The Experimental Determination of Heat Capacity Ratios of Gases, N2, CO2 and Ar, by the method of Adiabatic Expansion

ABSTRACT The heat capacity ratios [Cp/Cv] of Nitrogen, Carbon Dioxide and Argon were obtained by the method of adiabatic reversible expansion. Four studies were conducted for each gas using an 18.0 L carboy and phthalate manometer at 297K. The initial and final pressures were obtained and used to calculate the experimental heat capacity ratios. Based on the results, Argon has an experimental heat capacity ratio of 1.66 and a Cv value of 1.56, which agrees with the theoretical Cp/Cv and Cv values obtained from the classical equipartition theorem. Based on the results, the diatomic molecule, N2 and triatomic molecule, CO2 have heat capacity ratios of 1.45 and 1.35, respectively. The experimental values for N2 are very close to the predicted values obtained from the classical equipatition theorem for diatomic gases without vibrational contribution (7RT/5RT). The experimental Cp/Cv and Cv values for CO2 are less than the determined theoretical Cv values obtained from the classical equipartition theorem for triatomic gases. The differences in experimental and theoretical Cp/Cv and Cv values for CO2 molecules are perhaps due to the contribution of vibrational modes of freedom.

INTRODUCTION The heat capacity is the measurable physical quantity that specifies the amount of heat required to change the temperature of a substance by a given amount (1). The manner in which a substance takes up heat can provide information about its internal molecular structure (2). It is known that the number of atoms and physical structure of molecule plays a major role in its heat capacity. There are ways to determine the heat capacity of a molecule. For an ideal gas there is an association between the heat capacity at constant volume (Cv) and heat capacity at constant pressure, Cp= Cv + R (3). The difference between heating a gas at constant volume and constant pressure is equal to the work of expansion, Cp/Cv=y (3) In order to fully understand heat capacity and how it leads to increased temperature of a gas molecule, the method in which a molecule absorbs heat has to be considered. The heat intake is equipartioned amongst the possible modes of energy available to the gas (3). The gaseous molecule can gain energy by moving along the x, y, z axes, thus increase its translation energy (2,3,4). As previously notated the number of atoms a molecule is composed of is essential because molecule with two or more atoms can also rotate and vibrate, in addition to translational movement. The energy storage modes, also referred to as degrees of freedom is based on the number of independent coordinates needed to specify the position and configuration of all the atoms in the molecule at any give time point (3,4). In detail, a molecule of N atoms would have 3N degrees of freedom. By incorporating classical statistical mechanics it is possible to derive the expansion of work, via the movement of molecule in three-­‐ dimensional space. This is done with the use of the equipartition of energy theorem, which associates kinetic energy of RT/2 per mole with each quadratic term (3,4). By using this theorem, the effects translational, rotational, and vibration movement are accounted for in the molecules ability to absorb energy (3). The heat capacity ratio can be measured using the adiabatic expansion method. At first a gas with the initial pressure, P1, expands adiabatically and reversibly to atmospheric pressure P2 and then relaxes to original temperature, with resulting pressure

P3. The recorded pressures can be incorporated into the formula y=Cp/Cv=In(P1/P2)/In(P1/P3) to obtained the heat capacity ratios and heat capacity of a gas at constant volume, Cv (3). The experimental values can then be compared to the theoretical values obtained from the Cv values from the equipartition of energy theorem. In this experiment the experimental and theoretical values are determined and compared to explore the affects of vibrational and rotational mode of freedom contribution to the overall heat capacity of a gas.

EXPERIMENTAL PROCEDURE The adiabatic expansion experimental trials were obtained using an 18.0 L carboy at temperature 24 degree Celsuis, 0.99 bar. Each gas trial began with the sweeping of the gas to be studied through the carboy at a rate of approximately 4L/min for 15 mins. The flow rate was measured by displacing a 600 mL of water from an inverted beaker under water. Since air is eighty percent nitrogen gas, Nitrogen was studied first followed by Carbon dioxide and Argon. Although Carbon dioxide is a heavier gas it was studied before Argon due to gas availability. Argon was studied after Carbon dioxide and to account for residual CO2 the first reading for Argon was discarded from experimental data. For each trial and each gas studied, the initially pressure, P1, was obtained after the manometer achieved a constant reading after the tube connected to the manometer was opened, and the tubes connecting to the gas tank and atmosphere were closed. The adiabatic reversible expansion was carried out by quickly removing and replacing the stopper of the carboy. Once the gas remaining in the carboy was allowed to return to its initial temperature, the final pressure was recorded, P3. The initial and final pressure readings were than converted to cm of Hg by multiplying the values to the density ratio of dibutyl phthalate to mercury and by adding P2, 741 mm of Hg (3).

RESULTS

Trials P1 P3 P1 final P3 final 1 58.7 15.5 11.94 8.61 2 59.5 15.4 12.00 8.59 3 59.0 15.0 11.96 8.57 4 61.0 15.8 12.12 8.62 Table 1: Initial and final pressure changes for Nitrogen gas, N2. Four experimental trials were completed at a temperature of 24 degrees Celsius. The initial pressure, P1, and the final pressure, P3, were obtained from the adiabatic expansion method and recorded from an open-­‐tube manometer. P1 and P3 final values were obtained by converting manometer reading, P1 and P3 to millimeter of mercury (Torr), therefore P1 and P3 values were multiplied by the ratio of dibutyl phthalate density to the density of mercury, (1.046 g cm^-­‐3/13.55 g cm^-­‐3) followed by the addition of the atmospheric pressure, 741 mm of Hg, obtained from the barometer (3).

Trials In(p1/p2) In(p1/p3) Cp/Cv=In(p1/p2)/In(p1/p2) 1 0.477 0.327 1.46 2 0.482 0.334 1.44 3 0.479 0.333 1.44 4 0.492 0.341 1.44 Table 2: Heat capacity ratio (Cv/Cp) for N2. Cp/Cv ratios are computed using equation Cp/Cv=In(p1/p2)/In(p1/p2) (3)derived from the properties of reversible adiabatic expansion of a perfect gas. The values are calculate from the P1 final and P3 final values obtained from Table 1. Average(Cp/Cv) ¼(1.46+1.44+1.44+1.44) 1.45 Table 3: The Average Heat Capacity ratio for Nitrogen gas, N2. Using the values obtained from the four experimental trials in Table 2 an average value for the heat capacity ratio for N2 is determined, 1.45.

Trials P1 P3 P1 final P3 final 1 57.9 12.4 11.88 8.36 2 60.4 13.2 12.07 8.43 3 60.7 12.2 12.09 8.35 4 60.8 13.2 12.10 8.43 Table 4: Initial and final pressure changes for Carbon Dioxide, CO2. Four experimental trials were completed at a temperature of 24 degree Celsius. The initial pressure, P1, and the final pressure, P3, were obtained from the adiabatic expansion method and recorded from an open-­‐tube manometer. P1 and P3 final values were obtained by converting manometer reading, P1 and P3 to millimeter of mercury (Torr), therefore P1 and P3 values were multiplied by the ratio of dibutyl phthalate density to the density of mercury, (1.046 g cm^-­‐3/13.55 g cm^-­‐3) followed by the addition of the atmospheric pressure, 741 mm of Hg.

Trials In(p1/p2) In(p1/p3) Cp/Cv=In(p1/p2)/In(p1/p2) 1 0.472 0.351 1.34 2 0.488 0.359 1.36 3 0.489 0.370 1.32 4 0.490 0.361 1.36 Table 5: Heat capacity ratio (Cv/Cp) for CO2. Cp/Cv ratios were computed using equation Cp/Cv=In(p1/p2)/In(p1/p2) derived from the properties of reversible adiabatic expansion of a perfect gas. The values were computed from the P1 final and P3 final values obtained from Table 1. Average(Cp/Cv) ¼(1.34+1.36+1.32+1.36) 1.35 Table 6: Average Heat Capacity ratio for CO2. Using the values obtained from the four experimental trials in table 2 an average value for the heat capacity ratio for N2 was determined to be 1.35.

Trials P1 P3 P1 final P3 final 1 57.5 18.4 N/A N/A 2 59.9 20.2 12.03 8.97 3 63.2 21.9 12.28 9.10 4 62.1 20.8 12.20 9.02 Table 7: Initial and final pressure changes for Argon. Four experimental trials were completed at a temperature of 24 degrees Celsius. Although four trials were completed, trial one values were not included in the final calculations. Due to gas availability, Carbon dioxide was studied before Argon. Carbon dioxide is much heavier than Argon, therefore to account for residual CO2 the first experimental reading for argon was discarded from experimental data. The initial pressure, P1, and the final pressure, P3, were obtained from the adiabatic expansion method and recorded from an open-­‐tube manometer. P1 and P3 final values were obtained by converting manometer reading, P1 and P3 to millimeter of mercury (Torr), therefore P1 and P3 values were multiplied by the ratio of dibutyl phthalate density to the density of mercury, (1.046 g cm^-­‐3/13.55 g cm^-­‐3) followed by the addition of the atmospheric pressure, 741 mm of Hg, obtained for the barometer (3).

Trials In(p1/p2) In(p1/p3) Cp/Cv=In(p1/p2)/In(p1/p2) 2 0.485 0.294 1.65 3 0.499 0.300 1.67 4 0.499 0.302 1.65 Table 8: Heat capacity ratio (Cv/Cp) for Argon, Ar. Cp/Cv ratios were computed using equation Cp/Cv=In(p1/p2)/In(p1/p2)(3) derived from the properties of reversible adiabatic expansion of a perfect gas. The values were computed from the P1 final and P3 final values obtained from Table 1. Average(Cp/Cv) 1/3(1.65+1.67+1.65) 1.66 Table 9: The Average Heat Capacity ratio for Argon. Using the values obtained from the four experimental trials in table 2 an average value for the heat capacity ratio for N2 is determined to be 1.66.

Gases Cv (J/mol K) Cp Cv (theoretical) N2 18.48 26.79 20.8 CO2 23.74 32.05 29.1 Ar 12.59 20.90 12.5 Table 10: Heat Capacity at constant volume and constant pressure for N2, CO2 and Ar. Since for an ideal gas, Cp=Cv + R and Cp/Cv=y to obtain the Cv and Cp values the equation were rearrange to Cv= R/[y-­‐1] and Cp=Cv +R. Therefore using the y values obtain from each gas, Cv and Cp values were calculated. In addition, the theoretical Cv and Cp values were calculated via the gas atomic class, monatomic, diatomic or linear triatomic and by the

principle of Equipartition of energy. Thus, the Cv theoretical values for CO2=7/2R, N2=5/2R and Ar=3/2 R.

Gases Cp/Cv Theoretical -­‐vib Cp/Cv Theoretical + vib N2 γ=7/5=1.4 γ=9/7=1.28 CO2 γ=7/5=1.4 γ=9/7=1.22 Ar γ=5/3=1.66 N/A Table 11: Theoretical Cp/Cv values with and without vibrational contribution. The predicted Cp/Cv values were calculated using the equipartition theory and the relationship of Cp=Cv +R. Gases Vibration (J/mol k) N2

0.012 CO2 V2 Bend V1 V3 Symmetric asymmetric stretch stretch 3.72 0.459 .013

Table 12: Vibration heat capacity for N2 and CO2. Cv from the vibrational modes of a molecule can be calculated if the vibration frequency, v, of a molecule is known. Using v=2358cm-­‐1 for N2 and V2=667.3cm -­‐1 for the CO2 bends which are doubly generated and v1=1388.3cm-­‐1 for CO2 symmetric stretch along with V3=2349.3cm-­‐1 to account for CO2 asymmetric stretch, the vibration heat capacities were obtained from the following formula, Cv (vib)=R (u^2e^-­‐u/(1-­‐e^-­‐u), with u=1.43889(v)/T(298.15K) (3).

Gases Theoretical Value of y N2 1.39 CO2 1.29 Table 13: Theoretical value of y for N2 and CO2 using Van der waals. Using the formula y=1+R/Cv(1+2(a)/pV^2), where (a) equals the van der waals constant (a)(in dm^3/mol). The van der waals’ constant (a) for N is 1.408 and is 3.640 for CO2. V is equivalent to RT/p, where R is 0.083145 bar dm^3/mol and T is at room temperature, 298.15 K, and p is equal to atmospheric pressure, P2=0.99 bar (3).

Discussion In this study, the heat capacity ratios of gases, N2, CO2 and Argon, at constant pressure to that at constant volume were determined by the method of adiabatic expansion. The average experimental Cp/Cv ratio calculated for N2 is 1.45 (Table 3), which agrees with the predicted values obtained from the classical equipatition theorem for diatomic gases without vibrational contribution (7RT/5RT). This is expected due to the negligible contribution of the vibrational mode of freedom at room temperature(Table 12). N2’s calculated Cv value is slightly less than the theoretical value obtained from the classical equipartition theorem for diatomic gases (Table 10), this recorded difference in values may be due to vibrational mode contributions, however at such low temperatures it is unlikely. The difference may be due to experimental error. The timing in allowing the carboy to completely fill up with the gas, in addition to the rate at which the stopper was removed and replaced may have been miscalculated. The average Cp/Cv ratio for CO2 is 1.35 (Table 6) and is considerably smaller than the predicted values obtained from the classical equipartition theorem. Although this is the case the experimental Cp/Cv values are closer to the predicted value

without the contribution of vibrational mode of freedom, this is also due to the experimental temperature state. At room temperature the vibrational mode of freedom contribution is less than at higher temperature, where vibrational mode of freedom has a high contribution to Cp/Cv ratios. The calculate Cv is less than the theoretical value obtained from the classical equipartition theorem for polyatomic gases (Table 10), this recorded difference in values may be due to vibrational mode contributions, however at such low temperatures it is unlikely. The difference may be due to experimental error. The timing in allowing the carboy to completely fill up with the gas, in addition to the rate at which the stopper was removed and replaced may have been miscalculated. The average Cp/Cv ratio for Argon is 1.66 (Table 9) and agrees with the predicted Cp/Cv values obtained for the classical equipartition theorem for monatomic gases. In addition, the calculate Cv value also agrees well with the theoretical value (Table 10). Since monatomic gases can only absorb energy using the translational mode of freedom, rotational and vibration modes of freedom do not apply, thus cannot affect values. The experimental values for monatomic gases agree exactly with the predicted value. In Table 11, the affects of vibrational mode of freedom contribution to Cv/Cp values for N2 and CO2 were calculated. In the case of N2 the Cv vibrational contribution is much less than that of the Cv vibrational contribution of CO2 (Table 11). For CO2 it is shown that the two bending modes make a significant contribution to the overall heat capacity, 3.72 J/mol K (Table 11). Although there are differences obtained from experimental and theoretical that may be due to the affects of vibrational contribution, these affects are not as significant at room temperature as compared to their affects at higher temperatures. As previously mentioned, the configuration of a molecule plays a role it its heat capacity ratio. However, for CO2 theoretical Cv calculations it does not appear that a distinction can be made between non-­‐linear and linear configurations. The theoretical heat capacity ratio for nonlinear CO2 would be 6/2R. The theoretical Cv for linear carbon dioxide without vibrational modes is 5/2 and for a nonlinear configuration another degree of freedom would be added. Based of the formula Cp= Cv+R, it appears that Cp will also increase. With both Cv and Cp increasing it would be difficult to determine between nonlinear and linear carbon dioxide theoretical heat capacity ratios.

Conclusion

Based on the data from this study, the correlations of increasing atomicity of gases and their degree of freedom is a positive relationship. Polyatomic molecules can absorb energy in many ways and because of this it is shown that the diatomic and triatomic molecule Cp/Cv values are smaller than that of monatomic molecules. In addition, vibrational and rotational modes of freedom can have a significant impact on Cp/Cv ratios, however their impact is less significant at lower temperatures. Since monatomic gases only take into account translational mode of freedom their experimental and theoretical values are consistent. This exploration of heat capacity ratios of gases was very insightful and that it revealed that molecular properties play a major role in the heat capacity of molecules and these properties can be explored using many derived equations from the ideal gas law, classical equipartition theorem and the van der waal equation.

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