You should consult your Physical Chemistry text for a review of the meaning of heat capacity, its thermodynamic definition, and the relationships between molecular structure and heat capacity.
The heat capacity of a substance is a thermodynamically defined quantity expressing the relationship between heat absorbed and the temperature change of the sample; heat capacities can be measured directly in the laboratory, or they can be inferred via several thermodynamic relationships. Carefully evaluated heat capacities have been of great value in studies of the physical properties of matter. Heat capacities depend upon the manner in which they are measured. We are concerned here with the heat capacities referred to constant volume (Cv ) and referred to constant pressure (Cp ) for mono and diatomic gases; we will use helium and nitrogen as examples of the two types. Our measurements will not evaluate the heat capacities themselves, only the ratio Cp/Cv. We will make two very different kinds of measurements, to illustrate broad applicability of thermodynamic principles.
The basic device in this method is a large glass vessel initially containing the gas of interest at a pressure slightly above atmospheric. The apparatus is in effect a very sensitive gas thermometer, but analysis of the data does not require knowledge of temperature. The vessel is first purged for a few minutes with the gas of interest, the exit tube of the vessel is then closed and gas is allowed to flow until a modest pressure is established. Success depends upon the absence of leaks and the use of not too high initial pressure. Pressures are measured with an open arm manometer filled with a low density - low vapor pressure fluid (dibutylphthalate) to improve sensitivity. As soon as pressure is constant (this is pressure p1), the gas is allowed to escape rapidly from the vessel, approximating an adiabatic expansion against the constant external atmospheric pressure (you must read the laboratory barometer at the time of your measurements; this is p2). The adiabatic expansion is achieved by lifting and quickly replacing the stopper in the flask. Because work is done adaibatically, the gas in the flask is cooled. After the stopper is replaced, the gas remaining in the vessel quickly warms to room temperature; the manometer reading will be seen to increase from the zero difference which obtained while the stopper was removed. When a constant reading is seen, record its value as p3. Atmospheric pressure, the pressure at the end of the adiabatic expansion and before the stopper is replaced, is called p2. At least three expansions should be done with each of the two gases. A derivation of the function needed to analyze the data is given below.
We assume an ideal gas. In general, dU= dq - dw. If p-V work only is done, and if the process is adiabatic, then dU = -pdV. For an ideal gas the internal energy depends only on temperature, and dU = nCvdT = -pdV. Integrating,
Using V = nRT/p on the right hand side,
Consider the following two-step process involving an ideal gas denoted by A:
Step 1: allow the gas to expand adiabatically and reversibly until the pressure of the gas has dropped from p1 to p2:
Step 2: at constant volume, restore the temperature of the gas to T1:
From step 2 in the process,
Therefore, Cv or the ratio Cp/Cvcan be calculated from the three pressures, without knowledge of the temperature or volumes in these states.
The wave propagation of sound through a gas is longitudinal in nature, that is, it is associated with an oscillatory motion of the gas molecules in the direction of motion. A gas is compressible, and therefore the density of the gas is a function of the pressure variation in the longitudinal direction. The vibrational motion occurs quickly enough to classify the compressions and expansions as adiabatic rather than as isothermal, as might be first supposed. Derivation of the wave equation for a homogeneous gas in a tube leads to a partial differential equation characteristic of standing waves (1,2). The velocity v of the wave is related to pressure and density by the equation
which follows for an adiabatic process (1,2). R is the gas constant, T the Kelvin temperature, M the mean molecular weight of the gas, and gamma is the ratio Cp/Cv. The velocity v can be determined from knowledge of the frequency and wavelength of the wave.
In our experiment we use an audio frequency generator to drive a small speaker attached to one end of a glass tube. A microphone forms a piston in the tube, and can be moved by the operator as required. This arrangement causes the development of standing waves in the tube between the speaker and the microphone. The output from the audio generator and from the microphone are supplied to the x-y inputs of an oscilloscope. The experimental setup is shown here. The waveforms combine in this manner to form traces called Lissajous figures; the nature of the figure obtained is very sensitive to the frequency and phase relationships of the two signals supplied to the axes of the oscilloscope. A Mathcad program is available on the laboratory computers to help you understand how Lissajous figures are formed.
In our measurements the frequencies of the input signals are the same, but their phase and amplitude are different, depending upon the position of the microphone in the tube. Three kinds of figures will be seen : a straight line of slope +1, a line of slope -1, and ellipses ( one special case being a circle). The straight line figures are probably the best choice, since they are the most easily reproduced. By moving the microphone along the tube, the Lissajous figures can be reproduced at wavelength intervals, whose spacing is measured with a meter stick. If successive straight line figures of alternating slope are used, the intervals mark half-wavelengths. Since the frequency of the applied waveform is measured with a frequency counter, sufficient information is at hand to calculate the desired heat capacity ratio. Nitrogen and helium should be used in this experimental procedure. A frequency of 1 kHz should be used for N2; use 2 kHz for He.
The usual general requirements should be followed in preparing this part of the report. Compare your values of gamma for each gas to the results obtained by treating the gases as ideal. In addition, each person should work at least briefly with the Mathcad Lissajous document. Each person should also read the reference cited in the lab text (J. Chem. Ed.) regarding an alternative interpretation of the method of Clement and Desormes.
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