The heat capacity ratio of three gases will be measured: N2, Ar and CO2.
Read carefully the instructions for working with gas cylinders in SGN (in the chapter on "Miscellaneous Procedures") before coming to the lab.
Connect the wave generator to the frequency monitor and the scope. The "strong" output from the wave generator should be connected with the freqeuncy monitor, and the "weak" output should be connected with the scope. Set the frequency generator to 1000 Hz and record carefully the actual frequency. Set the filter on and average over 1 second on the frequency meter. Display a Lissajous figure on the scope to determine the distances between speaker and microphone that correspond to in phase and out of phase coherence.
Start with the microphone pulled out far from the speaker. Run N2 gas through the tube for at least 10 minutes before starting the measurements. You can monitor the gas flow by dipping the plastic hose into the silicon oil. Pull the hose up before measuring because theformation of bubbles in the oil causes noise in the measurement. Use the metal marker near the microphone to read the position accurately. Measure the positions corresponding to to in-phase and out-of-phase interference as the microphone is brought towards the speaker. Repeat the measurements as the microphone is pulled out again. Be careful not to pull the microphone out so fast that air gets sucked into the tube. This can be monitored by the gas bubbles in the silicon oil. Repeat the measurements as the microphone is pulled in and out again to check reproducability. If the same results are not obtained, repeat the measurements. After reliable measurements have been obtained for this frequency, set the frequency generator to 1500 Hz and measure again at least two rounds of pulling the microphone in and then out again. Which frequency gives more consistent measurements? Stick to that freqeuncy as you measure the Ar and CO2 gas. In the case of CO2 it is important to heat the valve of the CO2 gas cylinder with the hair-dryer (it can be mounted on the rack). Record the temperature of the gases in the tube and obtain the atmospheric pressure from the web page of the meteorology institute (Vedurstofa Islands).
By assuming the gases are ideal, find the heat capacity ratio in each case. Estimate the error in each of the measurements and estimate the uncertainity in the value you obtain for the heat capacity ratio. Then, use the van der Waals equation of state for N2 and CO2 and recalculate the heat capacity ratio (the van der Waals coefficients can be found in chapter 1 of the text book by Silbey and Alberty). How important is it to take non-ideality into account in this case? Is the uncertainity in your determination of the heat capacity ration small enough to make non-ideality detectable?
The proper theoretical discussion of the heat capacity of gases will be given later in the course (one needs quantum mechanics as well as statistical mechanics to do that, chapter 16 in the text book by Silbey and Alberty). It turns out that vibrational motion often does not contribute fully to the heat capacity until the temperature is quite high. This is because the quantization of vibrational energy gives large energy gaps between adjacent energy levels. For this report, you first of all assume the classical result is right for all degrees of freedom, i.e. use equipartition theorem. Recall that equipartition theorem says that each translational degree of freedom contributes kT/2 to the internal energy (k is the Boltzmann constant), each rotational degree of freedom also contribues kT/2, but each vibrational degree of freedom contributes kT (more than translation and rotational degrees of freedom because of the potential energy increase as the bond lengths and bond angles are distorted). What is the predicted heat capacity ratio for N2, Ar and CO2 using classical physics? Is it in agreement with your measured results (carefully taking the uncertainity of the measurement into account)? At room temperature, the vibrational degrees of freedom that have high frequency do not contribute the full kT to the internal energy, as you will see later from quantum statistical mechanics. It is often a better approximation to skip completely the vibrational contribution to the internal energy, rather than to include the classical, high temperature limit, kT. Repeat your calculation of the heat capacity of N2 and CO2 now skipping the contribution from vibration. Does this give better agreement with your measured values? Are they in agreement with your measurements, again taking into account the experimental uncertainity? Finally, assume the contribution of translation and rotation is given correctly by classical physics and evaluate, from the measured heat capacity ratio, the contribution of vibration to the constant volume heat capacity. What fraction of the full, classical value does the vibrational contribution turn out to be under the conditions of your experiment?
Derive an expression for the heat capacity ratio of a gas of bent molecules containing a total of N atoms, and of a gas of linear molecules with N atoms. Could you use your measurements to tell whether CO2 is linear or bent (considering just the estimated uncertainity in the measurement, not the agreement between the predicted and measured values)?
Consider the importance of the purity of the gases. What effect would it have on the estimated heat capacity ratio of the nitrogen gas had 1% oxygen in it? Calculate how much gamma would be affected. What if the Ar gas had 1% nitrogen gas mixed in? What is the purity of the gases in the cylinders you used?
Later on in the course, you will redo calculations of the vibrational contribution to the heat capacity of CO2 and N2 using quantum statistical mechanics and you will then be able to get very close agreement with the experimentally estimated values obtained from gamma, if all goes well in the experiment.