Guided Inquiry Analysis of Collected Gas Law Data
Reviewing the Characteristics of a Proportional Relationship.
Two properties are said to be proportional to one another when the ratio of one property over the
other is constant for all points. Consider the following example of a car travelling at 60 miles
per hour.
Time (hrs)
Distance Travelled (miles)
0
0
1
60
2
120
3
180
4
240
Notice that at every point (except 0 hrs) taking the ratio of distance travelled to hours yields the
same result.
Distance Travelled (miles) 180 mi
mi
Example calculation:

 60 or 60 mph
Time (hours)
3 hr
hr
Results:
60 mi
mi
 60
1 hr
hr
120 mi
mi
 60
2 hr
hr
180 mi
mi
 60
3 hr
hr
240 mi
mi
 60
4 hr
hr
Notice that if one variable changes by a multiplicative factor (2x, 3x, etc.) that the other changes
by the same factor. For example, if time is doubled from 2 to 4 hours, the distance travelled is
also doubled (from 120 mi to 240 mi) or time is decrease to a third (180 to 60 minutes), then
distance is also decreased to one third (180 to 60 miles).
Distance Travelled (miles)
Plotting this data and determining the best fit line reveals additional information about a
proportional relationship. Notice that the relationship is linear with the line going through the
origin so that the intercept (b) is
300
zero and the resulting linear
250
equation is in the form of
y = 60x
200
y = mx. The slope (60 in this case)
is referred to as the constant of
150
proportionality and is the constant
100
ratio between the two properties.
50
The slope has units equal to the unit
0
of the y-axis property divided by the
0
1
2
3
4
5
x-axis property and is a property
Time (hrs)
that is independent of x and y. In
mi
this example, the proportionality constant is the speed of the car in
or mph.
hr
Characteristics of a proportional relationship:
1
Guided Inquiry Analysis of Collected Gas Law Data
1. Ratio of properties is constant at all points and is referred to as constant of proportionality.
2. If one property increases or decreases by a multiplicative factor, the other increases or
decreases by the same factor, respectively.
3. A plot of the two variables results in a linear relationship with an intercept of zero (the graph
goes through the origin (0,0).
4. The two variables are related by the equation y = mx. The constant, m, is the constant of
proportionality and also the slope of the line.
5. The constant of proportionality can be used to calculate one variable, given the other is
known.
Documenting Group Work
Each group will submit one report for each section of today’s lab. In addition, all of the work
should be saved in an Excel file. In the Excel document, one tab should be designated for each
of the sections with the following headings
V vs n
V vs T
V vs P
Ideal Gas
In addition, you will be required to submit a printed Excel page (pages) as documentation of
your computer work for each section of this lab. Follow the directions provided for this
computer work documentation documents carefully.
2
Guided Inquiry Analysis of Collected Gas Law Data
Exploring the Volume-Mole Relationship
Open the spreadsheet that contains your volume-mole data from last week’s experiment.
Based on these observations, your data should indicate that the volume-mole relationship is a
proportional relationship.
Record your slope and intercept of your plot below. Include units
Slope __________ _________
Intercept __________ _________
Slope Units
Intercept Units
The intercept should be relatively small (< 10% of your smallest volume). If it is much larger
than this, you may want to consult your instructor before proceeding.
To obtain a proportionality constant that we can use
for further calculations, we will recalculate the
equation of the line by modifying the trend-line
calculation. In your graph of the volume-mole data,
right click on the trend-line and click “Format
Trendline . . .”
At the bottom of the “Format Trendline”
dialog box, put a check in the “Set Intercept =”
and make sure that the 0.0 is entered into the
input box to the right. Click “Close”. The
resulting equation should have only a slope
and the trend-line will be redrawn to go
through box. Enter the new slope below:
Zero Intercept Slope __________ _______
Slope Units
Calculate the % variation in slope =
Original Slope  Zero Intercept Slope
 100%  ________
Original Slope
If the % variation is greater than 10%, you may want to consult with your instructor.
3
Guided Inquiry Analysis of Collected Gas Law Data
Use this slope to write an equation that relates volume (V) and moles (n).
Use this equation to predict one of your volumes from the moles of sample.
Original Data: Moles ___________
Volume ____________
Volume calculated from moles using equation ____________
Error = __________
Error = Calculated Volume – Measured Volume
Use this equation to predict one of your mole values from the volume a of sample.
Original Data: Volume ___________
Mole ____________
Moles calculated from volume using equation ____________ Error = __________
Error = Calculated Moles – Measured Moles
Often, we want to calculate one value based on one other set of measurements. For example,
given that 0.595 moles has a volume of 695 mL, we want to know the volume that will be
occupied by 3.25 moles. We will derive a simple relationship that can be used for this purpose.
Pay careful attention to how this is done, as you will be asked to do this in later sections for other
relationships. First we write the generic volume-mole relationship for two sets of conditions
using subscripts to show what mole value and volume are associated with each other
V1 = m.n1 and V2 = m.n2 . (“m” is the slope from the graph)1
Since both equations have the slope (or constant) in common, we solve those equations for the
V
V
slope. m  1 and m  2
Since the right-hand terms are both equal to m, we can set them
n1
n2
next to each other
V1 V2
 . This equation can be used to solve the above problem and
n1 n2
similar problems without having to calculate m.
First, assign the numbers above to the variables: V1 = 695 mL; n1= 0.595 mol & n2 = 3.25 mol
V V
V2
695 mL
Next substitute into the equation: 1  2 becomes

n1 n2
0.595 mol 3.25 mol
Finally, solve for the unknown variable:
V2

(695 mL)(3.25 mol)
0.595 mol
Amedeo Avogadro proposed that equal volumes of a perfect gas at the same temperature and
pressure contained the same number of particles. Today we know these particles as molecules
(or in the case of the Noble Gasses, atoms). In honor of Avogadro’s work, the relationship
between volume and moles is often referred to as Avogadro’s law.
Try using this approach to solve the following problems.
1
Alternatively, you could write the equations as V1 = K.n1 and V2 = K.n2 , where K = m. In either case, the slope
is a constant.
4
Guided Inquiry Analysis of Collected Gas Law Data
a. If the volume of 0.25 mol methane gas is 4.25 L at a certain temperature and pressure, what is
the volume of 0.75 mol methane at the same temperature and pressure?
b. A balloon containing 1.00 g oxygen gas at a certain pressure and temperature has a volume of
804 mL. What would the volume of the balloon be if 3.32 g oxygen was added? Assume the
temperature and pressure remain unchanged.
c. A sample of gas containing 0.350 mols of gas has a volume 1.2 L. Additional gas is added to
system while keeping the temperature and pressure the same so that the final volume is 2.00 L
What is the final number of moles in the system? How many moles was added?
Documentation Document for Volume-Mole Relationship
To show your work in this section, prepare a one page Excel document that includes the
following. When printed this document should fit one(1) page.
Title: Exploring the Volume-Mole Relationship
Source of data: describe the source(s) of data used for this exercise. Justify any actions you took
to make the data more useful (removing data points, using data from more than one group, using
online data, etc.)
The data used in this experiment, including any calculations used to manipulate the data.
Graphs:
V vs. n plot with normal trend-line (Y=mX+b)
V vs. n plot with trend-line forced through intercept (Y=mX)
When printed this document should fit one(1) page.
5
Guided Inquiry Analysis of Collected Gas Law Data
Summary of Avogadro’s Law
At constant pressure and temperature, the volume of a sample of gas is proportional to the
number of moles of gas molecules in the sample.
This can be expressed mathematically in the following ways.
V1 V2
Where K is a constant of proportionality.
V n
V K n
and

n1 n2
A plot of V vs. n is a straight line with a positive slope and an intercept of zero.
450
y = 828.65x
400
Volume O2 (mL)
350
300
250
200
150
100
50
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Moles O2
When using
V1 V2
 , any units for volume can be used as long as you are consistent: always use
n1 n2
mL or always use pints, etc.
At constant temperature and pressure conditions, stoichiometric relationships can be used to
describe gas volume relationships. For example, in the reaction of nitrogen and hydrogen to
form ammonia N2 + 3 H2 → 2 NH3, one volume of nitrogen gas reacts with three times the
volume of hydrogen to produce twice the volume of ammonia.
Historical Note on the Volume-Mole Relationship
We began this study with some knowledge of the mole concept. Historically, it was observations
of the volumes of gas in stoichiometric relationships that led to the development of the concepts
of atoms and molecules. For example, when water is decomposed by electrolysis two volumes
of hydrogen are produced for every volume of oxygen produced (2 H2O → 2 H2 + O2). Many
other gas volume relationships that yielded integer volume ratios were observed. These types of
observations contributed to the hypothesis that substances were made of small indivisible objects
that recombined in chemical reactions to produce new substances. In other words . . . atoms.
6
Guided Inquiry Analysis of Collected Gas Law Data
Exploring the Volume-Temperature Relationship
Open the spreadsheet that contains your volume-temperature data from last week’s experiment.
Your data should have demonstrated a linear volume-temperature relationship. However, since
the intercept was nowhere close to the origin, there is no way that we can conclude that this was
a proportional relationship. In the case of the Avogadro’s Law, we observed that having a
proportional relationship between two properties facilitates convenient conversions between
those properties. As a result, it would be useful if we could come up with a way to convert the
relationship between temperature and volume to a proportional relationship. As the relationship
is already linear, we simply need to devise a way of changing the intercept. This can be done by
adding a constant value to one of the two properties, so that the intercept goes through the origin.
Since the Celsius temperature scale is an arbitrary scale anyway (0oC being arbitrarily set to the
freezing point of water and 100oC being arbitrarily set to the boiling point of water), it makes
more sense to adjust the temperature scale than the volume scale.
What value must be added to Celsius temperatures so that your gas law plots will go through
zero at a volume of zero? You may want to use the data you obtained from the online gas law
simulator to answer this question.
Write an equation that shows how you would convert a Celsius temperature to this new
temperature scale. What do we call this temperature scale?
Of course, it was observations and considerations identical to these that led to the development
of the absolute temperature scale.
In your spreadsheet, covert all of your temperatures to the Kelvin temperature scale and re-plot
your data to demonstrate that when the Kelvin scale is used, volume is proportional to
temperature. Select one of your two graphs and using the same approach used for the volumemole relationship, force the trend line to go through the origin.
Record the original slope and intercept of your plot below. Include units
Slope __________ _________
Intercept __________ _________
Slope Units
Intercept Units
The intercept should be relatively small (< 10% of your smallest volume). If it is much larger
than this, you may want to consult your instructor before proceeding.
7
Guided Inquiry Analysis of Collected Gas Law Data
Record the slope generated when the line forced to have an intercept of zero.
Zero Intercept Slope __________ _______
Slope Units
Calculate the % variation in slope =
Original Slope  Zero Intercept Slope
 100%  ________
Original Slope
If the % variation is greater than 10%, you may want to consult with your instructor.
Use this slope to write an equation that relates volume (V) and Temperature (T).
Use this equation to predict one of your volumes from the temperature at one your points.
Original Data: Temperature ___________
Volume ____________
Volume calculated from Temp. using equation ____________
Error = ______
Error = Calculated Volume – Measured Volume
Use this equation to predict a temperature value from the volume at one of your points.
Original Data: Volume ___________
Temperature ____________
Temp. calculated from volume using equation ____________
Error = ______
Error = Calculated Temperature – Measured Temperature
In a similar fashion to what was done with the mole volume relationship, develop a relationship
between two sets of volume-temperature data (V1,T1 and V2,T2). Use it to solve the following
problems.
a. Derivation of relationship between V1,T1 and V2,T2.
b. A sample of gas with a fixed amount and at a constant pressure at 25oC was heated to 117oC
where its volume was 87.5 mL. What was the volume of the original sample?
c. A sample (fixed moles) of gas that is held at constant pressure can be used as a thermometer.
If a sample of N2 has a volume of 237 mL at 42oC is heated until the volume is 512 mL, what is
the temperature after heating? Report your answer in Kelvin and oC.
8
Guided Inquiry Analysis of Collected Gas Law Data
Joseph Gay-Lussac, who was the first to publish this relationship between volume and
temperature in 1802, credits Jacques Charles with its discovery in the 1780’s. As a result, this
relationship is referred to as Charles’ Law.
Exploration of the Effect of Moles on Charles’ Law
You should have two sets of Charles’ Law data and should have plotted both of them already.
Look at the plots of volume vs. Kelvin temperature for this study. Make sure that you set the
trend-lines of both lines, so that the line passes through the origin (T = 0 K at V=0). Recall that
the difference between the two plots is a different starting volume in the syringe. Compare the
starting volumes and the slopes for these two sets of data.
V vs. T Data Set 1
V vs. T Data Set 2
(your data)
(obtained from other students)
Initial Volume _______
(Moles if using online data.)
Slope (V vs. T) _______
Initial Volume _______
(moles)
Slope (V vs. T) _______
Ratio of Volumes _______ 2
(Ratio of moles)
Ratio of Volumes _______
How do the ratios of initial volumes (or moles) compare to the ratios of the slopes?
Recall that in both experiments, the syringes were filled at room temperature and room pressure
(same T and P). Considering your earlier observations of the volume-mole relationship, explain
why the ratio of the moles of the two samples is equal to the ratio of the initial volumes.
The key observation here is that the difference between the two samples of gas was that they
contained different numbers of moles.
Consider the following and discuss it with your team members until you are confident you
understand it. If your group as a whole has trouble seeing this concept, consult with your
instructor.
2
Check to make sure both of your V vs. T plots intersect the x-axis close to 0 K (that your data is okay). If one or
both of them are cross the x-axis more than 20 or more K from 0, then you may want to use the “perfect” volume
versus temperature data that you acquired with the online gas law simulator. If you do so, you will will not be able
to calculate the volume ratio. In its place, calculate the ratio of moles used in generating the data online.
9
Guided Inquiry Analysis of Collected Gas Law Data
Consider the two samples of gas (Data Set 1 and Data Set 2). The slope of the curve (the
volume-temperature proportionality constant) for the two samples was different, so we can
express Charles’ with two separate sets of conditions.
V = K1.T and V = K2.T (K1 is the slope of Data Set 1 and K2 is the slope of Data Set 2)
Since K1 and K2 vary with the ratio of moles of the two samples, we are seeing that both
proportionality relationships are working in both samples. Volume varies proportionall with
both temperature and moles. To show this we can rewrite the Charles’ Law expressions above to
indicate that both are related to a more general constant that we will call K’ so that K1 and K2 can
be determined from K’ and the moles of gas in the sample: K1=n1.K’ and K1=n1.K’. As a
result, we can determine any K value (Kn) determined at the same pressure from the equation:
Kn=nn.K’.
The key factor here is to realize that relationships of the volume to both the temperature and
moles of gas are always operative in any sample of gas and can be combined into one equation.
We will explore this idea in more detail after we have examined the relationship between volume
and pressure.
Documentation Document for Volume-Temperature Relationship
To show your work in this section, prepare a one page Excel document that includes the
following.
Title: Exploring the Volume-Temperature Relationship
Source of data: describe the source(s) of data used for this exercise. Justify any actions you took
to make the data more useful (removing data points, using data from more than one group, using
online data, etc.)
The data used in this experiment, including any calculations used to manipulate the data.
Graphs:
V vs. T in Celsius, plot with normal trend-line (Y=mX+b)
V vs. T in K, plot with normal trend-line (Y=mX+b)
V vs. T in K plot with trend-line forced through intercept (Y=mX)
When printed this document should fit one (1) page.
10
Guided Inquiry Analysis of Collected Gas Law Data
Summary of Charles’ Law
For a sample gas of fixed size (number of moles) and at a constant pressure, the volume of the
sample is proportional to absolute temperature (Kelvin scale) of that gas.
This can be expressed mathematically in the following ways.
V1 V2
Where K is a constant of proportionality.
V T
V  K T
and

T1 T2
A plot of V vs. n is a straight line with a positive slope and an intercept of zero.
8
7
y = 0.00832x
Volume O2 (L)
6
5
4
3
2
1
0
-200
-1 0
200
400
600
800
1000
Temperature (K)
When using
V1 V2
, any units for volume can be used as long as you are consistent: always use

T1 T2
mL or always use pints, etc.. It is imperative that Kelvin be used for temperature.
11
Guided Inquiry Analysis of Collected Gas Law Data
Exploring the Volume-Pressure Relationship
Open the spreadsheet that contains your volume-pressure data from last week’s experiment.
A direct plot (V vs. P) of this data should have
demonstrated a non-linear decreasing
relationship.
However, a plot of 1/P (the reciprocal of
pressure) yields a linear relationship that has an
intercept of zero. In other words, volume is
proportional to 1/P; volume is said to be
inversely proportional to pressure. Again, we
will find it convenient to take advantage of the
simplicity of this relationship to simplify
calculations.
As a result we can express this relationship in terms of V and 1/P.
1
V K
Again, K is a constant of proportionality; K = slope of the V vs 1/P plot
P
One way to detect an inversely proportional relationship is that if one of the properties increases
(or decreases) by a multiplicative factor, the other property decreases (on increases) by that same
factor. So if the pressure increases by a factor of three, the volume will decrease by a factor of
three to 1/3 its original value. Likewise, if the volume were to decrease by a factor of 2 (to ½ its
original value) then a pressure increase of two times can be predicted.
The relationship between volume and pressure is named Boyle’s Law after Robert Boyle who
described it in 1661.
Compare the volume vs. temperature plots of everyone in your group. Based on the linearity of
these plots and the extent to which they seem to agree with trends of the ideal data that you
obtained online, select one set of data to use for this exercise. Enter it into your group
spreadsheet and plot V vs. 1/P.
12
Guided Inquiry Analysis of Collected Gas Law Data
Record the original slope and intercept of the V vs. 1/P of your plot below. Include units.
Slope __________ _________
Intercept __________ _________
Slope Units
Intercept Units
The intercept should be relatively small (< 10% of your smallest volume). If it is much larger
than this, you may want to consult your instructor before proceeding.
Force the line through the origin by setting the trend-line options. Record the slope generated
when the line forced to have an intercept of zero.
Zero Intercept Slope __________ _______
Slope Units
Calculate the % variation in slope =
Original Slope  Zero Intercept Slope
 100%  ________
Original Slope
If the % variation is greater than 10%, you may want to consult with your instructor.
Use this slope to write an equation that relates volume (V) and Temperature (P). Remember that
you plotted 1/P on the x-axis and not P.
Use this equation to predict one of your volumes from the reciprocal of pressure for one point.
Original Data: Pressure ___________
Volume ____________
Volume calculated from 1/P using equation ____________
Error = ______
Error = Calculated Volume – Measured Volume
Use this equation to predict a pressure value from the volume at a different point.
Original Data: Volume ___________
Pressure ____________
Pressure calculated from volume using equation ____________
Error = ______
Error = Calculated Preasure – Measured Pressure
In a similar fashion to what was done with the mole-volume and temperature-volume
relationships, develop a relationship between two sets of volume-temperature data (V1,P1 and
V2,P2). Though the approach will be similar, the resulting relationship will have a
significantly different form than what we saw for the effect of moles and temperature on
volume. Use the relationship just derived to solve the following problems.
a. Derivation of relationship between V1,P1 and V2,P2.
13
Guided Inquiry Analysis of Collected Gas Law Data
b. A sample of gas with a fixed amount (moles) and at a constant temperature had a volume 375
L at 0.900 atm. What will the volume be if the pressure is reduced to 0.300 atm?
c. A family is flying back from Disney Land in a non-pressurized private aircraft. A souvenir
balloon has volume 755 mL at the airport prior to take off. The atmospheric pressure at the
airport in 747 mm Hg. When the plane gets to its cruising altitude, the balloon’s volume has
increased to 980 mL. If the temperature is 25oC in the aircraft both on the ground and in the air,
what is the pressure at the cruising altitude?
Extra Credit: If time allows, search the internet to relate pressure to altitude and estimate the
cruising altitude of the aircraft.
Documentation Document for Volume-Pressure Relationship
To show your work in this section, prepare a one page Excel document that includes the
following.
Title: Exploring the Volume-Pressure Relationship
Source of data: describe the source(s) of data used for this exercise. Justify any actions you took
to make the data more useful (removing data points, using data from more than one group, using
online data, etc.)
The data used in this experiment, including any calculations used to manipulate the data.
Graphs:
V vs. P (atm), plot with normal trend-line (Y=mX+b)
V vs. 1/P (atm), plot with normal trend-line (Y=mX+b)
V vs. 1/P (atm), plot with trend-line forced through intercept (Y=mX)
When printed this document should fit one(1) page.
14
Guided Inquiry Analysis of Collected Gas Law Data
Summary of Boyle’s Law
For a sample gas of fixed size (number of moles) and at a constant temperature, the volume of
the sample is inversely proportional to the pressure of that gas.
This can be expressed mathematically in the following ways.
1
1
V
V K 
and
P1V1  P2V2 Where K is a constant of proportionality.
P
P
A plot of V vs. 1/P is a straight line with a positive slope and an intercept of zero.
When using P1V1  P2V2 , any units for volume or pressure can be used as long as you are
consistent: always use mL or always use pints, etc. For pressure, always using torr or always
using psi, etc.
15
Guided Inquiry Analysis of Collected Gas Law Data
Putting It All Together-The Ideal Gas Law
Recall that when we looked at Charles’ Law, we observed that the value of the Charles’ Law
Constant was dependent on the number of moles of gas. So that if Kn is Charles Law constant
under certain set of conditions, it can be related to the number of mole of gas by Kn=nn.K’.
Where K’ is a more general constant; it turns out that K’ is only dependent on the pressure.
Remember that Charles’ Law is only obeyed when both the moles and pressure of the gas sample
are held constant. This suggests that in examining the Charles’ Law constant, we could do a
similar experiment in which we performed the experiment under different pressure conditions.
The result would be similar to what we observed when we varied the moles except that we would
find that the constant was inversely proportional to the pressure. If we carry this one step
further, we can show that the Charles’s Law constant is dependent on both pressure and moles in
n
the following way: K n  R , where R is an even more universal constant called the “ideal gas
P
law constant”. If we substitute this expression for Kn into V = Kn.T we get
nRT
V 
This is most frequently written as PV = nRT and is known as the Ideal Gas Law.
P
The Ideal Gas Law Constant, R, is a universal physical constant (like absolute zero and π) and
is independent of the sample and its properties.
Use the data from your lab (n, T, P and V) to determine the value of R.
When we collected the mole versus volume data, we also determined the pressure and
temperature of the O2 for your samples. Each person in your group should calculate R using a
different set of data from that experiment.
Name
Mol O2
P (atm)
V (L)
T (K)
R
Units of R _________
Average Value of R ______________ (Discard any questionable values before averaging.)
Hints:
1. Solve the ideal gas law for R and substitute in the rest of the properties.
16
Guided Inquiry Analysis of Collected Gas Law Data
2. Temperature always has to be in Kelvin when doing any gas law calculations. R is
L  atm
normally reported in units of
, so convert each property to the appropriate unit
mol  K
before plugging into the Ideal Gas Law.
Does your data suggest that R may be a constant value?
Compare your average value of R to the value that can be found in your textbook or lab manual.
Try to use the ideal gas law to solve the following problems:
(Use R value from textbook or lab manual):
a. Find the volume of 0.35 mole of ethane gas at 35oC and a pressure of 18.1 psi.
b. A sample of Kr has a pressure of 0.877 atm, a volume of 452 mL when at 373 K.
What is the mass of this gas sample? Hint: find moles Kr and convert to grams.
17