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The speed of sound traveling through a gas is basically a series of minute pressure variations in the gaseous medium which oscillate as a wave function. These localized and alternating pressure variations will propagate from their source and be translated by the eardrum into sound by the resultant vibrations. Since there is no macroscopic flow of gas, but only localized differences in pressure, a relationship between heat capacity and speed of sound can be derived. At a localized point of higher pressure, the molecules will be accelerated past this point. This causes the localized pressure difference to move in a concerted molecular fashion, but it also indicates a relationship between density and velocity. A local increase in velocity must be offset by a local decrease in density. As a gas passes this local point, its increase in momentum implies an increase in pressure. Therefore, an equation can be mathematically derived which will relate velocity to the gradient of pressure with respect to density. This velocity, since there is no net gas movement, is the velocity of the pressure differential, or the velocity of sound. Pressure and density can be easily reduced to functions of the particular gas through the ideal gas law. Although the ideal gas law will be less accurate for non-ideal gases, especially at extreme temperatures, it should be accurate enough for the purposes of this experiment. This forms an expression which relates sound velocity to the heat capacity ratio as follows:
(1) m=(gRT/M)1/2
In this equation m is the speed of sound, g is the heat capacity ratio, R is the gas constant, T is the temperature, and M is the gram molecular weight of the gas. By experimentally measuring m and T, g can be calculated. The speed of sound through a specific gas can be determined through experimental observation of the wavelengths of a given frequency through a gas using a Kundt's tube. The formula for this determination of the speed of sound is as follows:
(2) m=lf
Here l is the experimental wavelength measurement, and f is the frequency. The heat capacity ratio is related to the constant volume heat capacity by the following formula1:
(3) Cv=R/(g-1)
In this formula, Cv is the constant volume heat capacity. All other variables remain the same.
The equipartition principle relates the heat capacity of a molecule to the number of degrees of freedom that the molecule has available to it. Degrees of freedom can be derived from translational, rotational, and vibrational motion. Translational motion is the movement of a molecule to different points in space. Everything in the physical world has three translational degrees of freedom relating to each spatial dimension (x,y and z axes). The energy contribution from translational degrees of freedom is RT/2 for each of the three degrees, resulting in a 3RT/2 total contribution in energy from translational motion for all molecules. This energy is strictly the kinetic energy derived from physical movement of the molecule.
Rotational degrees of freedom are assigned according to the number of different spatial axes (x,y and z) which you can rotate the molecule around1. Thus, a monatomic molecule has zero degrees of rotational freedom because is has perfect three dimensional symmetry. A linear molecule will have two possible spatial axes of rotation and thus two rotational degrees of freedom, and a non-linear molecule will have axes of rotation around all three spatial axes, thus three rotational degrees of freedom. Rotational degrees of freedom also contribute RT/2 energy each. Again, this is because rotational motion is kinetic motion, producing kinetic energy.
Vibrational degrees of freedom are somewhat more abstract. Vibrational motion arises from the motion between bonded atoms through the medium of a covalent bond. Atoms will vibrate around bonds, creating intramolecular motion. Bond stretching by elongating or compacting along the length of the bond or bending perpendicular to the direction of the bond also contributes to the vibrational motion of the molecule. Argon has no bonds, and thus no vibrational degrees of freedom, but N2 and CO2 both have bonds, and thus exhibit vibrational degrees of freedom.
The number of vibrational degrees of freedom in a molecule can be derived from the following formula which determines the total number of degrees of freedom in a molecule.
(4) 3N
Here N is the number of atoms in the molecule. To determine the number of vibrational degrees of freedom, one must then subtract the number of translational degrees of freedom (always three) and the number of rotational degrees of freedom (two for linear molecules, three for nonlinear).
The energy contribution of each vibrational degree of freedom is RT, twice the contribution of translational and rotational motion. This is because vibrational degrees of freedom account for intermolecular vibrations, or kinetic energy of motion, but also registers the potential energy of bond stretching and bending.
The equipartion principle states that by counting the total number of degrees of freedom and determining their energetic contributions, the Cv can be determined. The formula for this determination is as follows:
(5) Cv=(dE/dT)v
Where dE is the change in energy and dT is the change in temperature. The subscript v indicates a constant volume process. This formula can be simplified as follows:
(6) Cv=3R/2+nR/2+n1R
In this formula, the 3R/2 term represents the energy contribution from translational motion which is uniform in all molecules. The nR/2 term represents the energy contribution from rotational motion, and n is the number of rotational degrees of freedom. The n1R term represents the energy contribution from vibrational motion, and n1 is the number of vibrational degrees of freedom. R is the gas constant.
By calculating the Cv from the equipartion principle, the assumption is made that the energy contribution from the three types of bond motion is not a function of temperature. This assumption holds for translational motion, but rotational and vibrational motion are strongly dependent on temperature. This means that for all polyatomic molecules, the equipartion principle fails to accurately determine the Cv.
This results because a molecule receives most of the energy contribution due to translational motion even at room temperature. Higher temperatures are required for rotational energy contributions to count, and the temperatures needed for vibrational energy contributions to count is so high that rarely is equipartion valid for vibrational modes4. It is more accurate to calculate the vibrational energy contribution as it varies with temperature. Formulas have been derived for this2, and they provide an accurate theory for determining energy contributions and Cv's for polyatomic molecules.
Thus the theoretical energy with the vibrational and rotational energy contribution as calculated from the equipartion principle is a maximum energy for the molecule which is approached at high temperatures. Similarly, as the temperature decreases below that necessary for vibrational and rotational energy contribution to be significant, the theoretical energy of translational motion effectively represents an energy minimum for most experimental temperatures (including room temperature).
Viscosity can be viewed as the resistance to flow of or through a medium. For example, hexane is less viscous than water because of water's high number of hydrogen bonds. The resistance to flow through hexane is less than that through water because there are weaker intermolecular interactions which must be broken for hexane. The energy of the van der Waals forces in hexane is much lower than the energy of the hydrogen bonds in water. Thus it takes more energy to move through water, corresponding to the higher viscosity of water. This also translates into the resistance of flow of a medium. Hexane will flow much easier than water due to the lack of intermolecular interactions binding individual molecules closer together. This corresponds to a lower viscosity for hexane.
As you heat water, more molecules achieve enough energy to break the hydrogen bonds around it, thus making it easier to move through or less viscous. A gas, however, acts in the opposite manner. Although the concept of viscosity remains the same, heating a gas increases the viscosity. As a gas is heated, the molecules' movement increases and the probability that one gas molecule will interact with another increases. This translates into an increase in intermolecular activity and attractive forces - just the opposite effect of heating a liquid.
On a technical level, the viscosity of a gas is caused by a transfer of momentum between stationary and moving molecules. As temperature increases, molecules collide more often and transfer a greater amount of their momentum. This increases the viscosity. In either view, it is obvious that the molecular effect of heating a gas is drastically different - even opposite from the molecular effect of heating a liquid, producing the opposite effect.
In order to measure the viscosity of a gas, the gas must be timed as a given volume passes through a capillary of known dimensions. The following formula measures viscosity as a function of time:
(7) t=Kh
In this formula t is time, K is the apparatus constant which accounts for capillary tube length, bulb dimensions, room temperature, apparatus mineral oil pressure, etc., and h is the viscosity. In order to determine the constant K, the easiest method is to pass a gas of known viscosity through it. For this experiment, this gas was air. The viscosity of air was determined from the following formula6:
(8) hA=(145.8x10-7*T3/2)/(T+110.4)
In this formula, T is the Kelvin temperature and hA is the viscosity of air in poise at 1 atm. From formulas 7 and 8 the viscosities of several gases can be experimentally calculated and compared with literature values for accuracy.
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