Transcript Experimental Verification of Filter Characteristics Using

Experimental Determination of
Molecular Speeds
Stephen Luzader
Frostburg State University
Frostburg, MD
Outline of topics
• Purpose of the experiment
• A schematic representation of the apparatus
• Some theory assuming the molecules are all
moving in one direction
• A correction for motion in all directions
• Some data for our particular apparatus
• How to analyze the data
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Purpose
• The purpose of the experiment is to verify
kinetic theory predictions of how the
average speed of gas molecules depends on
the mass of the molecules
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The Basic Apparatus
Preservoir
P
Vreservoir
V
Nreservoir
N
Molecules move from a reservoir to a chamber that is
initially empty (P = 0). They flow through a small
opening with area A.
The gas in both chambers is at a constant temperature T.
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Some theory assuming the molecules are all moving from
left to right.
Preservoir
P
Vreservoir
V
Nreservoir
N
In the right hand chamber, the ideal gas law gives the
relationship between the number of molecules N and the
pressure P:
P 
Nk B T
V
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V and T are constant, so as molecules flow into the
chamber, the pressure increases. The rate of change of
the pressure is proportional to the rate of change of the
number of molecules:
k B T  dN 



dt
V  dt 
dP
If the number molecules in the chamber increases by dN,
the number of molecules in the reservoir must decrease
by the same amount.
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Preservoir
P
Vreservoir
V
Nreservoir
N
If all the molecules in the reservoir are moving to the
right at some average speed v, the number of molecules
that pass through the opening during time dt is
 N reservoir 
dN
 N reservoir 
 vA 

dN  Avdt 
 
dt
 V reservoir 
 V reservoir 
where
N reservoir
V reservoir
reservoir.
is the density of molecules in the
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We can substitute this expression into the one for
the rate of change of the pressure in the chamber
to find a relation between the rate of change of
pressure and the average speed of the molecules:
N reservoir 
k BT 

 vA

dt
V 
V reservoir 
dP
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We can use the ideal gas law to express the density of
molecules in terms of the temperature and pressure of
the gas in the reservoir:
N reservoir

Preservoir
V reservoir
k BT
This gives very simple relation between the rate of
change of the pressure in the chamber and the average
speed:
dP
dt

vAP reservoir
V

v
V
AP reservoir
 dP 


 dt 
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The equation we just derived is wrong because it was based
on the unrealistic assumption that all the molecules are
moving in the same direction.
A more detailed analysis that takes the random motions of
the molecules into account gives the correct result:
v
4V
AP reservoir
 dP 


 dt 
If we know V, A, Preservoir , and can measure the rate of
pressure increase, we can calculate a value for v.
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In our experiment, we know the following quantities:
•V = 60 cc plus the volume of the associated tubing
•Preservoir = atmospheric pressure, which we
measured (units!)
•
dP
is determined from the experimental
dt
graph of P vs t. (units!)
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We also need the following information about the
apparatus itself, which is provided by the
manufacturer:
Diameter of pinhole = 12.5 mm  25%
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For your analysis, you must carry out the following steps.
1. Prepare a table with all common data, including
atmospheric pressure, room temperature, data for tubing
volume, hole diameter (with uncertainty)
2. Show the calculation of the total volume.
3. Calculate the area of the pinhole, including its
uncertainty.
4. Tabulate the measured values of the rate of change of
pressure (determined by LoggerPro). This table must
include an identification of each gas.
5. Calculate experimental values of v for each gas,
including uncertainty.
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To test whether the experimental results agree with
predictions made by kinetic theory, do the following.
1. Calculate expected values for v using the measured
value of room temperature. These results must be in a
table which includes the molecular mass of the gas.
2. Compare the experimental values of v (including
uncertainty!) with the predictions. Account for
discrepancies.
3. Compare the ratios of the experimental values of v with
the ratios of the molecular masses of the gases and
explain whether these ratios agree with predictions
from kinetic theory. Again, you should consider
experimental uncertainty.
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