Čv = (δE/δT)v [eq. 2]
where (δE/δT)v is the rate of change at which the total energy changes as the temperature varies at constant volume. Thus, for diatomic molecules, Čv is equal to 7R/2.
However, vibrational energy is highly quantized and greatly depends on temperature. For gaseous diatomic molecules, to Čv would be very small at ordinary temperatures. For example, the vibrational contribution for N2 would only be significant on temperatures above 4000 K. So the vibrational contribution can be neglected since the experiment was done at 300.15 K. Thus, for diatomic molecules, Čv is equal to 5R/2 when the vibrational contribution is neglected.
Based on the theorem of equipartition of energy, Čv is equal to 20.785 J mol-1 K-1 when the effect of vibrational motion is neglected and 29.099 J mol-1 K-1 when the effect of vibrational motion is considered. To compute for the molar heat capacity at constant pressure, this equation can be used:
Čp = Čv + R [eq.3]
So when the effects of vibrational motion is neglected, Čp is equal to 29.099 J mol-1 K-1 and when the effects of vibrational motion is considered, Čp is equal to 37.413 J mol-1 K-1. It is very evident that Čp and Čv values are greater when the effects of vibrational motion is considered because it contributes to the total amount of energy produced. The specific heat ratio can then be determined by getting the ratio of Čp and Čv values. When vibrational motion is neglected, the specific heat ratio of N2 is equal to 1.40 and when vibrational motion is considered, the specific heat ratio is equal to 1.29.
To verify these theoretical values, the experiment was done with the use of the adiabatic expansion method that relates the change in energy to the change in volume by:
dE = -pdV = (-nRT/V)dV = -nRTd lnV [eq. 4]
As shown in [eq. 2], energy is dependent on temperature at constant volume so ČvδT can substitute E. With further integration,
Čv ln(T2/T1) = -R ln(V2/V1) [eq.5]
where Čv and V are molar quantities. This suggests that the gas is subjected to reversible adiabatic expansion until the pressure drops from p1 to p2: A(p1, V1, T1) –> A(p2, V2, T2) [step 1] and the temperature of the gas is restored at constant volume: A(p2, V2, T2) –> A(p3, V2, T1) [step 2]. [Eq. 5] also implies that a decrease in temperature should be expected when the gas expands adiabatically because work is done but no heat enters the system. As the internal energy falls, the temperature of the gas also decreases. Using the ideal gas law for the equivalent expression of T, V, and P for the two steps and [eq.3], the heat capacity ratio (γ) equation can be derived as:
γ = [ln(p1/p2)] / [ln(p1/p3)] [eq. 6]
where p1 is the initial pressure of the gas in the carboy, p2 is the barometric pressure, and p3 is the pressure of the gas when the stopper is returned. The pressure readings from the manometer, mm methyl salicylate, should be converted to mmHg by multiplying it by the density ratio of methyl salicylate and mercury which is 0.086.
Tables 3 and 5 show the experimental specific heat ratio of nitrogen gas and its comparison with the theoretical values with and without the consideration of vibrational contribution to Čv. At 95% confidence interval, the experimental value of nitrogen gas’ specific heat ratio is 1.28 (±0.02) which is near the theoretical value of 1.4 without vibrational contribution since it only generates 8.88% relative error. The experimental value is also close to the theoretical value of 1.29 when vibrational contribution is considered with 0.78% relative error. Moreover, the computed standard deviation of 0.01 reveals that the experimental values at all trials are highly precise. With a relatively high accuracy and precision, it only suggests that the experiment was done properly and accurately.
However, the experimental specific heat ratio appears to be closer to the theoretical value when vibrational contribution to Čv is considered. At 300.15 K, the temperature when the experiment was done, the energy contribution of N2 molecules’ vibrational motion is negligible. It is only at temperatures greater than 4000 K that N2 molecules would have a significant vibrational contribution to Čv. The experimental value is smaller than the theoretical value of 1.4 when vibrational contribution is neglected because the carboy might have been left open for so long that the conditions are not sufficiently adiabatic. On the other hand, the value of the specific heat ratio would be higher if the stopper weren’t removed for a long enough time to permit the pressure to drop momentarily to atmospheric. It is difficult to estimate how much time it should take for the stopper to be returned after removing it from the carboy for optimal results. Moreover, since perfect evacuation can’t be attained and the stopper was not returned to the carboy immediately, other gases might have contaminated the system like CO2 which behaves in a different way as compared to N2 contributing to the slight deviation from the theoretical value. Furthermore, there might have been leaks in the carboy that thwarted the reversibility of the gas expansion. Instrumental errors and parallax errors in reading the pressure measurement in the manometer might have contributed to the overall deviation from the theoretical value although these may only have a very slight and infinitesimal effect.
Although the experimental results are highly accurate and precise as proved by the computed standard deviation and relative % error, the technique used, which is the adiabatic expansion method, would not give the most optimal results in the determination of specific heat ratio of gases. A better method measures the velocity of sound in gases while other methods determine heat capacity ratios directly although this may be quite difficult and challenging.
Conclusion:
The experimental heat capacity ratio of N2 gas is computed as 1.28 (±0.02) with the help of the adiabatic expansion method. This value is closer to the theoretical value of 1.29 when vibrational contribution to Čv is considered, generating only 0.78% deviation, than the theoretical value of 1.4 when the vibrational motion of the gas molecules is neglected, which generates 8.88% deviation. However, the experimental value should be nearer to the theoretical specific heat ratio when vibrational contribution to Čv is neglected since only at temperatures greater than 4000 K that the vibrational contribution of N2 becomes significant. The experimental value is smaller than the theoretical value when the vibrational motion of the gas molecules is neglected because of errors in the experiment. More specifically, the carboy might have been left open for so long that the conditions are not sufficiently adiabatic.
Sample Calculations:
For trial 1:
∆H cm MS = height 2 – height 1 = 49.90 - 31.90 = 18.00 cm
∆H mmHg = (∆H cm MS * ρMS) / ρHg = (18*10*1.174)/13.55 = 15.6 mmHg
P1 = barometric pressure (P2, mmHg) + ∆H (mmHg) [before the stopper was removed]
= 755.25 + 15.6 = 770.85 mmHg
P3 = barometric pressure (P2, mmHg) [after the stopper was returned] + ∆H (mmHg)
= 755.25 + 3.47 = 758.71 mmHg
Heat capacity ratio = [ln(p1/p2)] / [ln(p1/p3)] = [ln(770.85/755.25)]/[ln(770.85/758.71)] = 1.29
References:
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Atkins, Peter and Julio de Paula. Atkins’ Physical Chemistry: 9th ed. Oxford University Press: UK. 2010.
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Skoog, Douglas A., Donald M. West, F. James Holler, Stanley R. Crouch. Fundamentals of Analytical Chemistry: 8th ed. Thomson: Brooks/Cole. 2004
