmaterial / study


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IDEAL GAS

BERTUMBUKAN

LENTING SEMPURNA

NO FORCEAMONGTHE

MOLLECULE

MOLLECULE + WALL

Have

characteritic

Except If

between

causes

EQUIPARTISITHEORYOF ENERGY

E

Moverandomly

di ka

TEMPERATURE

So

PRESSURE

dipengaruhimenyebabkan

MONOATOMICDIATOMIC

mollecule

Consist of

Can be among

have

DEGREEOF

FREEDOMRelated to the

CO


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[ Ideal Gas ] 4

Have you blown the balloon till the balloon blown?

Why did the balloon blow up? The balloon blew up be

cause the air that we have blown will press the

balloons wall again and again. If the balloons wallcant hold the pressure from the air so the balloon will

blow up. Another cause is the temperature of air in the

balloon increases perpetually. Why can it be

happened? Learn this chapter so that you will

understand.

In this chapter, you will learn about relationship

between gases temperature, gases volume, and gases

pressure, and then you will learn about kinetic theory

of gas and equipartition theory of energy.

A.Ideal GasTo learn about characteristic of gas, we use gas

ideal model. In fact, we cant find gas ideal model, but

in high temperature and low pressure real gas shows

characteristic like ideal gas.

1. Characteristic of Ideal GasHow does characteristic of ideal gas? And what

is difference between ideal gas and real gas? To

know, do the activity below.

AAAccctttiiivvviiitttyyy 111

Characteristic of ideal gas

What you need?

6 marbles, transparent box

What to do?

1. Put the 2 marbles into the transparent box.

2. Shake the transparent box, observe the marble

when collides the transparent boxs wall or collide to

the others.3. Repeat procedure no. 2 with 4 marbles and 6

marbles.

Analysis

Imagine that marbles is the gas molecule. How

does the moves of the marbles if you shake the

transparent box?


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[

d

l

s ] 5Bas

d on your obs

rv

which one that more

often happens marb e collides with other marble

or marblescollides with transparent boxs wall?

If many more marbles in the box, what do you

observe when you shake the box?Make conclusion aboutyour observe.

In activity 1, you have been shown how

characteristic ofideal gas is. Inideal gas model, is

assumed that:

1. A gas consists of parts call dmol cul .Eachmolecule is identical so that we can ! tdistinguishto each other

2. Mol" cul" s of gas id" al move randomly.Because of that so there are possible to

collide to each other or collide with wall#

sroom.3. Gas molecules are scattered in all part of

container. Gas molecule doesn $ t concentrateatcertain pointincontainer.

4. The distance among molecules is very largecomparedsize of molecules.

5. There is no interaction force between mole-cules,unless thereis collisionbetweenmole-cules or between molecule and walls con-tainer.

6. All collision between molecules or betweenmolecule andwalls container are completelyelastic. Gas molecules can be seen likeslipperyhardball.

7. Gas molecules satisfy Newtons law of mo-tion.

Because ideal gas has characteristics above soideal gas satisfies gases laws precisely.

2. Boyle% s LawHow does the relationship between gases

pressure and gases volume if the gases tempe-ratureis constant & To understand the relationship

between gases pressure and gases volume, do the

following activity.


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[ Ideal Gas ] 6


RELATIONSHIP BETWEEN GASES PRESSURE AND

GASES VOLUME

Figure above is image of tool which be used on expe-

riment to know relationship among volume and pressure.

Tool at set on constant temperature and amount of gases

molecules to make an abode. Container room can be varied

by decrease and increase the weight on the piston.

Changing volume and pressure gets at observes on grader.Acquired data of experimental that as follows:

PREASURE

(psi)VOLUME (mL)

17,27 29,8

17,98 28,62

18,39 27,98

19,38 26,55

20,59 24,99

22,33 23,04

24,64 20,88

27,06 19,01

28,69 17,93

30,63 16,8

32,48 15,84

33,17 15,51

37,07 13,88

44,74 11,5

'

ol( )

e 0 gauge

scale

Te) 1

e 2 ature

gauge

Pressure

gauge

pistonweig 3 t

Room t 3 at contain

gases molecules

Gases molecule

Figure.


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[IdealGas] 7Of that data makes graph PRESSURE versus

1/V4

LUME on graph paper.

1. How isgraphscurve form thatyou make?

2. How is relationship amongpressure ofgases and

volume ofgases?3. Make conclusion of graph that already youve

made!

There is ideal gas in enclosed container like

figure . Containers volumechangeablebyadd or

reduce weight that put on the piston, so that the

piston can moves up and down. If weight on the

pistonincreases and gases temperatureis constant

so the piston will move down. So containers

volume decreases,because of it, moved room of

gas molecules decreases and then gas molecules

collide more often, between gas molecules and

also with containers wall. Because gases mo-

lecules collide more often so that pressure in the

container thatcontains of gas increases.

Ifthe weight on the piston is decrease, so the

piston will move up and containers volume de -

creases. Because of it, gas molecules more free to

move than last condition. So that the collision of

gas molecule decreases and gases pressureincon-

tainer will decrease. From this result canbe con-cludebystatement as follow:

If gasestemperature in closedroom is

constant, so gases pressure reverse

equalto the gases volume

Statement above canbe written mathemati-

callyas follow

Vp

1g

or

kp 5 !

2211 VpVp !

V1 = initial volume;V2 = final volume

P1= initial pressure;P2 = final pressure

[ 1 ]


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[IdealGas] 8A scientist that investigated relationship

between gas volume and gas pressure is Robert

Boyle (1627-1691). Boyle has investigated rela-

tionship between gas pressure and gas volume in

closed container. Ne6

t, statement above is calledBoyle 7 s Law.

3. GayLussac 8 s LawHaveyouever seen airballoon

9

Before airbal-loon flies, gas in airballo0nmustbeheated so e@ -

pand and finallyairballooncan fly. And ifyou put

balloonunder the suninhot day,thatballoon will

blow up finally. So there is relationship between

gases temperature and gases volume. To show re-

lationship between gases temperature and gases

volume do activitybelow.

Problem Solution

1. A gas at temperature of 27oC andpressure of 105 Pa hasvolume of 30 liters.Determine the volume of the gasgiven that the pressure becomes 2,5 105 Pa and

the temperature isconstant?

Solution :

Given that: T1 = 27oC ; T2 = T1 ; V1 = 30 liters ; P1 = 10

5 Pa ;

P2 = 2,5 105 Pa

Question : V2Answer:

Thisgas is in constant temperature, so V2can be foundby Boyles Law

literP

VPV

VPVP

12105,2

30105

5

2

11

2

2211

!

!!

!

Exercise 1

1. A gas atconstant temperature of 37oC andpressure of 1,5 105 Pa hasvolume ofV1. Determine the volume V1 ofgasgiven that the pressure becomes 2,5 10

5 Pa

andvolume becomes 8 liters? (13,33 liter)

2. A gas has initial pressure of 3P. If final volume of a gas is 30% decrease of initialvolume, what is initial pressure ofgas? (4,29P )


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[ Ideal Gas ] 9


RELATIONSHIP BETWEEN GASES

TE

MPER

ATURE

AND GASE

S VOL

UME

We need:

Ping pong ball, electric heater, AC voltage source, water

What to do:

1. Fill the electric heater with some water.

2. Connect the electric heater to the AC voltage

source.

3. Dent the ping pong ball and then put into the

electric heater.

4. Lets the ping pong ball in electric heater till the

water in electric heater has boiled. Observe theping pong ball.

Analysis

Based on the result of the experiment, what

happens to the ping pong ball? How do you think it

happens?

Make a conclusion from the result of the

experiment.

A scientist that investigated relationship

between gases volume and gases temperature isJos A ph GaB Lussac. From his experiment, he

finds a fact that C

If gas D nclosed contaD ner D s heated at

constant pressure gases volu E eF D

ll

D ncrease proportional to absolute

gases temperature.

G tatement above called Gay Lussacs laws.

Mathematically can be written as followH

constant!

T

V

Or

2

2

1

1

T

V

T

V!

V1= initial volume I VP = final volume

T1= initial temperature I T2= final temperature

[ 2 ]


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[IdealGas] 10In Gay Lussac

Qs Law, we use absolute gases

temperature. If gases temperature intoC, absolute

gases temperatureTis K (Kelvin) and written as:

T= t+273

Relationship between temperatureand volumecanbe drawnin graphic,like

figure . Graphic of relationshipbetween

temperature and volume is linear with

gradient. If graphic is drawntilllowtem-

perature, it will cut out in -273oC or 0 K.

This is meantthatif all of gas is cooled till

273oC so volume willbe 0.

Problem Solution

1. A gas in closedroom hasvolume of 7 cm3 and temperature of 33oC. Then,volume ofroom decreasesbecome 14 cm3. Determine the final temperature of

thatgas ifgaspressure isconstant.

Solution:

Given that: V1 = 7 cm3 T1 = 33

oC V2 = 14 cm3

Question: Gay Lussacs law

Answer:

We use Gay Lussacs law, because in constantpressure condition. And we have

to use absolute temperature to solve thisproblem.

T1 = 33+ 273 = 310 K

KV

TVT

T

V

T

V

6207

31014

1

12

2

2

2

1

1

!

!

!

!

Exercise 2

1. A gas at temperature of 37oC andconstantpressure of 0,5 atm hasvolume of 2m3. Determine the volume V2 ofgasgiven that the temperature becomes 43

oC?

(2,06 m3)

2. Amount ofgas experience isobaricprocessso that absolute temperaturebecomes twice of initial. What isvolume ofgas now? (twice of initial volume)

T(oC)

V(m3

)

T2T1

V1

V2

-273

FiR

ST

e .R T

aphicof relationship between

temperatS

reand v0lS

meatconstantpressure


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[ Ideal Gas ] 114. General U quation of Ideal Gas

If equation [1] and [2] are combined, the result

is Boyle Gay Lussacs Law, which is

2

22

1

11

T

VP

T

VP!

Or

TpVg

V quation [1] W [3 ] hold only on gas in an en W

closed container ( with no leakage), in a way that

mass of the gas is constant. What about gas that

leaks or gas with changing mass? To answer this

equation, we need a gas equation that gives soW

lution when condition of the three variables is

changing.Have you blown a balloon? When you blow a

balloon so you are adding molecules to the balloon

and balloons volume and pressure will change.

Based on this fact, gases temperature, volume and,

pressure is influenced by number of molecules

(N).

NTpVg

Different gas has difference volume, pressure,

and temperature although has same number of

molecule. X o we need a constant that used for all

gas, that constant is Bolt Y mann Constant (k). Then

proportionality NpVg can be written as

kNTpV!

p = pressure

V= volume

N= amount of molecule

T= temperature

K= Bolt a mann constant (1, 3 b v10-23 J/ c )

We can change amount of molecule Nwith

nNd

N=nNd

Nd is number of Avogadro. This number is use

for state how many particle is, that called mol,n.

Based on experiment, number of Avogadro is

6,022 1023. For example, 1 mol marble

[3]

[4]

FigureBe

of ing a bae e

oong

eansf

e add nug

ber ofg

oe

eh

ue

e.


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[IdealGas] 12contain 6,022 1023 marbles. 1 mol water contain

6,022 1023 waters molecule.

NA = 6,022 1023molecules per mole

= 6,022 1026molecules per kilomole

n is number ofmol of gas. Mol isntmass but

sizes that statehowmanyparticleis.

Number ofmol of gas ,n,is proportion ofmass

of gas,m, and relativemass of gases molecule,M.

rM

mn !

m is mass of gas in kilogram.Mr is relative

mass of gases molecule.Mr.is mass of gas in kilo-gram of 1 kilomol substance (kg/kmol). For

ei ample, 1 kmol of Carbon C-12 has relativemass

Mr= 12 kg/kmol.

If we substitute N = nNA to equation [4], we

get

TknNpV A!

Number of Avogadro multipliedbyBoltzmann

Constant, kNA, in equation above is general

constant of gas,R, so equation [4] canbe written

as

nRTpV!

p = pressure

V= volume

n = number ofmole of gas

T= temperature

K= Boltzmanconstant (1,38v 10-23 J/K)

[ 5 ]

Problem Solution

1. A rubberballoon with volume of 20 liters filled with oxygen atpressure of 135atm and temperature of 27oC. Determine the oxygen massgiven that R = 8,314

J/(mol K).(Mr=32 gr/mol)

Solution:

Given that: T = 27oC = 300 K ; V = 20 liter ; P = 135 atm

R = 8,314 J/(mol K) = 0,0812 L atm/(mol K)

Question: mass of oxygen


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[IdealGas] 13Answers: we use general equation in this form pV = nRT

np

TpV! First, we look for number of mol of oxygen thatcontain in balloon

molRT

pVn 6,109

3000812,0

20135!

!!

To find mass of oxygen, we multiplied number of mol with relative mass of

oxygen

Then oxygens mass in the balloon is

m =n Mr = 109,6 mol v 32 gr/mol = 3507,2 gr

2. Atstandardcondition where the temperature is 0oC andpressure is 1 atm,what is the volume of oxygen of mass 4 gr? (Mr = 32 kg/kmol ; R = 8,314

J/(mol K); 1 atm = 105 N/m2 ).

Solution:

Given that:dont forget, we mustchange into proper unitT = 0oC = 273 K

m = 4 gr = 4.10-3kg

p = 1 atm = 105 N/m2

R= 8,314 J/(mol K) = 8314 J/(kmol K)

Mr = 32 kg/kmol

Question:volume of oxygen

Answers:remember that number of mol is mass ofgas in kg orgdividedby

relative mass of molecule or atom in kg/kmol org/mol

nRTpV!

RTM

m

pVr

!

33

5

-3

108,21032

2738314104m

pM

mRTV

r

v!

v

vv!!

3. Determine the massdensity of an gas ideal that has massrelative of 28,8kg/kmol at 20oC and at atmosphericpressure (1 atm).

Solution :

Given that: Mr = 28,8 kg/kmol

P = 1atm = 105 Pa

T = 20oC = 293 KQuestion : massdensity

Answer:

35

/1202938314

8,2810

;

mkgRT

pM

RT

pM

V

mRT

M

mpV

r

r

r

!

!!

!!

V


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[ Ideal Gas ] 14

5. The Application of Ideal Gas Lawa)Pressure Gauge

Gauge pressure is difference among

pressure which unknown with atmosphericpressure. Pressure that measured in

generally is gauge pressure.

Pressure gauge consists of hollow cylinder

that contains springing piston. When you

press the edge of it to the air hole of wheel,

gas molecule from the wheel will into

hollow cylinder. Based on kNTpV! ,

number of gas molecule in hollow cylinder

increases then pressure in it will increases

too. This pressure causes a force to the

springing piston as equal air pressure in

cylinder multiplied by cross-sectional area

of piston (F = pA). If spring satisfies

Hookes law, piston is pulled in certain

distance that proportional with force,F, and

pressure, p, too. q cale on pressure gauge

can be calibrated so will only show pressure

in the hollow cylinder or in the other word

show pressure in the wheel.

b) Manometer "U"-TubeUsing a "U"-Tube enables the pressure of

both liquids and gases to be measured with

the same instrument. By use linked vault,

pressure in A as same as in B.

BApp !

ghppoA

Vr

Exercise 3

1. An amount of gas has volume of 600 liter, temperature of 27oC and pressure of 5atm possesses mass of 1,95 kg. Determine the relative mass of the gas.(16

kg/kmol)2. The mass density of an ideal gas at temperature T and pressure P is V. If the

pressure becomes 2P and the temperature decrease into 0,5T, what is the mass

density now?(V final is 4V)

A B

Figures

anos

et

er Ut

ube


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[ Ideal Gas ] 15The "U" is connected as in the figure. and

filled with a fluid called the manometric

fluid. The fluid whose pressure is being

measured should have a mass density less

than that of the manometric fluid and thetwo fluids should not be able to mix readily

- that is, they must be immiscible.

This tool can be used to measured at-

mospheric too. We vary high of tube, h,

such as h1and h2, so we get two value of gas

volume, V1and V2. Then, we use Boyles law

to find atmospheric pressure.

c) Pressure RegulatorAp

u vssu

u v

u vgu

w

ax

ou is a valve that auto-

matically cuts off the flow of a liquid or gas

at a certain pressure. A pressure regulator's

primary function is to match the flow of gas

through the regulator to the demand for gas

placed upon the system. If the load flow

decreases, then the regulator flow must

decrease also. If the load flow increases,

then the regulator flow must increase in

order to keep the controlled pressure from

decreasing due to a shortage of gas in the

pressure system.

A pressure regulator includes a restrictingelement, a loading element, and a

measuring elementy

y The restricting element is a type of valve. It can be a globe valve, butterfly

valve, poppet valve, or any other type of

valve that is capable of operating as a

variable restriction to the flow.

y The loading elementapplies the neededforce to the restricting element. It can

be any number of things such as a weight, a spring, a piston actuator, or

more commonly the diaphragm

actuator in combination with a spring.

y The measuring element determineswhen the inlet flow is equal to the outlet

flow. The diaphragm is often used as a

Figure. Oxygen

y

inders

i

h

ressure regu

a

ors


16/29

[ Ideal Gas ] 16measuring element because it can also

serve as a combine element.

In the pictured single-stage regulator, a

diaphragm is used with a poppet valve to

regulate pressure. As pressure in the upperchamber increases, the diaphragm is

pushed upward, causing the poppet to

reduce flow, bringing the pressure back

down. By adjusting the top screw, the

downward pressure on the diaphragm can

be increased, requiring more pressure in

the upper chamber to maintain equilibrium.

In this way, the outlet pressure of the

regulator is controlled.

B.Kinetic Theory of GasWe have learned about characteristic of ideal gas.

One of characteristic of ideal gas is move randomly.

And concept that learn about characteristic of gas

based on its moving randomly particle called kinetic

theory of gases.

Kinetic theory explains macroscopic properties of

gases, such as pressure, temperature, or volume, by

considering their molecular composition and motion.

Gas that learned in theory kinetic of gases is ideal

gas. Like ideal gas, the theory for ideal gases makes the

following assumptions

The gas consists of very small particles, all withnon-zero mass.

The number of molecules is large such thatstatistical treatment can be applied.

These molecules are in constant, random motion.The rapidly moving particles constantly collide

with the walls of the container.

The collisions of gas particles with the walls of thecontainer holding them are perfectly elastic.

Except during collisions the interactions amongmolecules are negligible (they exert no forces on

one another).

The total volume of the individual gas moleculesadded up is negligible compared to the volume of

the container. This is equivalent to stating that

Figure. sing

e-s

age regu

a

or


17/29

[IdealGas] 17the average distance separating the gas particles is

largecompared to their size.

Themolecules are perfectlysphericalin shape, andelasticinnature.

The average kinetic energyof the gas particlesdepends onlyonthetemperature ofthe system.

Relativisticeffects arenegligible. Quantum-mechanicaleffects are negligible. This

means that theinter-particle distanceis much

larger thanthethermal de Broglie wavelength and

themolecules aretreated as classical objects

The time during collision of molecule with thecontainer's wall is negligible as comparable to the

timebetween successivecollisions.

1. Monoatomic Gases Pressure Based onKinetic Theoryof Gases A gas consists of a large number of particles

that moves randomly. Because of its random

motion, gases particles collides others and

containers wall. This collidecauses pressurein gas.

Everyobjectinmotionhas kineticenergy. A gas

consists of moving particles, meaning that gas

particles possess kineticenergies.

Figure. .Shows a cube of side l where

its wall is completely elastic. The cube

contain a number of gas particles. A particles

withmass m is moving toward the wall with

velocityvx at negativex-axis direction. After

collide the wall, the particles momentum

changes.Since the wall is completely elastic,

the particle willbebounced with the same

speed vxbut in the opposite direction. The

momentum changed experienced by the gas

particleis

(p = p2,x p1,x = mvx-(-mvx) = 2mvx

After colliding to theleft wall with speed vx,the

particle is bounced to the right with speed vx,until

it collides to the right wall. The particle is then

bounced to the left until itcollides to the left wall,

and so on.

z

Y

Fi

ure .acubicalcontainerwhichhasthe sidelengthlcontains gasparticle

x

l

l

l

vx


18/29

[IdealGas] 18The time interval needed by the particle for

doing a back-and-forthmovement in the container

is

(t= 2l/vx

Based onthe Newtons second law,the average

forceexerted bythecontainers wallto the particle

is

l

mv

vl

mv

t

pF x

x

xx

2

2

2!!

(

(!

The average pressure ofthecontainers wallcan

be calculated as the force per unit area. With the

area ofthecontainers wallA = l2, wehave

V

mv

l

mv

l

lmv

A

FP xxx

2

3

2

2

22!!!!

In the calculation above we assume that the

particles motionis parallelto thex-axis. The other

particleis in fact different direction and speed. And

the particles speed is will varywithtime due to the

collision withthe other molecule. Thequantitythat

is constantinthis caseis the speed distribution.

The speed distribution of gas particles is invest-tigatedbyJame Clark Maxwell. Formulation of

the speed distribution contains several statistical

value such as probabilityand average value.

Agreeing with Maxwells statistics, the value 2xv

inequation [5] mustbe replaced bythe average 2xv

whichcovers the whole particles ofthe gas.

Ifthere areNgases molecules with speed inx-

axis directionis v1x,v2x,v3x,,vNx, so total pressure

ofcontainers wallis

)...(22

3

2

2

2

1 Nxxxx vvvvV

mP !

2

xvNV

mP!

With 2xv is average value of square of speed inx-axis

direction.

[ 6 ]

[ 7 ]


19/29

[IdealGas] 1 In fact, gases molecules moves in all direction,

so it not only has 2xv but also

2

yv and 2

zv . Then its

resultantis2222

zyx vvvv !

Inthemodel of kinetictheoryof gas,thereis no

difference of vx, vy, and vz therefore it applies

222

zyx vvv !! , so

2222

zyx vvvv !

22 3xvv !

22

3

1vvx !

Thenequation [ 7 ] canbe written as

2

3

1v

V

mP!

withp = gases pressure (Pa)

V= gases volume (m3)

N= number ofmolecules (particles)2v = average of square of speed (m/s2)

m = mass of a molecule (kg)

Wecan writeequation [8] as

2

2

1

3

2vm

V

P!

From equation above, we now that if there is

gas in enclosed container (value of N, and V is

constant), so gases pressure is only depend on

speed of gases molecules.

Theterm2

2

1

vm is average of kineticenergy, KE

Cause ofit, wecan write gases pressureinterm

of average of kineticenergy

KE

V

NP

3

2!

[ 8 ]

[ 9 ]


20/29

[IdealGas] 20withP= gases pressure (Pa)


N= molecules (particles)

KE = average of kineticenergy(Joule)

For kineticenergytotal

!

!

N

PVNE

ENE

K

KK

2

3

PVEK

2

3!

withP= gases pressure (Pa)


KE = average of kineticenergy(Joule)

Problem Solution

1. Gasespressure in enclosedcontainerdecreases of 81% from initial condition.What about the speed ofgases molecule now?

Solution:

In enclosedcontainer there is no change in volume and number of molecule.

So, pressure onlydepends on speed of molecule. From equation [ 8 ]

112

12

2

1

2

2

2

1

2

2

12

%909,0

%81

%81

3

1%81

3

1

%81

vvv

vv

vv

v

V

mv

V

m

PP

!!

!

!

!

!

So, speed ofgases molecule decrease of 90% from initial condition

2. Determine average kinetic energy of molecule from 2 mole of Neon that hasvolume 22,4 L atpressure of 101 kPa. (at normal condition, neon is

monoatomic)Solution:

Given that: n = 2 mole

V = 22,4 L = 22,4.10-3 m3

P = 101 kPa = 101.103 Pa


21/29

[IdealGas] 21

2. Temperature and KineticEnergyFromuniversalequation of gas

kNTpV!

And fromequation [8],

22

3

1or

3

1vmNPVvN

V

mP !!

Question: average kinetic energyK

E

We need to know how manyparticles in 2 mole of Neon, so

N = n NA

= 2 mole v 6,021023

particles/mol = 12,04 1023

particles

joulePN

VE

EV

NP

K

K

213

23

3

1082,2101011004,12

104,22

2

3

2

3

3

2

!

!!

!

Exercise 4

1. In enclosedcontainer, speed ofgases moleculesbecomes 1,2 from initialcondition. Determine itspressure now.(pressure will be 1,44 of initial

pressure)

2. Kinetic energy of 2 mole monoatomicgas in volume of 20 L is 1,01v 10-20 J.Determine itspressure.(3,37v 10

-17Pa)

3. Kinetic energy of a monoatomicgas in tank with volume of 30 L andpressureof 1 atm is 2,52v 10-21 J. How much gas in tank, in mole?(1 atm = 1,01v 105

Pa )

(3 mole)

4. Based on kinetic theory ofgas, pressure causedbycollision ofgasparticles asfollows

2

3

1vN

V

mP!

What are term (mN)/Vphysically?


22/29

[IdealGas] 22We get

kNTvmN !2

3

1

kNTvmN !2

2

1

3

2

kTEEk

kTKK

2

3atau

3

2

2

3!!

Fromequation [10],canbe said that

1. Thats equationnotcontainterm

V

N,

thats meanmolecule per unit volume,

V

, does notinfluence gas temperature.

2. Equation [10] said thattemperatureisonlyrelated withmolecules move (kinetic

energyor molecules speed).

In equation [8], there is term 2v .Whats the

meaning of term 2v ? Because of gas molecules

arentmoves with same velocity, so wemust define

themeaning of 2v .

Ifinenclosed container there areN1mole-cules

with speed v3, N2 molecules with speed v2, N3

molecules with speed v3, and so on, so average of

square of speed 2v is

N

vN

N

vN

NNNN

vNvNvNvNv

ii

i

ii

i

ii

!

!

!

2

2

321

22

33

2

22

2

112

...

...

Square root of average of square of speed, 2v ,

is effective value of speed or root mean square

(RMS) of speed

2vvv RMSeff !!

vRMSis a kind of average speed ofmolecular .

[ 10 ]


23/29

[IdealGas] 23Fromequation [10],

kTEK

2

3!

kTmv

kTvm

RMS2

3

2

1

23

21

2

2

!

!

m

kTvRMS

3!

Withk = Boltzmannconstant

T= temperature (Kelvin)

m = mass ofeachmolecule (kg)

Equation [10] can alsobe written as

r

RMSM

RTv

3!

Based onequation [12], gas molecule, atcertaintemperature,lighter molecules move faster, ontheaverage,than do heavier molecules.

Then,howis the relationshipbetweenvRMS and

pressure? The relationship between effective speed

and pressure of the gas be obtained by using theexpression

2

3

1vN

V

mP!

TermV

Nmis mass density,V,then wehave

2

3

1vP V!

V

Pv

32!

As we know before, square root of average

square speed is effective speed so

V

Pvveff

32!!

[ 11 ]

[ 12 ]

[ 13 ]


24/29

[IdealGas] 24Equation [13] isntbase equation of effective

speed. Because of ityou cant say that effective

speed is proportionalto square root of gas pressure

and reverse equal to square root of gas mass den-

sity. Why? BecauseVP concern with and depend on

absolute temperature T. So, base equation of

effective speed is equation [12] or [11].

Problem Solution

1. A tankcontains argon gas with relative atomic mass of 40 kg/kmol attemperature of 27oC. Determine the average translational kinetic energyper

molecule and its effective velocity.

Solution:Given that: T = 27oC = 300 K

Mr = 40kg/kmol

Question: EK andveff

Answer: kTEK

2

3! = J1021,63001038,1

2

3 2123 v!v

An argon molecule consist of an atom so the value of Mrcan be replacedby Ar

smA

RTv

r

RMS /5.43240

300831433!

v!!

Exercise 5

1. Helium (Mr = 4 kg/kmol)gas iskept in a container at temperature 273 oC.Calculate (a)the average kinetic energy and (b)value of the effective speed.(a.

1,13v10-20 J, b. 1845,15)

2. Determine the ratio of the effective speed of hydrogen gas molecules (Mr = 2gr/mol) and oxygen gas molecules at the same temperature. (vRMSHidrogen:

vRMSOksigen = 4:1)3. A gas tank hassmall leakage. Pressure in the gas tank is fasterdecrease ifgas

tank filledby helium than filedby oxygen. Why?

4. A gas tube with certain volume contains ideal gas with pressure of 2p. If atconstant temperature, the same gas is inject into gas tube, so itspressure

becomes 2,5p, determine effective speed of it.


25/29

[IdealGas] 25C.TheoremofEnergyEquipartition

The average kinetic energy of gas molecules is

expressedbyequation

!!

kTEkTE KK 2

1

3or2

3

Multiplier factor 3 appears first at 22 3 xvv ! , it

appears because equivalency of average square of

velocitys components

22222 3 xzyx vvvvv !!

This equivalency shows fact that characteristic

of gas doesnt depend on which orientation of

coordinate systemXYZ, and wecan write

kTmvmvmvzyx

2

1

2

1

2

1

2

1 222!!!

Summation of 3 contributions results

equation [8]. Multiplier factor 3 relate with

degree of freedom of monoatomic gas. Each

degree of freedom relates with ability of a

molecule to participate in a 1 dimensional

motion that gives contribution to mechanicenergyofthis molecule.

Amoleculehas a velocitys componentinx-

axis direction that gives contribution to

mechanic energy 22

1xmv . Because there are 3

direction, x; y;z, where molecule moves, so

kineticenergyof a moleculeis

222

2

1

2

1

2

1zyx mvmvmv

. Because of that, monoatomic ideal gas has 3

degree of freedoms, and the mechanic energy per

molecule equal to the average kinetic energy per

molecule

!! kTEE

KM2

13

z

x

y

Figure .monoatomicgas,

translationalmotion in x-axis,


26/29

[IdealGas] 26General statement fromthe result aboveis known

as theorem ofequipartitionenergy

For a gas molecules system at absolute

temperature T, with each molecule hasY degreeof freedom, mechanic energy per molecule or

average kinetic energy per molecule is

!! kTEE

KM2

1Y

Equation [14] state that, averagely, mechanic

energy kT2

1relate witheach degree of freedom.

Whats about diatomic gas? Molecules of

diatomic gas canbeconsidered as a dumbbell(two balls connected by rod) as a shown in

figure. In this model, centre of mass can

move inx-axis direction, y-axis direction,z-

axis direction. This molecule can also rotate

toward one ofthe axis which is perpendicular

to each other. But rotationalmotionto y-axis

is negligible.So thatmolecule of diatomic gas

has 5 degree of freedom: 3 relate with

translational motion, and 2 relate with

rotationalmotion.

!! kTEE

KM2

15

Ideal gas inenclosed container consists of so many

molecules. Each molecule has average kinetic energy

! kTE

K

2

1Y . Internal energy is defined as sum-

mation of allmolecules kineticenergythatcontain in

container. Ifthere is Ngas molecule inenclosed con-

tainer, so internalenergyof gas Uis

!! kT

ENUK

21Y

!!

!!

kTNENU

kTNENU

K

K

2

15gasdiatomic

2

13gasmonoatomic

[ 14 ]

z

x

y

Figure . diatomicgas, translational

motion in x-axis,y-axis, z-axis and


27/29

[IdealGas] 27

General statement of theorem of equipartition

energyis

For a gas molecules system at absolute

temperature T, with each molecule hasY degree

of freedom, mechanic energy per molecule or

average kinetic energy per molecule is

Problem Solution

1. 5 moles of monoatomic ideal gas has temperature of 900 K. Determine theaverage kinetic energy and internal energy.

Solution:Given that: T = 900 K

n = 5 moles

Question:K

E and U

Answer: kTEK

2

3! = J1086,19001038,1

2

3 2023 v!v

J

kTnN

NkTU

A

3,56076

9001038,11002,6523

2

3

2

3

2323

!

vv!

!

!

Exercise 6

1. Neon is monoatomicgas. What is the internal energy of 5 gram neon gas attemperature of 80 oC, given thatrelative molecular mass is Mr = 10gr/mol?

(2199,44 J)

2. Internal energy of 2 molesdiatomicgas at 700 K is 7,25v10-21 J. Determine thekinetic energy of the gas at that temperature. (6,02v10-52 J)

3. Based on figure, rotational motion of diatomic molecule to the y-axis isnegligible. Why?

4. Internal energy of a ideal gas iskinetic energy total of all its molecule. Is it trueor false? Why?

Subchapter Summary


28/29

[IdealGas] 28

!! kTEE

KM2

1Y

Degree of freedom relates with abilityof a molecule

to participate in a 1 dimensional motion that givescontributionto mechanicenergyofthatmolecule.

Monoatomic gas has 3 degree of freedom and

diatomic gas has 5 degree of freedom.

Internalenergyis defined as summation of all mole-

cules kinetic energy that contain in enclosed con-

tainer.

------------------------------------------------------------

1. Monoatomic gashasdifference number ofdegreeof freedom with diatomic gas. Why?(bwt

evaluasi)2. Ideal gases with same number of molecule and

same temperature will have same internal

energy. Isittrue or false?Why.?(bwtevaluasi)

The theory for ideal gases makes the following

assumptions:

The gas consists of verysmall particles, all withnon-zero mass.

The number of molecules is large such thatstatisticaltreatmentcanbe applied.

The total volume of the individual gasmolecules added up is negligible compared to

the volume ofthecontainer.

Themolecules are perfectlyspherical in shape,and elasticinnature.

The average kinetic energyof the gas particlesdepends onlyonthetemperature ofthe system.

Relativisticeffects arenegligible. Quantum-mechanicaleffects are negligible.

This means that theinter-particle distanceis

much larger than thethermal de Broglie

wavelength and the molecules are treated

as classical objects.

The time duringcollision of

molecule with

the container's

wall is negligibleas comparableto

thetimebetween

successive

collisions.

The equations ofmotion of the

molecules are

time-reversible.

These molecules

are in

constant, ran-

dommotion.

The rapidly

moving particles

constantly

collide with the

walls of the

container.

The collisions ofgas particles

with the walls ofthe container

holding them are

perfectlyelastic.

Except duringcollisions

theinteractions

among

molecules

arenegligible

Monoatomic

ases

Pressure

Based on

Kinetic

T

eory of

ases

review

Subchapter Summary


29/29

[IdealGas] 29Based ontheorykineticenergy, gas pressurecaused

by motion randomly of gas molecules and collides

one another and with walls container. If speed of

molecules is faster, so gas pressure willincrease.

Mathematically relationship between motionsvariable (speed), and gas pressure as follows

2

3

1vN

V

mP!

withp = gases pressure (Pa)


N= number ofmolecules (particles)2v = average of square of speed (m/s2)

m = mass of a molecule (kg)

Gas pressure is proportional with the average ofkineticenergyof gas molecules,mathematically

KEV

NP

3

2!

WithP= gases pressure (Pa)


N= molecules (particle)

KE = average of kineticenergy( Joule)

Temperatureand Kinetic Ener yTemperature is only related with molecules move

(kinetic energy or molecules speed). And the ave-rage kinetic energy is only depends on absolute

temperature,T.Mathematically

kTEK

2

3!

Effective speed, vRMS or veff, is a kind of average

speed ofmolecular. Gas molecule atcertaintempe-

rature, lighter molecules move faster, on the ave-

rage,than do heavier molecules.Mathematically

r

RMS

M

RT

m

kTv

33!!

1. In kinetic theory of gas, Quantum-mechanicaleffects are negligible. Why?(pertanyaan evaluasi

ahir)

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