CHM 347 Physical Chemistry II Lab:
Rüchhardt’s Method for Determining γ, the Ratio of Heat Capacities
Objective:
You will determine the ratio of the constant pressure heat capacity to the
constant volume heat capacity for a number of gases. This parameter is
known as γ. To accomplish this you will use the method developed by
Rüchhardt in 1929.
Background: For ideal gases, the two heat capacities are related by Cp,m = Cv,m + R. The
constant volume heat capacity can be calculated from the derivative of the
energy of a molecule with respect to temperature. According to the
equipartition theorem, which we studied in Chapter 32, each term in the
energy expersion that is quadratic with respect to x contributes ½kT to the
total energy. A monatomic ideal gas would thus have a constant volume
heat capacity of
⎛ ∂E ⎞
C v,m = ⎜
⎟ = 32 R
⎝ ∂T ⎠V
This expression results because a monatomic gas has only three
translational degrees of freedom. Diatomic and polyatomic molecules will
also have vibrational and rotational degrees of freedom, and thus the total
energy is the sum of these separate contributions.. Further, the total
constant volume heat capacity of these molecules is described by the
expression
C v ,total = C v ,trans + C v ,rot + C v ,vib
(The electronic contributions have been neglected in this expression since
they contribute an insignificant quantity to the total energy at room
temperature.) At “high” temperatures (relative to the energy separation
between adjacent rotational levels) the rotational contribution to Cv for a
linear molecule is R and for a non-linear molecule it is 32 R. The
vibrational contribution varies much more with temperature, though. In
Chapter 33 we derived the vibrational contribution in terms of the
vibrational temperature Θv:
(C v )vib = ∑ (C v )vib,m where (Cv )vib
m
ΘV
eT
⎛Θ ⎞
= R⎜ V ⎟
2
⎝ T ⎠ ⎛ e ΘTV − 1⎞
⎜
⎟
⎝
⎠
2
In this expression m represents the number of vibrational degrees of
freedom present in the molecule (either 3N-5 or 3N-6). This expression
can be used to determine the vibrational contribution to Cv, which then
allows for the calculation of theoretical overall constant volume heat
capacity of a gas.
There are several experimental techniques that have been employed to
measure heat capacities of gases, including the adiabatic expansion
method and the sound velocity method. In our
experiment we will make use of the Rüchhardt method,
which is based on elementary mechanics. A gas is
contained in a large vessel, like the one shown
schematically at right. Into the vessel is fitted a glass
tube containing a steel ball whose purpose is to slightly
compress the gas in the vessel. The ball is coaxed into
an oscillatory motion in the glass tube. As it oscillates
there are observed changes – albeit small – to the
volume and pressure of the gas. These changes can be
described in terms of the mass of the ball, the radius of
the glass tube, the volume of the vessel, the gas pressure, and the identity
of the gas itself (thus γ is introduced). Ultimately, since the motion of the
ball satisfies the conditions of simple harmonic motion, the period of
oscillation can be described in terms of these quantities, as well. Finally,
the period expression can be solved for γ.
τ = 2π
mV
γPA 2
and
γ =
4π 2 mV
A 2 Pτ 2
We can simply re-express the equation above in terms of the period and an
apparatus constant:
γ =
c app
τ2
By calibrating the system using an ideal monatomic gas whose γ = 53 , we
can find capp, and then use it to determine experimental values of γ for
other gases. Using this simplification, though, is not without the
introduction of error. In the case of the Rüchhardt method the three
sources are (1) that the gas is ideal, (2) that there is no friction, and (3) that
the volume changes accompanying the oscillatory motion of the steel ball
are adiabatic.
Procedure:
You will use the Rüchhardt apparatus to determine γ for three gases –
nitrogen, carbon dioxide, and oxygen. Prior to doing so you use argon to
determine the value defined above as capp.
The instructor will describe the operation of the apparatus to you prior to
beginning your work. Please be careful as this apparatus is on loan to us
from another institution.
Once you understand how to operate the apparatus, follow the steps listed
below to collect your data.
1. Slowly open the valves to introduce argon gas into the apparatus.
Allow the ball to rise up in the glass tube until it reaches the slot cut
into its side. Once the ball reaches this point it will vent the gas
through the slot and the ball will drop quickly. The goal is to have
enough pressure to prevent the ball from dropping all the way back
into the vessel.
2. Allow the system to vent gas through the slot in this manner for
approximately 50 times.
3. Adjust the pressure so that the ball begins to oscillate in the tube
without any further venting, if possible. If not, venting should be
reduced to occurring on every other trial, or less.
4. Once a consistent oscillation pattern is achieved, use a stopwatch to
time the period required to complete ten full oscillations. (This is
easier than timing a single cycle, and introduces less error in the
results – just divide this time by ten to get the period of a single cycle.)
5. Collect a total of ten times – that is ten “periods” for argon.
6. When you have collected all your data for argon slowly reduce the
pressure using the control valve so that the ball gently descends into
the vessel.
7. Disconnect the “plumbing” from the argon source and reconnect it to
another gas source.
8. Repeat steps 1 – 7 for the remaining gases to be observed.
Analysis:
Based on your average argon period, first determine the apparatus
constant, capp.
Knowing the apparatus constant, use the average periods of the other gases
studied to determine their experimental values of γ.
Use the method described in the background above to determine a
theoretical value of γ for each gas studied and compare your results. The
translational and rotational contributions to Cv should be quite easy to
determine. The vibrational contribution is more challenging.
Report:
The report for this experiment focuses on the theory and experimental
work. To that end, you should write a few pages discussing heat
capacities, upon what they depend, and how they can be determined.
Further, you should address, in more detail than the background
information included above does, how the Rüchhardt apparatus functions
and describe the procedures used to collect your data and calculate your
values of γ. Some references that might be helpful are listed below.
As usual, you should also attach a brief appendix of your data, example
calculations, and results.
The following sources may prove useful in developing a thorough description of the
theory and methods employed in this experiment:
Physical Chemistry, Engel and Reid, Chapter 33
Physical Chemistry Methods, Techniques and Experiments, Sime; Experiment 2:
“Measurement and Calculation of the Heat Capacity of Gases”.
Experiments in Physical Chemistry, Garland, Nibler, and Shoemaker; Experiment
3: “Heat Capacity Ratio for Gases”.
Rüchhardt apparatus handout from Dr. Steel.