Ideal Gas Law
Jane Doe
Title
Title
Physics 16, Tuesday Section
Partner: Michelle Smith
Sept. 12, 2006
Introduction:
Name, Date, Partner
Name,
Date, Partner
and
Lab Section
and Lab Section
The purpose of this experiment is to study the conditions under which air behaves as an
ideal gas. Measurements of air pressure and temperature are made at fixed volume and fixed
moles. The analysis of the measurements will show that air obeys the ideal gas law. From the
analysis an experimental value for the universal gas constant R is obtained.
Theory: The Ideal Gas Law
Subheading
Subheading
Ideal gases are gases in which the particles of the gas only interact through elastic
collisions; there are no intermolecular forces between the particles. Real gases can approximate
ideal gases at conditions of low pressure and high temperature where the intermolecular forces
are negligible. Under those conditions the equation of state for an ideal gas holds:
Equation reference number
PV =nRT
Equation reference number
(1)
where the pressure (P) and volume (V) are directly proportional to the temperature (T). The
constant R is the universal gas constant and n is the number of moles in the volume. If V and n
Variables defined in text
held fixed
Variablesare
defined
in textand a series of measurements of pressure and temperature are made, then Eq. (1)
predicts a linear relationship between the pressure and temperature. It is therefore expected that a
plot of P vs. T will be a straight line with slope proportional to R. Using a regression analysis of
the plot, the slope can be found, and an experimental value of R determined.
To test the ideal gas law, we used an air-tight cylinder with a movable piston. Due to its own
mass, the piston exerts pressure on the gas in the cylinder below it. Once the system is in
equilibrium, the pressure exerted by the cylinder is equal to the pressure of the gas. The total
pressure is:
P TOT =P atm
gM
A
(2)
Here g is the acceleration of gravity, M is the mass of the piston, A is the cross sectional area
of the piston and Patm is the atmospheric pressure.
The volume of the gas is proportional to the height ∆h of the piston above the floor of the
cylinder:
V =A  h
(3)
The number of moles n is determined by the volume of gas at standard temperature and pressure
(STP) of 273.15 K and 1 atm. At STP one mole gas occupies of volume of 22.4 liters. We can use
this to define a unit of molar density:
 STP =
1 mole
22.4 liters
(4)
Therefore the number of moles is:
n= V
(5)
By substituting Eq. (2) , (3) and (5) into Eq. (1), we find the equation of state in terms of
the measured quantities M and T:
M=
P atm A
 AR
T−
g
g
 
Note: no long derivation.
Note:
no long
derivation.
Just
reference
equations
Just
reference
equations
and
give
result. Keep
detailed
and
give
result.
Keep detailed
calculation in notebook.
calculation in notebook.
(6)
=aT b
M can be changed by placing additional mass on the piston. In this experiment, M is adjusted to
keep the height constant at each temperature T. A measurement of M vs. T at fixed height is
equivalent to a measurement of P vs. T at fixed V. The slope (a) and intercept (b) can be found
from a regression analysis of M vs. T, allowing R to be determined.
Experimental Technique: The Double-Walled Cylinder
Figure with relevant
Figure
with relevant
parts
labeled
parts labeled
Figure
Figure and
number
number and
caption
caption
Figure 1: Double-walled cylinder and piston
The experiment setup consist of a double-walled cylinder shown in Figure 1. An air-tight
piston of (5.000 ± 0.001) kg and cross sectional area A = (78.54 ± 0.31) cm2 holds the gas inside.
The space between the inner and outer walls is filled with water to maintain the temperature of
the gas in the cylinder. The cylinder is placed on top of an electric heating coil which heats the
water. After the setting of the coil is adjusted, the water and the gas in the cylinder are allowed to
reach thermal equilibrium before making further measurements. This typically takes about 15
minutes. The water bath temperature is then measured (Table 1) using a thermometer, with an
estimated uncertainty of ±2 K.
The apparatus was initially setup at STP. The water bath between the two cylindrical walls
was filled with ice water. The valve on the top of the piston was opened to allow air to flow
freely between inside and outside the cylinder. A support block was placed between support
stand and the top of the cylinder. This ensures that the only pressure on the gas inside the
cylinder was the atmospheric pressure. Two thermometers were used to measure temperature:
one of the ice water bath and the other the temperature of the air inside the cylinder. The second
thermometer was narrow enough to be inserted in the open valve without blocking air flow
through the valve.
After sufficient time had pass, the air in the cylinder was at STP. The molar density of the
air is then given by Eq. (4). The thermometer in the valve was removed and the valve was closed
off. The support block was removed and the piston was allowed to slowly descend into the
cylinder. The total pressure on the air was given by Eq. (2). Assuming the temperature of the air
remains constant, the molar density of the air was calculated using:
 gas = STP
PTOT
P atm
(7)
We measured the initial height h of the piston to be (18.90.1)cm from the base of the cylinder
(see Fig. 1). The heating coil was turned on and a series of measurements was made by varying
the temperature of the water bath using the heating coil. As the water bath warmed, the air inside
expanded and the piston rose. The air inside the cylinder was allowed to reach thermal
equilibrium with the water bath. Mass was then added to the support stand until the piston
compressed the air to the initial height of 18.9cm. We determined the uncertainty in the mass by
applying masses of 1 gram increments to the support stand to see if there was an observable
change in the height. We estimated uncertainty in the mass to be ± 5 grams.
Data, Analysis and Results:
Measurements were made at a total of eight different temperatures. Table 1 is a summary of
the data taken.
Table 1: Measurements of Temperature and Mass
Table number
Table
and
Titlenumber
and Title
Temperature (±2 K)
Mass (±0.005 kg)
300
7.201
310
10.430
320
14.550
330
16.256
340
20.109
350
23.034
360
26.322
370
28.178
Note: table has less than 10 entries.
Note:
table tables,
has lesslike
than
10 entries.
Keep
longer
motion
sensor
Keep
longer
tables,
like
motion
sensor
data, in spreadsheet.
data, in spreadsheet.
Column heading with
Columnunits
heading
variables,
and with
variables, units and
uncertainty
uncertainty
Figure 2 is a plot of mass vs. temperature. The plot demonstrates the linear relationship between
the mass and the temperature as predicted by the ideal gas law.
Title – vertical (y) vs. horizonal (x)
Title – vertical (y) vs. horizonal (x)
Axes scale divided into
Axes scale
divided
into
intervals
of 1,2,5
or 10
intervals of 1,2,5 or 10
Axes labeled with
Axes labeled
with
variable
and units
variable and units
Figure 2: Mass vs. Temperature - solid line is best
Figure number
Figure
number
and
caption
and caption
fit to data
A regression analysis was performed to determine the best fit slope and intercept.
Table 2 contains the results of this analysis:
Note: just present results
just present
results
ofNote:
regression
analysis.
Keep
of regression analysis. Keep
summary output from Excel 
insummary
notebook.output from Excel
in notebook.
Table 2: Regression Analysis Results
Slope a
(0.304 ± 0.012) kg/K
Intercept b
(-83.7± 4.0) kg
Using the best fit value of the slope a and Eq. (6), the universal gas constant R can be
determined. The result for R is:
Results always presented
Results
always Keep
presented
with
uncertainty.
detailed
J
with
uncertainty.
Keep
detailed
Rmeas =8.50±0.40
(8)
uncertainty analysis in notebook
mole⋅K
uncertainty analysis in notebook
The uncertainty in Rmeas does not include systematic effects, such as possible deviations from
standard temperature and pressure while the apparatus was being prepared. The atmospheric
pressure can also be calculated, using the best fit value of the intercept b:
P atm =1.03±0.05atm
(9)
This agrees well with the anticipated value of about 1 atm.
Conclusion:
Under the present experimental conditions, air behaves like an ideal gas. At fixed volume
and fixed moles, measurements of pressure vs. temperature exhibited a linear relationship,
consistent with the equation of state for an ideal gas. From the analysis of mass vs. temperature
an experimental value of Rmeas = (8.50 ± 0.40) J/(mole ∙ K) was obtained. This experimental value
is consistent, within uncertainty, with the accepted value of R = 8.31 J/(mole ∙ K).
The largest source of uncertainty in Rmeas comes from the slope of the regression analysis.
The uncertainty in the slope is 3.9%, compared to the uncertainty in A of 0.4%. This uncertainty
arises from the spread of the data in Fig 2, rather than from a systematic deviation from linearity.