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Revisiting ideal gases and proposal of a simple experiment for determining atmospheric pressure in the laboratory |

Leonardo M. Da Silva Departamento de Química, Universidade Federal dos Vales do Jequitinhonha e Mucuri, Rodovia MGT 367, km 583, 5000, Alto da Jacuba, 39.100-000 Diamantina – MG, Brasil Recebido em 15/12/2017 Endereço para correspondência
*e-mail: lsilvamorais@hotmail.com; leonardo.morais@ufvjm.edu.br RESUMO
Relevant historical aspects concerning ideal gases were briefly reviewed to provide a condensed resource for the undergraduate student of chemistry. The importance of the barometer and the concept of atmospheric pressure were reviewed and discussed. A combination of Boyle's law with the well-known equation for determining the pressure produced by a column of stationary fluid was used to obtain a graphical method for determining atmospheric pressure under isothermal conditions in the laboratory. Important aspects related to the study of ideal gases were reviewed in light of the pressure-volume data that can be obtained by physical chemistry students using a simple apparatus composed of a commercial hypodermic syringe connected to a mercury manometer. Relevant concepts associated with the measurement of a physical quantity, the importance of linear regression, as well as the use of a graphical method to test the validity of a theory were reviewed from a pedagogical perspective through the experimental study of Boyle's law during regular experimental physical chemistry classes. Palavras-chave: ideal gases; Boyle's law; atmospheric pressure; hypodermic syringe; experiments in physical chemistry.
Matter can be roughly divided into three main categories, namely gases, liquids, and solids. In this classification scheme, the 'rarefied gases' are a simplified gas state, commonly known as 'perfect' or 'ideal' gases, which obey some basic limiting laws. Significant advances in the study of the rarefied gases were first obtained in the 17th century from 1643–1662.
The invention of the 'air pump' by Otto von Guericke (1602–1686) in 1650 and his many experiments created an impetus for researchers to study the "vacuum" (actually, a gas confined at a pressure lower than atmospheric pressure). The first experiments leading to a 'gas law' were performed by Boyle during 1660–1662 using a large J-tube made of glass and partially filled with mercury. Boyle reported in 1662 that the volume ( where The validity of equation (1) is verified for several gases since the pressure-volume product remains nearly constant over a low/moderate range of pressure, as long as temperature ( Boyle's law represents one of the first natural laws recognized by scientists. The law discovered by Boyle, which is strictly valid for rarefied gases, describes a rectangular hyperbola (e.g., a reciprocal curve) in the
Important studies after Boyle's work considered the influence of temperature on gases kept at constant volume or pressure. where Therefore, if equation (3) is valid at all temperatures, the volume at a given temperature in Celsius From the above considerations, the Charles-Gay-Lussac's law represented by equation (3) can be rewritten using the 'absolute (thermodynamic) thermometric scale' proposed in 1848 by William Thomson (Lord Kelvin) (1824–1907). where According to Boyle's law and Charles-Gay-Lussac's law, three parameters can be changed for the rarefied (ideal) gases: volume, pressure, and temperature. The preceding laws deal with the relationship between two pairs of variables while the third is held constant. There remains to consider the relationship between the third pair of variables, pressure and temperature, with volume held constant. where Charles-Gay-Lussac's law given by equation (5) can alternatively be written as: where Equation (6) is sometimes known as Amontons' law.
From the basic laws discussed above a general equation for the rarefied gases that incorporates the three state variables ( where A rigorous (rational) derivation of equation (7) was presented in 1934 by Roseman and Katzoff using the concept of the total differential of the function
Boyle extensively examined effects on air volume at pressures greater than atmospheric (positive relative pressures) and less than atmospheric (e.g., vacuum – negative relative pressures). In his studies, pressures varied from 3 to 300 cm of Hg. For a long time, Boyle's law was applied only to atmospheric air. Therefore, it is evident that Boyle's law is, in fact, a 'limiting law', strictly valid only at low pressures. For instance, for a given mass of gas confined under isothermal conditions in a cylinder-piston system connected to a manometer, the degree of deviation from Boyle's law can be detected by comparing the product of pressure and volume ( Linearization of equation (2) can also provide a means of identifying deviations from Boyle's law: According to equation (8), a Another useful graphical method is based on the linear model given by equation (9): where In typical cases at low pressures (
Experiments with air or other gases are common in universities for the study of basic gas properties by undergraduate students of chemistry, physics, and engineering. This is evident given the various different experiments involving gases described in laboratory textbooks. The study of gases is always of interest to teachers who sometimes have ingenious ideas about a new (alternative) configuration for an experimental apparatus. Therefore, in addition to the well-established experiments with gases described in several laboratory textbooks, Blanco and Romero
In several cases, undergraduate physical chemistry students are not aware of the correct treatment of experimental findings for gaseous systems. For instance, analysis of Boyle's plot is sometimes not accompanied by graphical tests discussed previously (see eqs. (8) and (9) and the discussion therein). This is an important point, since students observe in the laboratory that 'apparent' and/or 'true' deviations from Boyle's law (see eqs. (1) and (2)) can occur for the following reasons: (i) experimental errors that occur during volume and pressure measurements; (ii) collection of data under non-isothermal conditions, and (iii) intrinsic limitations of the ideal gas model at moderate/high pressures. In this context, important questions and concepts related to the quality of experimental findings (accuracy and precision) can be discussed with the teacher. Also, during analysis of experimental findings, students must use mathematics and statistics to prove or disprove a theoretical model such as Boyle's law. For instance, when confirming the validity of Boyle's law for a given range of pressures, the number of moles ( Therefore, it is expected from students after carrying out the experiments proposed in the present article the acquaintance with some basic 'analogic measurements' involving temperature (e.g., conventional mercury thermometer), volume (e.g., commercial syringes), and pressure (e.g., mercury manometer). As a result, students can obtain important laboratory skills and they will have the opportunity to obtain a good comprehension of the experimental difficulty inherent to the precise measurement of physical parameters using classical methods without the use of digital instruments.
Air in the atmosphere is a complex system in constant random motion due to local differences in density and temperature. In fact, motion of the air in the atmosphere in relation to the earth's surface is a complex function of altitude and latitude. 'Atmospheric or barometric' pressure is thus defined as the pressure exerted by a column of air rising from a level of reference (e.g., the laboratory environment) to a very high altitude where air is extremely rarefied and the influence of gravity can be disregarded. Since many laboratory experiments involving gases depends on the 'true value' of the atmospheric pressure, it is important to measure Very precise determinations of the atmospheric pressure using a barometer requires a precise measure of the density of mercury (ρ While the density of mercury (e.g., 13.5364 g cm However, in ordinary experiments in teaching laboratories, correction for latitude and altitude is unnecessary and the 'standard acceleration of gravity' found in physical chemistry textbooks
An alternative experiment using a very simple apparatus is proposed in the present article for determining the local (laboratory) atmospheric pressure. In this sense, Boyle's law can be combined with the well-known equation Relative pressure ( Obviously, this experimental setup can be assembled using a commercial hypodermic plastic syringe connected to a manometer. The pressure exerted on the gas ( where Using equation (12) and the generalized ideal gas law denoted by equation (7), one obtains the following equation: Rearranging equation (13) yields the following linearized equation: Therefore, it is predicted that the To the best of our knowledge, this particular type of graphical method for determining local atmospheric pressure ( The pedagogical objectives of the present article include: First, important concepts relating to the properties of ideal gases, especially Boyle's law, are reviewed and discussed from an experimental viewpoint. Second, undergraduate students measure atmospheric pressure using a non-standard procedure. Third, students use linear regression analysis to analyze pressure-volume data obtained in the laboratory with different graphical methods to verify Boyle's law.
All experiments were carried out using dry air at 297 K. The experimental apparatus was composed of two commercial hypodermic plastic syringes (25-mL and 60-mL) used separately (see Figure 1). The syringe plunger was previously lubricated using a thin layer of silicone grease to avoid any leakage of the confined gas (air).
Figure 1. Apparatus for determining local atmospheric pressure and verification of Boyle's law
The experiments were conducted in two parts: (i) positive and (ii) negative relative pressures ( The 'true' volumes ( where All graphics and linear regression analysis were obtained using Origin
Figure 2 shows the
Figure 2. Dependence of relative pressure (p^{*}) of the confined gas on the inverse of volume. Conditions: V_{0} = 37.5 cm^{3} at p* = 0 and T = 297 K
The physical chemistry laboratory at the UFVJM, located in the city of Diamantina (State of Minas Gerais, Brazil), is at 1350 m above sea level. The laboratory atmospheric pressure was measured to be 0.773 atm on the same day using a Torricelli's barometer. Therefore, the relative error (R.E.) obtained with the proposed graphical method is R.E. (%) = [(0.773 – 0.749)/0.773] × 100 = 3.1 %. This small error demonstrates that the graphical method used in this work can be used to measure atmospheric pressure with a simple experimental apparatus. The number of moles of air present in the apparatus (see Figure 1) can be evaluated from the slope of the line in Figure 2 (see equation 14): Therefore, with
After determination of
Figure 3. Hyperbolic behavior of pressure-volume data (p = p_{atm} + p*). Conditions: V_{0} = 37.5 cm^{3} at p* = 0 and T = 297 K
It is worth mentioning that using water instead of mercury did not yield the characteristic hyperbolic behavior (data not shown). In fact, the small change in pressure caused by small increments in the fluid column (water) did not permit to distinguish a rectangular hyperbola from a straight line with a negative slope. Obviously, simple visual inspection of experimental findings shown in Figure 3 is not sufficient to detect deviations from Boyle's law. In addition, visual analysis of findings does not allow identification of systematic errors due to using an incorrect atmospheric pressure. Based on these considerations, other linearized plots must be used to assess the precision and accuracy of experimental findings. The first rigorous graphical test to verify Boyle's law is derived from analysis of the product of pressure and volume (
Figure 4. Dependence of the product pV on p. Conditions: V_{0} = 37.5 cm^{3} at p* = 0 and T = 297 K
Another graphical treatment is based on linearization of Boyle's law (see equation 8), where the
Figure 5. Dependence of total pressure of the confined gas on the inverse of volume. Conditions: V_{0} = 37.5 cm^{3} at p* = 0 and T = 297 K
Analysis of equation (18) shows that the atmospheric pressure obtained using the graphical method presented in Figure 2 was accurate, since the linear coefficient (5.21 × 10 The last graphical test used the Figure 6 shows the linear plot obtained using equation (9). As can be seen, a good linear behavior was verified. Linear regression analysis yielded the following equation:
Figure 6. Linear behavior of ln(p) vs. ln(V) plot. Conditions: V_{0} = 37.5 cm^{3} at p* = 0 and T = 297 K
The very good correlation coefficient (
Analysis of pressure-volume data obtained with simple analogic instruments using linear models (linearized equations) can clearly aid undergraduate physical chemistry students in several different ways. Important concepts such as the precision associated with measuring a physical quantity using classical (analogic) instruments, the importance of linear regression, as well as use of a graphical method to test a theory are presented from a pedagogical perspective during the study of Boyle's law using a simple and inexpensive experimental apparatus. In addition, students carry out an alternative experiment proposed for determining atmospheric pressure in the laboratory. It is worth mentioning that the study of the fundamental properties of ideal gases using a simple apparatus containing only analogic instruments that can be handled by students themselves is an effective method for teaching several important concepts related to physics, chemistry, and mathematics.
A brief review of relevant historical aspects of ideal gases was presented. After a critical discussion of Boyle's law using the pressure-volume data obtained in the physical chemistry laboratory, a graphical method was proposed for determining atmospheric pressure in the laboratory using a simple apparatus composed of a commercial hypodermic plastic syringe connected to a mercury manometer (U-tube). In addition, the analysis of pressure-volume data, obtained at constant temperature, was presented and discussed to emphasize the importance of mathematics and statistics for students in undergraduate physical chemistry courses. Graphical methods using commercial software (Origin
L.M. Da Silva wishes to thank the "Conselho Nacional de Desenvolvimento Científico e Tecnológico – CNPq" (PQ-2 grant). Fruitful discussions with my undergraduate students that have occurred over the last 9 years in the classroom and in the physical chemistry laboratory at the UFVJM motivated writing the present work. Therefore, I (L.M. Da Silva), express my sincere gratitude to all of them.
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