Names:
Group #:
Key
Due 9-10-10 bring to class!!!
Gas Properties: Tutorial I
-Only one set of answers needs to be turned in per group. Make sure everyone’s name in
your group is on the tutorial
-Every group will answer the questions in Section 1.
-In Section 2, answer ONLY your group number’s specific set of questions.
-Next class will include 5 minute presentations from each group. The goal of each
presentation should be to teach the class the main points from your specific set of
questions in Section 2. This can include a prepared simulation demonstration, graphs,
equations, real life examples, etc. If your presentation includes power point, please bring
the file on a memory stick.
Equipment Needed:
Computer with internet access
Web site for simulations:
http://phet.colorado.edu/en/simulation/gas-properties
**Note: You must have JAVA loaded on computer for simulation to work.** You
can find a link to the JAVA download on the “downloads” page of the site above.
Section 1: EVERY GROUP DOES THIS!!!
The heavy species you will see are a model representation of N2 gas molecules. The N2
gas molecules are shown as blue spheres for simplicity. In reality, we know that N2
molecules are not blue spheres.
Note for Section 1: Steps to perform with simulation are in BOLD, questions to answer
are in ITALICS.
Part I: What units are used to measure temperature, and pressure? How does
speed and size of molecules compare to speed and size of familiar objects?
Add 100 N2 molecules to the box. To do this, go to the window next to “Heavy
Species”. Enter “100” and hit enter key. You can also pump the handle on the
bicycle pump by clicking on it and dragging.
What is the temperature of the gas molecules?
Tgas(K) = __300K___________________
The temperature is given on the Kelvin scale. What would this temperature be on the
Celsius scale?
Tgas(°C) = ___26.85 °C_________________
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How does this compare to the temperature of the room you are in?
This is very comparable to room T! Room T is usually around 25°C or 75°C
What is the pressure in the box?
Pbox(atm) = ____0.52 atm________________
How does this compare to the pressure exerted on you by the air around you?
This is less than the external pressure, which is closer to .8 atm (80% of sea level
pressure)
Pause the simulation. Under “Measurement Tools” choose “ruler”. The ruler is in
units of nanometers. Click and drag the ruler to measure the diameter of a heavy
molecule.
What is the diameter of one heavy blue molecule in nm? In m?
diametermolecule(nm) = ___0.2 nm__
diametermolecule(m) =__2E-10 m___
The tip of a pencil is about 0.5 mm across.
How many heavy molecules could line up across the tip of a pencil?
0.5mm or 5E-4m/2E-10m = 2.5E6 molecules
Hit “PLAY” to restart motion.
Under “Measurement Tools”, choose “Species Information”.
What is the average speed of the molecules?
Avg. Speed (m/s) = ____420 m/s________________
Are these molecules moving faster or slower than a car driving 60 miles/hour?
420 m/s * (3600s/hr) * (102cm/m) * (in/2.54cm) * (ft/12in) * (mi/5280ft) ~ 940 m/s
Therefore, the molecule is moving faster by almost a factor of 2.
Are these molecules moving faster/slower than a supersonic jet traveling at 2000
miles/hour?
The molecule is moving slower by about a factor of 2
How long will it take a heavy molecule to travel 100 meters? (This is approximately the
length of a football field.)
Time to travel 100 meters (s) = ____.11s__________
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Part II: Relationships between pressure and volume, pressure and temperature, and
pressure and number of moles
In looking for a relationship between any 2 properties, we hold all other variables
constant. One property (the independent variable) is adjusted, and the impact on another
property (the dependant variable) is observed. You do not directly control the dependant
variable, but it will change as a result of adjustments you make to the independent
variable.
Predictions: For the following questions, make predictions about what would happen to
a gas in a container which is sealed, but which has an adjustable volume.
Predict: What will happen to the pressure if you increase the volume, while keeping the
temperature and number of molecules in the container constant?
The pressure will decrease and will go as 1/V
Sketch the relationship between pressure and volume on the graph below. When
graphing variables, the independent variable goes on the x-axis and the dependant
variable goes on the y-axis. Be sure to label the axes with the variable name and
corresponding units, example: pressure (atm) and volume (liters)
Predict: What will happen to the pressure if you add more molecules to the container,
while holding temperature and volume constant?
The pressure will increase linearly if you increase the moles/molecules of gas
Sketch the relationship between pressure and number of molecules on the graph below.
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Predict: What will happen to the pressure if you raise the temperature of the gas, while
holding the volume and number of molecules in the container constant?
The pressure will increase linearly
Sketch the relationship between pressure and temperature on the graph below.
Part III: Collecting Data with the Simulation: You will do a number of experiments
using the simulation
First experiment at constant temperature.
Measure the width of the box using the ruler.
Width box (nm)=______6.6 nm_______________
Under “Measurement Tools”, choose “Layer Tool”. Move the bar to the top of the
box. What is the height of the box?
Height box (nm) = _____5.4 nm (may have 5.3 nm)_______
Click “Pause” and then “Reset”
In the “CONSTANT PARAMETER” section click “TEMPERATURE”.
Change the width by clicking the handle on the left wall, and moving it left or right.
Using the ruler, set the box width to 3 nm.
Add 100 heavy molecules and hit enter.
Once pressure and temperature have stabilized record the following data. Then
move the wall to a new width and record the temperature and pressure. Make sure
pressure and temperature have stabilized before you record new values.
Table 1.
Run 1
Number
T (K)
P (atm)
Width (nm)
Volume (nm3)
100
300
1.1
3
1620
4
Run 2
Run 3
Run 4
Run 5
Run 6
100
100
100
100
100
300
300
300
300
300
0.85
0.70
0.53
0.50
0.43
4
5
6
7
8
2160
2700
3240
3780
4320
How do you calculate the volume of a 3 dimensional box?
Width x Height x Length
Calculate: If the width of the box is doubled, what happens to the volume?
The volume doubles
Here only two dimensions, width and height, are visible. You have already measured
both of them. Assume the depth of the box is 100 nm.
Calculate the volume of the box for each run and enter the values into Table 1.
What variable(s) did you adjust?
Adjusted the volume
Besides the variable(s) you adjusted, what other variable changed?
The pressure changed
Write one sentence describing how volume and pressure are related.
As the volume is increased the pressure decreases. The change in pressure goes as 1/V
What is the mathematical relationship between volume and pressure?
PV = constant
Use EXCEL to make a graph of your Pressure vs. Volume data.
Does the graph you sketched in the Predictions section agree with the graph of your
data? Describe any differences. Yes they agree
A Second Experiment at constant temperature.
Hit the “RESET” button.
Make sure “TEMPERATURE” is clicked under constant parameters.
Add 100 heavy molecules to box.
Once pressure has stabilized record the following data. Then change the number of
molecules, and record the new data.
Table 2.
Number
T (K)
P (atm)
Width (nm)
Run 1
100
300
0.37
4320
Run 2
200
300
0.78
4320
Run 3
300
300
1.12
4320
5
Run 4
400
300
1.55
4320
Run 5
500
300
1.82
4320
NOTE numbers could change depending on the width of the box and therefore, the
volume
What variable did you adjust?
The number of N2 molecules
Besides the variable you adjusted, what other variable changed?
The pressure increased
Write one sentence explaining how pressure and number of molecules are related.
As you increase the number of molecules the pressure responds in a linear fashion
What is the mathematical relationship between pressure and number of molecules?
P/no. molecules=constant
Use EXCEL to make a graph of the relevant variables.
Does the graph you sketched in the Predictions section agree with the graph of your
data? Describe any differences. Yes, they are similar
A third experiment investigating Temperature and Pressure.
In the first two experiments you investigated the relationship between pressure and
volume as well as pressure and number of molecules. Design a procedure to investigate
how pressure is related to temperature. Write a short step by step procedure below:
Keep the volume constant
Add 100 heavy molecules
Start the temperature at 100K and increase the T in 100K increments
record the pressure and average speed for each temperature setting
What variable will you adjust?
Temperature
Besides the variable you adjust, what other variable(s) change?
The pressure and average speed
What variable(s) will you hold constant?
The volume and number of molecules in the simulation
Table 3. Use this table to record your data.
Run 1
Run 2
Run 3
Run 4
Run 5
T (K)
100
200
300
400
500
P (atm)
0.20
0.34
0.55
0.69
0.84
Width (nm)
3564
3564
3564
3564
3564
# molecules
100
100
100
100
100
Average Speed (m/s)
245
345
420
480
545
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NOTE numbers could change depending on the width of the box and therefore, the
volume
Write one sentence explaining how temperature and pressure are related.
As the T doubles so does the pressure
What is the mathematical relationship between temperature and pressure?
P/T = constant
Use EXCEL to make 2 graphs. In one, demonstrate the relationship between pressure
and temperature. In the second, show the relationship between average speed and
temperature. The first should be linear just like in the predictions and the second should
be v α (T)1/2
Does the graph you sketched in the Predictions section agree with the graph of your
Pressure vs Temperature data? Describe any differences. They are similar
******************************************************************
Section 2: Exploring Further:
- ONLY answer the questions for your group #
-These questions are not as step-by-step as section 1 and require more thought and
creativity. Stay organized and use tables and graphs where necessary.
- As instructed at the beginning, prepare a 5 minute presentation to be given to your peers
on what you learned
Group 1
Goals: difference between ideal and real gases
Applying the volume correction to the ideal gas law
1. Place 1 and 100 heavy particles in the box with interactions (molecules collide)
turned off. Deduce the following for each case: How do the mean speed and
temperature vary with time? How does pressure vary with time? Why?
For n=1, the mean speed and T don’t vary, and the P = 0 atm
For n=100, the mean speed and T don’t vary, and the P varies drastically
Since the molecules do not interact with each other their momentum is constant.
Therefore, the mean velocity and T don’t change.
The P varies because the F/A is not constant at every ∆t. The # of collisions and the
momentum is not constant at every ∆t.
2.
Place 1 and 100 heavy particles in the box with interactions (molecules collide)
turned on. Deduce the following for each case: How do the mean speed and temperature
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vary with time? How does pressure vary with time? Why?
For n=1, the mean velocity and T don’t change, and the P = 0 atm
For n=100, the mean velocity varies around an average value and the T is constant, and
the P varies a little
The mean velocity will change with collisions (elastic collisions) but will change around
a constant value. The T is constant since the mean velocity doesn’t deviate far from a
constant velocity value.
The P varies because the F/A is not constant at every ∆t. The # of collisions and the
momentum is not constant at every ∆t.
3. Obtain 100 particles in the box with a temperature near room temperature and
randomized trajectories. Turn off the interactions (molecules collide). Using this
ensemble, investigate the dependence of p and V and PV with T. What is an
appropriate equation of state for this gas?
Hint: Think about what parameters need to be held constant for each experiment!
As T increases, the P increases (V and n are held constant)
As T increases, the V increases (P and n are held constant)
As T increases, PV increases (hold nothing constant) and P will increase
PV=const*T
4. Place 100 particles in the box with a temperature near room temperature and
randomized trajectories. With the interactions still present, record the pressure, the
mean speed and the temperature. Turn off the interactions. Record the same
properties. What happened? Why? Do this with 500, 200, 50, and 10 particles in the
box. Suggest the form for an appropriate equation of state for the gas when the
particle – particle interaction is turned on. The van der Waals equation of state is a
good place to begin to think about this problem.
The pressure consistenly went up as the interactions were turned off. The speed and T
stayed roughly the same. This occurred because as the molecules collide they are no
longer volumeless (as the ideal gas law says they are). There must be some
correction for volume of the particles
In VdW:
P=nrt/(V-nb) – (an2/V2)
Need to correct the volume term. The molecules have less space to roam if they cannot
go through other molecules. Need to reduce the volume term
So will you use the P=nRT/(V-nb) to decrease the amount of volume accessible for
roaming
Group 2
Goals: irrev and rev processes
1. Place a small number of particles in the box with the interactions turned on
(molecules collide) and constant held constant. Move the left wall of the box in and
out. Do so with (fast) and without (slow) allowing particles to collide with the
moving wall. What is the difference in the gas properties in these two circumstances?
Do this again with no parameters held constant. What is the difference in the gas
properties when you move the wall with and without collisions? How does this
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compare to holding the temperature constant?
When the T is held constant and the walls are moved slowly and quickly, there is no
difference in the gas properties between these two scenarios. But when the T is not
constant and the walls are moved quickly, there is a change in T and velocity. When
the T is not constant and the walls are moved slowly, there is no change in T or
velocity. Holding the T constant is like moving the walls slowly: aka there is no
transfer of E from wall momentum to particle momentum. Moving the wall slowly is
a reversible process; the pressure and velocity can always be brought to their starting
value. Moving the wall quickly is an irreversible process; the pressure and velocity
cannot be brought back to their starting value.
2. In the real world, the speed of the gas particles is much greater than that of the piston,
and the direct kinetic energy transfer from the piston to the gas is not significant.
Under this very slow motion condition, the there is no heat transfer into or out of the
gas, and the process is referred to as being adiabatic. The motion of much of our
atmosphere is quasi adiabatic, with individual “parcels” of the atmosphere behaving
nearly adiabatically. In a Colorado windstorm, parcels of air from the continental
divide (p= 0.6 atm) descend rapidly into Boulder (p=0.8 atm). Assuming that the air
on the divide is pretty cold (-20º), what would be the thermal effect of a windstorm in
Boulder?
Assuming ideal gas law, PV=nRT.
P1 = 0.6atm P2= 0.8atm
T1 = -20°C or 253K T2 = ?
P1/P2 = T1/T2 if P1< P2 then T1 must be < T2, The windstorm brings warmer weather to
boulder: T2~ 337K or 64°C
Group 3
Goals: higher collision freq increases diffusion, molecules diffuse faster at lower
concentrations, how <KE> and <v> are related
1. Turn on the center of mass markers. Inject some 100 particles without molecule
interactions (uncheck molecules collide). Allow time for the distribution to appear
relatively uniform spatially. At this point, turn on molecular interactions. Estimate
how long it takes for the energy and speed distribution to become stable. How long
does it take for the spatial distribution to become uniform? Are the times different?
Why? Do any of the following make a difference in the times?
a) particle – particle interactions
b) number of particles
c) temperature
d) mass of the particles
You may want to use a watch to estimate the equilibration times. Plot the results.
Can we draw any general conclusions?
Yes <KE> and <v> equilibrate at the same time. Because KE is proportional to v2.
The molecular distribution appears uniformly spatially distributed the entire time.
All of the above make a difference.
a) of course! the KE and v will not distribute until interactions are turned on
b) the more particles the faster the system equilibrates. More molecules, higher
collision frequency
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c)The higher the T the faster the system equilibrates. Faster molecules, higher
collision frequency
d) lower the mass, faster the molecules move, the faster the system equilibrates.
Faster molecules, higher collision frequency
Summary: higher collision frequency will equilibrate the system faster
Plot: exponential decay of <v> or <KE> as a fxn of time
2. Let us look at the same set up to study particle diffusion. Equilibrate 300 heavy
particles at, say 300ºK. Inject about 10 light particles. Using the center of mass
markers, how long does it take the light particles to diffuse and become uniformly
mixed? Use a watch to estimate the equilibration time. How does this time scale with
the initial number of particles (varied from 10 to 300) and initial temperature (varied
from 1 to 500 K)? Plot these results and see if you can draw any conclusions/
takes longer to equilibrate as the number of He atoms increases. This is because more
He atoms had to make their way through a greater number of molecules.
Takes less time to equilibrate as the T is increased. This is because the molecules
have a higher average velocity.
3. We want to try to look at the approach to equilibrium in another manner. With
interactions turned off, inject approximately equal numbers (say 75) of heavy and
light particles. Note the energy and speed distributions after each injection. Turn on
interactions. What happens to the distributions? Are they different in shape? Mean?
Which one(s) are fastest to approach their equilibrium? What is driving the system to
equilibrium?
The E and speed distributions equilibrate very quickly and at same time.
N2: narrower dist and lower <v> same <KE>
He: broader dist and higher <v> same <KE>
The collisions are driving the system to equilibrium.
Group 4
Goals: evaporation and the impact of gravity
1. Let’s look at the process of evaporation. Place 200 particles in the box and
equilibrate them at 300 K. Open the cover (make sure you open the lid the same
amount each time) and record the number of particles remaining and the temperature
as a function of time. Plot the results and deduce a scaling law. Does it make sense?
What is the effect of initial temperature? What is the importance of moleculemolecule interactions (molecules collide)? How does the evaporation change with the
hole size (amount that the lid is open)?
plot looks like exp decay of number of particles or T as a fxn of time
Plot makes sense. The faster molec get out first and leave the slow ones behind thus
lowering the T.
Initial T is important. The higher the initial T the faster evaporation occurs because
the molecules have a higher <v>
Molecule-molecule interactions are important. If turned off, molecules are slower to
evaporate and T drop is very slow because slow ones are left behind. With no
transfer of energy, side-to-side trajectories will take a long time to get out of the box.
2. Let’s now do this with gravity turned on and the other parameters identical to
question 1. What conclusions do you reach about the cooling process here? Without
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turning the gravity off, how can you increase the rate of evaporation?
With gravity on, it takes longer to evaporate the sample. At the same parameters as
problem 1 and gravity turned all the way up, nothing escapes the sample. To increase
the rate of evaporation, increase T or decrease the volume
Group 5
Goals: how do spatial equilibrium and speed equilibrium relate at room T and at
~0K
1. Let’s look at diffusion in another fashion. Equilibrate 400 heavy molecules at 300 K
with collisions on (molecules collide). Inject 10 light particles and estimate the time
required for the light particles to reach an equilibrium spatial distribution and
equilibrium speed distribution? Are they the same? Why or why not? How do these
times scale with the number of initial heavy particles?
The time to reach an equilibrium spatial distribution takes much longer than the time
to reach an equilibrium speed distribution. The light molecules are rapidly
equilibrated in their speed because of the number of collisions.
The time scales for spatial equilibration takes longer if you have more initial heavy
particles because the light particles have to fight their way through more particles to
get to the other side of the box. However, the time scales for speed distribution
decrease if you have more initial heavy particles because the concentration is greater
and the collision frequency is greater.
2. Finally equilibrate 500 heavy particles in the box, and then cool them to near
absolute zero. (Easy in a simulation, but hard in the real world!) Now add perhaps 5
light particles and again estimate the time required to diffuse halfway across the box
and to reach an equilibrium speed distribution. What if we added 10 particles? 300
particles?
It’s going to take a very long time to spatially equilibrate and reach an equilibrium speed
distribution. However, the equilibrium speed distribution will not take as long as the
uniform spatial distribution.
The average speed of the molecules is very low. The more light particles that are injected
into the box, the longer it will take to spatially equilibrate since the molecules are moving
so slow. This is because the density is greater and it takes longer to move through the
dense sample. This is true at any T. However, it will take less time to equilibrate in
speed.
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