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Heat-Capacity Ratios for Gases
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Introduction & Theory
Gases, and all other substances, can exchange energy with their surroundings, frequently as heat. This absorption of heat may be measured in terms of temperature. The relationship between the amount of heat absorbed and the resulting temperature change varies for different substances, leading to a concept known as heat capacity.
For ideal gases, heat capacity is dependant upon the number of ways the gas has to store energy. These are known as degrees of freedom. There are three different types of degrees of freedom: translational, rotational, and vibrational. By translating (moving), rotating, and vibrating, gas molecules store energy. All gas molecules have three translational degrees of freedom (x, y, and z), but the number of rotational and vibrational degrees of freedom may vary. For instance, a He molecule has no rotational degrees of freedom (as it has no moment of inertia), nor any vibrational degrees of freedom. H2 and other diatomic molecules have two rotational degrees of freedom, and nonlinear molecules have three rotational degrees of freedom. Linear molecules have 3N-5 (Eq. 1) vibrational degrees of freedom, and nonlinear molecules have 3N-6 (Eq. 2), where N is the number of atoms.
As a result, molar heat capacities of gases depend largely on the number of atoms in the molecule, and to a lesser extent the geometry of the molecule.
The goal of this experiment is to determine the ratio of heat capacity at constant pressure to heat capacity at constant volume for an ideal gas through measurement of the speed of sound.
A Kundt's tube, ~150cm long and ~5cm in diameter is fitted with a speaker on one end and a Teflon piston in the other. A microphone is attached to the piston such that it picks up sound from the inside of the tube. Also, a tube is put through the piston so that gas may flow into the tube. A rod is attached to the piston as well, allowing remote movement and measurement.
In this experiment the amplitude of a sound wave will be observed at different locations in a tube, allowing a half-wavelength to be measured between an adjacent maximum and minimum.
Using a version of Kundt's tube, the wavelength of standing waves of frequency f is found by electronic means. The apparatus is set up as shown in figure 1, with an audio oscillator driving a speaker unit. The output from the oscillator and the input from the microphone are displayed on an oscilloscope on opposing axes. When the Lissajous figure on the scope moves from a 45 line tilted to the right to a 45 line tilted to the left, one-half the wavelength of the sound can be measured as the displacement of the piston.
After assembling the apparatus in figure 1 and calibrating the electronic devices at frequencies between 1 and 2 kHz, move the piston to the end away from the speaker. Allow the gas being studied to flow through the tube for 10 minutes to displace the air. Reduce the flow to a very slow stream to prevent ambient air from entering the tube. Adjust the electronics so that a nearly perfect circle can be seen on the scope (not an ellipse). If using Helium, set the oscillator at 2kHz, otherwise set it at 1kHz. Slowly push the piston toward the speaker until a 45 line is seen on the scope. Note the position of the piston, as well as the phase shift. If the line is angled to the right, the phase shift is 0, if to the left, the phase shift is 180. Record as many locations as possible through the length of the tube. Make note also of the temperature of the exiting gas several times throughout the run. Using helium (or argon), nitrogen, and carbon dioxide, repeat this procedure until the results are consistent. Note the ambient barometric pressure.
For an ideal gas, the heat capacity ratio is related to the speed of sound c, molecular weight M, and the temperature T by the following equation:
For a van der Waals gas, the molar volume, pressure, and the van der Waals constants are required, and have the following relationship:
The speed of a wave (in this case, the speed of sound in the experimental gas), c, is a function of wavelength and frequency f.
Mercury and mercury vapor are highly poisonous. Inhalation of vapor may lead to fever, nausea, vomiting, diarrhea, headache, chest pain, and possibly death. Skin contact may lead to a rash or allergic reaction, and if extensive may also cause the same effects as inhalation of vapor. Affected persons should be removed to fresh air and contaminated clothing removed. Necessary first aid techniques should be performed. Seek medical attention immediately.
Inhalation of carbon dioxide gas in low to medium concentrations can cause nausea, dizziness, headache, affect blood circulation, and acidify bodily fluids. High concentrations can lead to death, especially if oxygen is displaced. Persons affected should be removed to fresh air and necessary first aid techniques performed. Seek medical attention immediately.
Inhalation can lead to asphyxia with any, all, or none of the following: dizziness, tightness in the forehead, tingling in tongue, fingertips, or toes, loss of vocal abilities, loss of movement, reduced consciousness, loss of tactile sensations, and heightened mental activity. Persons affected should be removed to fresh air and necessary first aid techniques performed. Seek medical attention immediately.
Same as argon, see argon hazards (gas, compressed).
Same as argon, see argon hazards (gas, compressed).