Kinetic Theory of Gases Part II

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Kinetic Theory of Gases Part IIThe ideal Gas Pressure


If an ideal gas rests in a closed container, then the gas will exert pressure on the container wall because the gas particles are continously moving and colliding the container wall.

vx

l

vy

vzX

Z

Y

Suppose the length of container side is l, the mass of particle is m, the velocity of particle in the direction of x-axis, y-axis, and z-axis are vx, vy, and vz. Respevtively, the pressed area is A, the pressure force is F and the pressure against the wall is p

Obserb the direction of the particle motion in the direction of x-axis :

A


A B

mm

vx-vx

l

The time required by the particle to collide the wall B and return to its original position (A) is determining by the following equation :

xv

lt

2

Every time the particle collides the wall, then it will give its momentum that is equal to 2 mvx to the right wall.

The change of momentum on the particle is

p = I = F . t , so :

I = -mvx – mvx

F.t = -2mvx

Because F = p.A, where p = pressure, then :

2

2..

22..

22

x

x

x

xx

mvpV

mvlAp

mvlAp

mvv

lpA


Because of the velocity of the gas particle to all direction is assumed to the same vx = vy = vz , then :

2222

2222

333 zyx

zyx

vvvv

vvvv

However, because we observe in x-axis direction, we use :

2312

22 3

vv

vv

x

x

So the equation pV = mvx2 become :

V

mvp

mvpV2

31

231

For N particles of ideal gas, then we obtain the following equation :

V

Nmvp

2

31

Where :

N : number of gas particles

m : mass of gas particles

v : velocity of gas particles

V : volume of container

S.A


Average Kinetic energy :

221 vmEk

If this equation is subtitude into the previous equation, that is :

V

Nmvp

2

31

Then will be obtained the equation as follows :

V

NEkp

V

Nvmp

V

Nvmp

32

221

32

231

This equation express the relationship between gas pressure and average kinetic energy of gas particles.

The greater the average kinetic energy of gas particle, the greater the pressure on the gas

Presure and Kinetic Energy


If the gas is heated, then the temperature of the gas will increase, This increase of temperature causes the speed of gas particle motion to increase, so that the average kinetic energy of the gas particle will also increase, The amount of average kinetic energy of the gas particle can be obtained from the following equation ;

V

NkTp

NkTpV

Enter the value of p into the equation :

V

NEkp 3

2

So that obtained the following equation :

kTEk

EkkT

V

NEk

V

NkT

23

32

32

This equation expresses that the average kinetic energy of gas particle is only influenced by its absolute temperature

It should be note that the equation above only holds for monoatomic gases

Absolute Temperature and Average Kinetic Energy


The velocity of gas particles in container is not all the same, Because of that, the concept of effective velocity of gas particles is needed.

Suppose a closed container contains are a number of N gas particles, each particle moves at a certain velocity. N1 particle moves at velocity of v1, N2 particle moves velocity of v2 and so on, Then the average (mean) velocity square v2, can be expressed as follow :

...

...

321

233

222

212

22

NNN

vNvNvNv

N

vNv

i

i

ii

The effective velocity of gas is defined as the root mean square of gas velocity, and it is determined as follow :

2vvRMS

The Velocity of Gas


Hence, the average kinetic energy of gas particle can be expressed using the effective velocity of gas particle as follows :

2

21

RMSmvEk By subtituting that value into the

kTEk 23

Then obtained the following equation :

m

kTv

kTmv

RMS

RMS

3

232

21

Another expression for effective velocity :

P

v

M

RTv

RMS

RMS

3

3

p = gas pressure (Pa)

= density of gas (kg/3)


Student Activity #1 A closed vessel contains 20 L oxygen gas.

If the gas is at temperature of 27oC and atmospheric pressure of 1 atm (1atm = 105Pa) determine the number of moles of the oxygen gas in the vessel.

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Student Activity #2 How many molecules are there in 6 grams

of hydrogen gas ?


Student Activity #3 A container of hydrogen of volume 0.1 m3

and temperature 25oC contains 3.20 x 1023 molecules. What is the pressure of the container ?


Student Activity #4 One mole of gas occupies 100 dm3 volume,

its temperature at the moment is 127oC Determine the pressure of the gas


Student Activity #5 The kinetic energy of 2 moles of

monoatomic gas in a 10 liter tube is 2.3 x 10-23 joule. What is the pressure of the gas in the tube ?


Student Activity #6 Determine the average kinetic energy of 5

moles of neon gas which its volume is 25 liters and its pressure of 100 kPa


Student Activity #7 If the kinetic energy of gas molecules

become twice of the kinetic energy of the gas molecules at 127oC, what is the gas temperature now ?


Student Activity #8 The speed of 20 particles are distributed as

follows :

Determine the rms speed !

Speed m/s 1.0 2.0 3.0 4.0 5.0 6.0

No. of molecules 1 3 4 5 2 5


Student Activity #9 What is the temperature at which the rms

speed of oxygen molecules is twice as great as their rms speed at 300oK


Student Activity #10 Determine the effective velocity of gas

particles at normal state, if the gas density is 10 kg/m3 and its pressure is 3 x 105 N/m2


Student Activity #11 A tank of 2.4 m3 in volume is filled with 2 kg

gas. The pressure tank is 1.3 atm. What is the effective velocity of the gas molecules ?


Student Activity #12 The pressure of a gas in a closed tube

decreases 81% from the initial pressure. Does this change the speed of the gas particles ?


Internal EnergyIn a closed container, an ideal gas only has kinetic energy. The total kinetic energy of the gas particles in the container is called the internal energy.

NkTEkU2

3

The equation of the internal energy above is valid for monoatomic gases, for example : He, Ne and Ar

For diatomic gasses such H2, O2, and N2. at low temperature + 300 K holds :

NkTEkU2

3

At medium temperature + 500 K holds :

NkTEkU2

5

At high temperature + 1000K :

NkTEkU2

7


Student Activity A container with temperature of 67oC and

pressure of 1.2 x 105 Pa contains 2 g helium gas with molecule mass being of 4g/mole. Calculate the internal energy of the gas


Student Activity Determine the internal energy of one mole

of gas at temperature of -37oC ( k =1.38 x 10-23 J/K, No = 6.02 x 1023 molecules/moles)


Student Activity What is the internal energy of 2 grams of

neon gas (Ne) at temperature of 27oC (known that neon gas has M = 10 g/mole)


Thanks for your attention

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Kinetic Theory of Gases Part II