Determination of Boltzmann constant by gas expansion method
Musaj Pacarizi, Arber Zeqiraj, Ibrahim Hameli, Isuf Tredhaku, Sefer Avdiaj
11 Determination of Boltzmann constant by gas expansion method
MUSAJ A. PAÇARIZI , ARBËR Z. ZEQIRAJ , IBRAHIM M. HAMELI , ISUF A.TREDHAKU , SEFER R. AVDIAJ
3* 1
University of Prishtina, Faculty of Natural Sciences and Mathematics, Department of Chemistry, Prishtina 10000, Kosovo University of Mitrovica, Faculty of Geosciences, Department of Materials and Metallurgy, Mitrovica 40000, Kosovo University of Prishtina, Faculty of Natural Sciences and Mathematics, Department of Physics, Prishtina 10000, Kosovo *Corresponding author: [email protected]
ABSTRACT
In this paper we present a new method to determine the numerical value of the Boltzmann constant k and its uncertainty. We have used Nitrogen gas in different pressure values in the range 6 kPa – 100 kPa, for three different volumes. In this experiment we have used a simple idea called static expansion method, simple equipment and simple equations for calculations in order to determine the Boltzmann constant. This method is suitable for students of high-school level as well as introductory higher education. Keywords : gas expansion, Boltzmann constant, universal constants, Avogadro number, ideal gas, molar volume.
INTRODUCTION
All basics equations of physics and chemistry have some fundamental invariant quantities called fundamental constants, like h -Planck’s constant, c -universal light speed, G-universal gravity constant, etc. These constants have definite symbols and their numerical values should be measured as accurately as possible. The role of the fundamental constants in science is to describe the relationship between parameters with different dimensions in equations. The numerical value of universal constants tells us the relationship between physical quantities. For example, the value of the universal gravitational constant gives us the information for the intensity of the interaction force between two bodies with certain masses. The values of fundamental constants are supposed to be accepted as a known value in countless experiments during working with students. This is usually the case when they are asked to perform experiments like, the gas-law experiment, or experiments related to semiconductor conductivity. So, it is very interesting for students to measure those fundamental constants in order to understand physical laws much better. In this paper we describe an experiment to measure Boltzmann’s constant. The experiment is suitable for the introductory laboratory of chemistry and physics, as the theory of this measurement method is quite simple and is given in many elementary textbooks of chemistry and physics . Teaching experimentally the gas laws allows students to discover the relationships between the pressure, volume, temperature, and number of moles (or number of molecules) of a given gas and serves as a foundation to understand kinetic molecular theory. Theoretical background of the Boltzmann constant: what is the Boltzmann constant?
We can study the Boltzmann constant in different phenomena in nature. The Boltzmann constant appears for the first time where it is defined as proportionality constant between the macroscopic entropy of a system and the multiplicity of states for that system:
𝑆 = 𝑘 ln 𝑊 where S is the system entropy, W is the number of possible microscopic particle configurations for a certain macroscopic state (multiplicity) and k is the Boltzmann constant. Thus, the Boltzmann constant is a proportionality constant which relates microscopic phenomena with macroscopic ones. Hence, each time that a macroscopic effect results from the states of a huge number (of the order of Avogadro's number) of microscopic particles, the mathematical model of that effect, will contain the Boltzmann constant in it.
MATERIALS AND METHODS
The Boltzmann constant in our case was determined by using ideal gas law : 𝑃𝑉 = 𝑁𝑘𝑇 (1) where P – is pressure of the gas, k - is Boltzmann’s constant, T –is the temperature of the gas in kelvins. From Eq. (1) the V becomes: 𝑉 = 𝑁𝑇𝑘 ∙ (2) By collecting the data for P , V and T the Boltzmann constant k, can be calculated if the amount of substance n or number of particles N is known. The problem in this case using this simple equation is how to determine ( N ), the number of molecules (particles) in the closed volume . This is equal to the quotient (3) of the volume V and the standard molar volume V m . 𝑁 = 𝑁
𝐴 𝑉𝑉 𝑚 (3) where N A is the Avogadro number. At temperature, T =273.15 K and pressure, P =101325 Pa (standard conditions) the molar volume is V m = 0.022414 m . The volume V, measured at different pressure P and temperature T must first be reduced to these conditions by using the equation: 𝑃 𝑉 𝑇 = 𝑃𝑉𝑇 (4)
The experimental scheme of our measurement method is shown in Figure 1. The idea of measurements is very simple: the gas was expanded in series from a small chamber CH1 to the bigger chamber CH2, and after that to the chamber CH3. Since amount of substance (number of particles) before and after gas expansion should remain the same because the same number of particles (atoms or molecules) was expanded from chamber CH1 to the chamber CH2 after that to the camber CH3. The volume of the gas before expansion is V1, after the first expansion is V1+V2 and after last expansion V1+V2+V3. In the Table 1 are shown the values of the chamber volumes used for this experiment. The chamber volumes are measured by gas expansion method described in reference . The pressure was measured by calibrated vacuum gauge CDG 1000 and the temperature was measured by calibrated thermometer Pt100. Figure 1. Scheme of the experimental setup.
Table 1. Values of the volumes of three chambers and three different expanded volumes
CH1 CH2 CH3 CH1+CH2 CH1+CH2+CH3 Volume (m ) 6.143x10 -5 -4 -4 -4 -4 RESULTS AND DISCUSSION
By using Eqs (3) and (4), is possible to determine the number of particles in the chamber for different values of pressure. For example, when the pressure is P = 400.8 Torr ( P =53435.458 Pa), volume V1 = 6.143x10 -5 m and temperature 297.35 K the calculated number of particles is 7.99612x10 . All measurements were done in the same temperature 297.35 K. As the gas expands it will cool, for this reason, the apparatus was left to return to ambient temperature. The same way was used to calculate the number of particles for different values of initial pressures enclosed in chamber CH1. The measurements were performed by filling CH1 with Nitrogen gas at pressure range 50 Torr – 800 Torr (6666.12 Pa – 106658 Pa). As we mentioned above this number of gas particles were expanded to the chambers CH2 and CH3. So we had two series of expansions. After expansion the pressure is decreased and the volume is increased but the number of particles should remain constant. So each initial pressure was expanded to two following pressures. The measurements were repeated for nine series, each of them with three different pressures and volumes. All the data are included in the appendix. Since the gas is expanded from one volume to another, it means that the volume was increased while the pressure decreased. The detailed procedure is given in Appendix. From Fig. 2 and Eq.(2) we can conclude that a graph of V against 𝑃 −1 is a straight line with a gradient NTk , so that k can be determined using: 𝑘 = 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡𝑁𝑇 (5) The equation of the graph given in Fig. 2 tells us there is a small positive intercept on V axis, −6 𝑚 . This intercept is very small compared with smallest experimental value of V . There are two possible reasons for this small value of intercept, (a). the gas is considered ideal gas, (b). the possibility of systematic error in V . From the gradient in Fig. 2, 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 = 3.2819 𝑚 𝑃𝑎 and number of molecules 7.99612x10 the following value have been calculated for the Boltzmann constant k, according to the equation (5): 𝑘 = 1.380 × 10 −23 JK −1 and the relative standard uncertainty for this measurements is: 𝑢(𝑘) = 0.78% The detailed procedure how those values were calculated is given in Appendix. The value obtained from the CODATA (https://physics.nist.gov/cgi-bin/cuu/Value?k) is: 𝑘 = 1.380649 × 10 −23 JK −1 Figure 2. Volume V as function of the reciprocal pressure 1/ P for constant number of particles. y = 3.2819x0.01.02.03.04.05.06.07.08.0 0.0 0.5 1.0 1.5 2.0 2.5 V / ( - m ) (P/10 Pa) -1 Serie 5
CONCLUSIONS
When students determine a fundamental constant they will be removed from the “secret” things to be believed without knowing them. The experiment is using simple equipment and simple equations. These equations are included in every basic book of chemistry and physics. For sure there are much more accurate and precise methods to measure the Boltzmann constant , but they are not appropriate for educational purposes. Also the handling with data in this experiment is very simple.
ACKNOWNLEDGEMENT
This work was supported mainly by Ministry of Education, Science and Technology of Kosovo. The authors thanks Professor Lars Westerberg from Uppsala University – Sweden for providing some vacuum chambers and vacuum valves.
REFERENCES Halliday, D.; Resnick, R.; Walker, J.
Fundamentals of Physics , 10 th ed.; John Willey & Sons: ; p. 551 2. Chang, R.
General Chemistry: the essential concepts , 5 th ed.; McGraw-Hill, New York, ; p. 179 3. Cornely, K., Moss, D. B. Determination of the Universal Gas Constant, R. A Discovery Laboratory.
Journal of Chemical Education, , Ivanov, D., Nikolov, S. Measuring Boltzmann’s constant with carbon dioxide,
Physics Education, , Avdiaj, S.; Setina, J.; Erjavec, B. Volume determination of vacuum vessel by gas expansion method,
MAPAN , , Fellmuth, B.; Fischer, J.; Gaiser, Ch.; Jusko, O.; Priruenrom, T.; Sabuga, W.; Zandt, Th. Determination of the Boltzmann constant by dielectric-constant gas thermometry.
Metrologia . , , 382 7. Gaiser, C.; Fellmuth, B. Low temperature determination of the Boltzmann constant by dielectric-constant gas thermometry.
Metrologia , , , 1 8. Pitre, L.; Plimmer, D.M.; Sparasci, F.; Himbert, E.M. Determinations of Boltzmann constant.
Comptes Rendus Physique , , , 129-139 Appendix:
Table 1. Rough data used for this experiment
Measured pressure (Torr) Volume (m ) Temp.(K) P1 P2 P3 50.56 7.92 4.094 V1 6.143•10 -5 -4 -4 -4 -4 Explanation:
How the number of molecules in a certain volume was calculated? Example: Suppose the gas in enclosed in a chamber being held in temperature T =297.35 K and gas pressure is P = 53435.46
Pa. The volume of the chamber where the gas is enclosed is
𝑉 =6.143𝑥10 −5 m . How many gas molecules are in this chamber? Answer: The volume of one mole gas in standard conditions (temperature T θ = 273,15 K and pressure P θ = 101325 Pa) is V m = 22.414 liter ( V m =0.022414 m ). One mole of gas in standard conditions has N A = 6.022x10 molecules. If we want to find the number of molecules N in another volume and for different conditions, we can compare 𝑁 𝐴 𝑉 𝑚 = 𝑁𝑉 (1) From Eq. (1) we find, 𝑁 = 𝑁
𝐴 𝑉 𝑉 𝑚 (2) V – is the volume V ( 𝑉 = 6.143𝑥10 −5 m ) reduced to standard conditions. How to calculate the volume V (How to reduce V to standard conditions)? Simply by comparing standard conditions with new conditions: 𝑃 𝜃 𝑉 𝑇 𝜃 = 𝑃𝑉𝑇 (3)
So the volume V from Eq. (3) should be, 𝑉 = 𝑃𝑉𝑇 𝑇 𝜃 𝑃 𝜃 (4) Inserting the given data yields, 𝑉 = −5 m = 2.976 × 10 −5 m (5) Substituting for V from Eq. (5) into Eq. (2) gives, 𝑁 = 6.022 × 10
23 2.976×10 −5 m = 7.996 × 10 (6) This is the calculation for the fifth series of measurements. The eight other measurements for number of molecules are made in the same way. How did the Boltzmann constant was calculated? By using ideal gas law: 𝑃𝑉 = 𝑁𝑘𝑇 (7) From Eq. (7) the V becomes,
𝑉 = 𝑁𝑇𝑘 ∙ (8) From Eq. (7), 𝑘 = 𝑔𝑟𝑎𝑑𝑁𝑇 (9) where gradient is: ∆𝑉/∆(𝑃 −1 ) = 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 . The value of gradient is taken from Figure 2. From the Figure 2 the gradient is: 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 = 3.2819 m Pa By using data from Eq. (6), temperature T = 297.35 K and the value of gradient we obtain, 𝑘 = 3.2819 𝑚 Pa7.996 × 10 × 297.35 K = 1.380 × 10 −23 JK −1 How was calculated the uncertainty in k, u ( k )? We will write gradient with letter α. From Eq.(9) the k , depends on, 𝑘 = 𝑘(𝛼, 𝑁, 𝑇) k should be, 𝑢(𝑘) = √( 𝑢(𝛼)𝑁𝑇 ) + ( 𝛼𝑁 𝑇 𝑢(𝑁)) + ( 𝛼𝑁𝑇 𝑢(𝑇)) (10) From Eq.(2), 𝑁 = 𝑁(𝑁 𝐴 , 𝑉 , 𝑉 𝑚 ) , since N A and V m are constants, the uncertainties in N A and V m are neglected 𝑢(𝑁 𝐴 ) = 0 𝑑ℎ𝑒 𝑢(𝑉 𝑚 ) = 0 , then the uncertainty in N should be, 𝑢(𝑁) = 𝑁 𝐴 𝑉 𝑚 𝑢(𝑉 ) (11) By putting the Eq. (11) to Eq. (10) the uncertainty in k will be , 𝑢(𝑘) = √( 𝑢(𝛼)𝑁𝑇 ) + ( 𝑁 𝐴 𝑉 𝑚 𝛼𝑁 𝑇 𝑢(𝑉 )) + ( 𝛼𝑁𝑇 𝑢(𝑇)) (11) The uncertainty in V is calculated from Eq. (4), 𝑢(𝑉 ) = 𝑇 𝜃 𝑃 𝜃 √( 𝑉𝑇 𝑢(𝑃)) + ( 𝑃𝑇 𝑢(𝑉)) + ( 𝑃𝑉𝑇 𝑢(𝑇)) (12) The uncertainty of measured pressure, volume and temperature were: u(p) = u(V) = 0.75% and u(T) = 0.05 K. The uncertainty of pressure and temperature were taken from calibration certificates of the instruments whereas the uncertainty of the volume was taken from reference 5. The uncertainty in V should be, 𝑢(𝑉 ) = 273.15 K101325 Pa × √(6.143 × 10 −5 m + (53435.46 Pa297.35 K × 4.61 × 10 −7 m ) + (53435.46 Pa × 6.143 × 10 −5 m (297.35 K) × 0.05 K) 𝑢(𝑉 ) = 2.31 × 10 −7 m From Figure 2, the uncertainty in the gradient was calculated to be, 𝑢(𝛼) = 0.0012 m Pa . Using the Eq. (11) and the data for α, u (α), N , T , N A , V m , u ( V ), u ( T ), the uncertainty in k should be, 𝑢(𝑘) = 1.06 × 10 −25 JK −1 The relative uncertainty of k , 𝑢(𝑘)𝑘 = 0.78 % The other values of k and u(k) can be evaluated through similar calculations. 0 Figure 3. Volume vs reciprocal pressure for all series of measurements V / m (P / Pa) -1-1