Download Heat Capacity Ratio by Kundt`s Method

CHEM 477: Physical Chemistry Lab II
Heat Capacity Ratio by Kundt’s Method
Used and adapted with permission from Paul Ayers (McMaster University)
Background
By definition, 𝐶𝐶𝑣𝑣̅ (or 𝐶𝐶𝑝𝑝̅ ) denotes the amount of heat energy that must be absorbed by one mole of a gas at
constant volume (or pressure) to raise the temperature of the gas by one degree. The absorbed heat energy
causes the molecules to move faster (increase in translational energy), to rotate faster (increase in rotational
energy) and to vibrate faster (increase in vibrational energy). Thus, the knowledge of heat capacities plays a role in
understanding the complexity of gaseous molecules. Unfortunately, the easiest method for determining the
individual heat capacities of gases is beyond the scope of an undergraduate laboratory class. The heat capacity
ratio, 𝛾𝛾 = 𝐶𝐶𝑝𝑝̅ ⁄𝐶𝐶𝑣𝑣̅ (1), is just as useful in understanding the structure of gaseous molecules and is more accessible
experimentally.
The equipartition theorem states that 𝐶𝐶𝑣𝑣̅ of an ideal gas depends on the types of motion the molecule is able to
perform. The different types of motion are called degrees of freedom. The more degrees of freedom a molecule
possesses, the more energy a molecule can absorb before its temperature increases (𝐶𝐶𝑣𝑣̅ is larger).
Theoretical calculations, experimental measurements, and common sense suggest that monatomic gases have the
smallest heat capacities, while complex molecules have larger heat capacities. Each type of molecular motion –
translations, rotations, and vibrations – contribute to the heat capacity. As per classical statistical mechanics,
translational and rotational degrees of freedom contribute ½RT per mole, while vibrations contribute RT.
The number of translational and rotational degrees of freedom available to a molecule depends on the molecule’s
shape (linear or non-linear) and the number of atoms in the molecule. Diatomic molecules have two rotational
degrees of freedom and one vibrational degree of freedom. Linear polyatomic molecules have 3N – 5 vibrational
degrees of freedom, where N is the number of atoms that make up the molecule. Non-linear polyatomic molecules
have 3N − 6 vibrational degrees of freedom.
For example, an ideal monatomic gas can move along the x-, y-, and z-axes, and therefore has three translational
degrees of freedom. Because it has no rotational or vibrational degrees of freedom, its 𝐶𝐶𝑣𝑣̅ is 3/2R. A diatomic gas
has three translational degrees of freedom, two rotational degrees of freedom, and one vibration. It has a 𝐶𝐶𝑣𝑣̅ of
7/2R.
The energy level spacing for translations and rotations are very close together, and at room temperatures, many
translational and rotational energy levels are available to gas molecules. Vibrational energy levels are spaced
further apart, and these levels are not as likely to be populated at room temperature. While classical theory
predicts a 1R contribution to 𝐶𝐶𝑣𝑣̅ for vibrations, quantum calculations and experimental measurements show much
smaller contributions.
Spring 2009
Revised 01/27/2009
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Heat Capacity Ratio by Kundt’s Method
CHEM 477: Physical Chemistry Lab II
Using Speed of Sound to Measure Heat Capacity Ratio
Sound waves require elastic media for their propagation, as a train of sound waves consists essentially of a
sequence of elastic displacements of elements of the medium. These displacements take place in the direction of
the propagation of the wave: sound waves are longitudinal in nature. A stimulus, when applied to the medium at
rest, causes the elements to vibrate about their mean positions and, because they do not vibrate in phase, this
causes pressure (and hence density) differences in the medium, resulting in compressions and rarefactions along
the wavefront. These compressions and rarefactions take place with such rapidity that the process is – for all
intents and purposes – adiabatic. That is, there is no time for heat exchange between the vibrating elements and
their surroundings, and 𝑞𝑞 = 0. The adiabatic compressions and expansions of ideal gases are governed by
𝑃𝑃𝑃𝑃 𝛾𝛾 = constant (2).
The speed of sound in any medium depends on the elasticity of the medium and its density. It can be shown that
the speed of sound 𝑐𝑐 in any medium is given by:
𝐾𝐾
𝑐𝑐 = �
(3)
𝜌𝜌
where 𝐾𝐾 is the bulk modulus of elasticity of the medium and 𝜌𝜌 its density. For a gas undergoing abiabatic pressure
changes, the bulk modulus of elasticity is simply 𝛾𝛾𝛾𝛾, where 𝑃𝑃 is the absolute pressure of the gas. Thus:
𝛾𝛾𝛾𝛾
𝑐𝑐 = �
(4)
𝜌𝜌
If an assumption is made that the gas behaves ideally, then 𝑃𝑃𝑃𝑃 = 𝑛𝑛𝑛𝑛𝑛𝑛 (5) also applies, which upon
rearrangement gives:
𝑃𝑃
𝜌𝜌
=
𝑅𝑅𝑅𝑅
(6)
𝑀𝑀
where 𝑀𝑀 is the molar mass of the gas.
Substitution of Equation 6 into Equation 4 and rearrangement gives
𝛾𝛾 =
𝑀𝑀𝑐𝑐 2
(7)
𝑅𝑅𝑅𝑅
If the distance between corresponding points on the wave front (e.g. maximum compressions) is called the wavelength 𝜆𝜆, and the rate at which such compressions pass a given section is called the frequency 𝑓𝑓, then the speed of
sound is also given by
and
(8)
𝑐𝑐 = 𝜆𝜆𝜆𝜆
𝛾𝛾 =
Spring 2009
𝑀𝑀𝜆𝜆 2 𝑓𝑓 2
(9)
𝑅𝑅𝑅𝑅
Revised 01/27/2009
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Heat Capacity Ratio by Kundt’s Method
CHEM 477: Physical Chemistry Lab II
In this experiment, which is based on a modified Kundt's tube (Figure 1), a sound-wave of known frequency is sent
down a long plastic tube and is made to reflect from a movable piston, creating a standing wave with a series of
maxima, called antinodes, and minima in between, called nodes.
When the standing waveform is plotted on the oscilloscope, it appears as a diagonal line, which is a result of
interference between two waves of the same frequency travelling with the same speed in opposite directions. The
distance between nearest nodes (or anti-nodes) is equal to 𝜆𝜆/2. The successive nodes (or anti-nodes) are in
opposite phase, i.e., they differ in phase by 180° (a phenomenon of which we will take advantage in this
experiment), with maximum sound intensity occurring at the nodes and minimum at the anti-nodes.
Figure 1. Diagram of a Kundt’s tube similar to that used in this experiment.
Experimental
Fill the heat exchanger with water, turn it on, and set the temperature to 25° C. Allow the temperature to
equilibrate before beginning the experiment.
Move the piston as far from the speaker as possible and purge the Kundt’s tube with one of the gases to be
studied (N2, CO2, or Ar). Flow the gas for 15 – 30 sec to purge the tube completely. Reduce the flow of the gas to a
slow stream that is still sufficient to prevent air from diffusing into the tube during the experiment. Note: be
careful while purging with CO2, as it cools considerably upon expansion from the bulk cylinder.
Connect the output of the signal generator to the speaker leads and channel 1 (X) of the oscilloscope. Connect the
microphone leads to channel 2 (Y) of the oscilloscope. Connect the battery to the microphone power leads. Select
an appropriate frequency on the signal generator (3000 Hz for CO2, air, and N2; 4000 Hz for Ar). Finally, switch the
signal generator on. A high frequency noise should be heard from the speaker and a waveform pattern will appear
on the oscilloscope. Details on adjusting the oscilloscope will be provided by the T.A. Ideally, the components
should be adjusted so that the two signals to the oscilloscope are about equal in magnitude.
Spring 2009
Revised 01/27/2009
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Heat Capacity Ratio by Kundt’s Method
CHEM 477: Physical Chemistry Lab II
Slowly move the piston toward the speaker until a diagonal line appears on the screen. Record the position of the
piston. Continue to move the piston inward until you see another diagonal line that is perpendicular to the first
diagonal line. Again, note the position of the piston. Continue taking as many readings as the geometry of the
tube allows. If necessary, repeat the entire procedure until reproducible results are obtained.
Repeat with the other gases provided (CO2, N2, air, and Ar). Repeat the measurements at a minimum of 3 different
frequencies for each gas.
Waste Disposal
There are no waste concerns for this experiment.
Calculations
Given the values of 𝜆𝜆/2 and using the known frequency calculate 𝛾𝛾 for each gas from Equation 9. Using 𝐶𝐶𝑝𝑝̅ − 𝐶𝐶𝑣𝑣̅ =
𝑅𝑅 and Equation 1 determine both 𝐶𝐶𝑣𝑣̅ and 𝐶𝐶𝑝𝑝̅ for each of the gases (you have two equations and two unknowns).
Discussion
Compare your experimental 𝐶𝐶𝑣𝑣̅ values, the literature values, and the theoretical values and discuss the role of
vibrational energy levels in explaining any difference. Discuss the relationship of the value of 𝛾𝛾 with respect to the
structure of the gas molecules.
Spring 2009
Revised 01/27/2009
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