Generalization of Eight Methods for Determining R in the Ideal Gas Law Donald B. Macnaughton* The ideal gas law of physics and chemistry says that PV = nRT. This law is a statement of the relationship between four variables (P, V, n, and T) that reflect properties of a quantity of gas in a container. The law enables us to make accurate predictions of the value of any one of the four variables from the values of the other three. The symbol R (called the “molar gas constant”) is the sole parameter or constant of the law. R stands for a fixed number that has been shown through experiments to equal approximately 8.314 472. Eight methods are available to analyze the data from a relevant experiment to determine the value of R. These methods are specific instances of eight general methods that scientists use to determine the value(s) of the parameter(s) of a model equation of a relationship between variables. Parameter estimation is one step in the study of a relationship between variables. KEY WORDS: Relationship between variables; Fitting a model equation to data; Molar gas constant. Part A of this tutorial paper discusses eight methods for determining R (the molar gas constant) in the ideal gas law. Part B discusses how the ideas can be can be general- ized. The generalization permits us to view many empirical research projects in terms of the same simple concepts. Ten appendices discuss some details behind the ideas. PART A: DETERMINING THE MOLAR GAS CONSTANT 1. The Ideal Gas Law The ideal gas law is a relationship between four vari- ables that says that for a quantity of gas held in a container, PV = nRT. Here the variable P reflects the pressure, V re- flects the volume, n reflects the amount, and T reflects the temperature of the gas. Many experiments have shown that this law is “true”. That is, if we measure the numeric values of P, V, n, and T in a “standard” situation in which a gas is held in a container, we find that PV is almost exactly equal to nRT. The ideal gas law is based on the work of eminent sci- entists in the 17th through 19th centuries: Boyle, Charles, Gay-Lussac, and Avogadro. Following convention, this paper assumes that P, V, n, and T are measured in the units of the International System *Donald B. Macnaughton is president of MatStat Research Consulting Inc. Email: [email protected] of Units, known as the SI (from the French Système Inter- national d’Unités). The SI is universally accepted among scientists and engineers and is defined and discussed by the International Bureau of Weights and Measures (BIPM 2006). In SI units the pressure of a gas is measured in pascals, the volume is measured in cubic meters, the amount is measured in moles, and the temperature is meas- ured in kelvins. The equation PV = nRT is called the “model equation” or “model” of the relationship between the four variables. Model equations are used to model relationships between variables throughout all branches of science. The R in the model equation of the ideal gas law is called the “molar gas constant”. (It is also sometimes called the “universal gas constant,” or simply the “gas constant”.) R is the sole parameter of the equation and is a fixed num- ber that serves as a scaling factor for the relationship be- tween the variables. That is, R makes the equation consis- tent with the (SI) units of measurement of each of the vari- ables in the equation. As with almost all parameters in model equations, the value of R must be determined through performing an ap- propriate empirical research project. Mohr, Taylor, and Newell (2008, p. 684) discuss how two recent experiments imply that the current “official” value of R is 8.314 472. The ideal gas law is important to theorists because it reflects a key relationship between variables in the theory of gases. In addition, the law is important to anyone work- ing with gases because it gives us the valuable ability to accurately predict or control for a gas (the numeric value of) any of the four variables in the law by measuring or controlling (the values of) the other three. The ideal gas law is highly accurate at predicting or controlling in many situations. Thus the law is regularly used in scientific work and in practical applications that require knowledge of the relationship between P, V, n, and T. However, as with most laws of science, the law isn’t the final word on the relationship between the four variables. This is discussed further in Appendix A. 2. Eight Methods for Determining R If we wish to use the ideal gas law to predict or con- trol, we need to know the value of R. Suppose that we are physical scientists and suppose that we wish to perform a research project to determine the precise value of R. How should we proceed? A reasonable approach is to collect multiple sets of values of P, V, n, and T for a quantity of gas under different Table of contents at end. Best printed on a color printer.
Generalization of Eight Methods for Determining R in the Ideal Gas Law
Jul 05, 2023
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