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1/38 PHYWE Systeme GmbH & Co. KG D-37070 GttingenLaboratory Experiments Chemistry

Kinetic Theory LEC 01

Principle and tasksBy means of the model apparatus forkinetic theory of gases, the motion of gas molecules is simulated and thevelocity is determined by registrationof the throw distance of the glassballs. This velocity distribution iscompared to the theoretical MAX-WELL-BOLTZMANN equation.

Distribution of molecule velocities of oxygen at 273 K.

What you can learn about

Kinetic theory of gases

Temperature

Model kinetic energy

Average velocity Velocity distribution

Kinetic gas theory apparatus 09060.00 1Receiver with recording chamber 09061.00 1

Power supply, variable, 15 VAC/ 12 VDC/ 5 A 13530.93 1Laboratory balance, with data output, 620 g 45023.93 1Digital stroboscope 21809.93 1Stopwatch, digital, 1/100 sec. 03071.01 1Tripod base -PASS- 02002.55 2Connecting cord, l = 750 mm, red 07362.01 1Connecting cord, l = 750 mm, blue 07362.04 1Test, tube, d = 16 mm, l = 16 cm 37656.10 1Test tube rack for 12 tubes, wood 37686.00 1

What you need:

Velocity of molecules and the MAXWELL BOLTZMANNdistribution function P3010101

01.01 Velocity of molecules and the MAXWELL BOLTZMANN distribution function

Experimental and theoretical velocity distribution in the model experiment.

theoretical distributionexperimental distribution

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Related conceptsKinetic theory of gases, model kinetic energy, average velocity,velocity distribution.

Principle A kinetic gas theory apparatus is to be used as a model to sim-ulate the motion of gas molecules and to determine their veloc-ity by determining the distance that glass balls are thrown. Thevelocity distribution found is to be compared with the theoreticalMaxwell-Boltzmann equation.

TasksMeasure the velocity distribution of the model gas and com-pare the result to theoretical behaviour as described by theMaxwell-Boltzmann equation.

EquipmentKinetic gas theory apparatus 09060.00 1

Receiver with recording chamber 09061.00 1Power supply, variable, 15 VAC/ 12 VDC/ 5A 13530.93 1Precision balance, 620 g 48852.93 1Digital stroboscope 21809.93 1Stopwatch, digital, 1/100 sec. 03071.01 1Tripod base -PASS- 02002.55 2Connecting cord, l = 750 mm, red 07362.01 1Connecting cord, l = 750 mm, blue 07362.04 1Test tube, d = 16 mm, l = 16 cm 37656.10 1Test tube rack for 12 tubes, wood 37686.00 1Glass beaker, 50 ml, tall 36001.00 5Spoon 40874.00 1

Set-up and procedure

Set up the experiment as shown in Fig. 1.

Fit the receiver with recording chamber to the apparatus asdescribed in the operating instructions for the determination ofparticle velocities.First determine the average weight of one glass ball by weighing

out a known number of balls (e.g. 100) to avoid the time con-suming counting of glass balls during the experiment.Following this, determine the average number of glass ballsexpelled from the apparatus during 1 minute. To do this, fill theapparatus with 400 glass balls and set it to the following condi-tions: height of the upper piston: 6 cm oscillator frequency: 50 s -1 (controlled by the voltage and

the stroboscope).

Now open the outlet for 1 minute and determine the number ofballs expelled by weighing them. Refill the apparatus with theseballs and repeat the experiment twice.Prepare for the simulation experiment by calculating the average

number of balls expelled per minute and filling this number ofglass balls into each of four glass beakers. Make the followingapparatus settings: height of the upper piston: 6 cm height difference between outlet and receiver: 8 cm number of balls: 400 oscillator frequency: 50 s -1 .

When the frequency is stable, open the outlet for 5 minutes. After each minute, pour the balls from one of the beakers into theapparatus to maintain a constant particle density. Determinethe number of glass balls in each of the 24 compartments of thereceiver by weighing.Refill the glass beakers and repeat the experiment four times.

PHYWE series of publications Laboratory Experiments Chemistry PHYWE SYSTEME GMBH & Co. KG D-37070 Gttingen P3010101 1

LEC01.01

Velocity of molecules and the Maxwell-Boltzmanndistribution function

Fig. 1. Experimental set-up.

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Theory and evaluation According to definition, the kinetic energy of the molecules of anideal gas is given by

(1)

Average kinetic energym Mass of the molecule

Average velocity of the molecule

From kinetic theory, the pressure of an ideal gas can bedescribed by

(2)

p Pressurer Density

Combining equations (1) and (2) together with the ideal gas law

p Vmol = R T (3)

Vmol Molar volumeR Gas constantT Absolute temperature

leads to the following expression for c:

or

(4)

k Boltzmann constant

This means that the average kinetic energy is directly propor-tional to the absolute temperature of the gas which is the inter-pretation of temperature on the molecular level.The direct determination of the velocity of a certain molecule isimpossible because collisions with other molecules cause it tochange incessantly. For a great number of molecules, a distrib-ution function for molecular velocities can be derived by meansof statistical methods. This was done by Maxwell and Boltzmannwith the following result:

dc (5)

This equation describes the probability that the velocity of a mol-ecule is within the interval {c,c+dc}. The corresponding distribu-tion function for oxygen at 273 K is shown in Fig. 2 as an exam-ple.For the velocity at the maximum cw of the curve (velocity withhighest probability) the following relation can be derived:

(6)

Introducing of equation (6) into equation (5) leads to

(7)

Note that cw c and

In the model experiment with glass balls, the velocity of the ballscan be determined from the distance thrown s:

(8)

g Acceleration at the earth surface (= 9.81 ms)h Height difference between outlet and receiver

Now the experimental results (number of balls per distancethrown interval) can be displayed graphically in the form

(9)

N i Number of balls in the interval i, i = 123 c Velocity interval corresponding to s = 1 cm

(0.078 ms)

as shown in Figure 3.

The theoretical distribution function can be evaluated by meansof equation (7) using the velocity at the maximum of the experi-mental distribution as cw. The result for the example in Fig. 3 isalso shown in the diagram. The agreement between the twocurves is reasonably good taking into account the model char-acter of the experiment.

1 Ni

Ni c

f 1c2

c s g

2 h K s

cw : c : 2 c2 2 2 : B 8p : 2 3 1 : 1.13 : 1.22

a c2cw2 bdNN

42 p

a1cw2 b3>2

c2 e

cw a2 k Tm b1>2

1m c22 kT 2 d NN

B 2p a mk T b3>2

c2 e

c a3 k Tm b1

>2

c a3 R TM b1>2

p13

r c2

c

E k

E km

2 c2

PHYWE series of publications Laboratory Experiments Chemistry PHYWE SYSTEME GMBH & Co. KG D-37070 GttingenP30101012

LEC01.01

Velocity of molecules and the Maxwell-Boltzmanndistribution function

Fig. 2: Distribution of molecule velocities of oxygen at 273 K

Fig. 3: Experimental and theoretical velocity distribution in themodel experiment