The Plane of Existence

Equations are mathematical expressions in which we substitute letters for numbers. In scientific equations these letters are instructions for measurements, and the equations are instructions for experiments. These measurements must be taken as part of the experiments and are not simply done as we might if we took the measurements independently. Experiments whose instructions can be given in equations are experiments whose predictions are proportions.

 

Here is an example:

 

PV = nRT

 

Called the ideal gas law, this equation, of course, can be rewritten as a proportion. PV/nRT = 1. Like all equations that are instructions for experiments, this fits the form A/B = 1. As with all proportions the numbers produced by the measurements can vary without the proportions doing so. Thus the letters are called variables. Both the numerator and denominator are mathematical manipulations of measurements, measurements taken as part of the experiment. For example in this case the pressure and volume, P and V, are measured within a closed space. And n is the “molar mass” of the gas, R a constant, and T the temperature. R is chosen to make the proportion equal 1 rather than 1/R. Such constants are for convenience. Otherwise, if we wanted the proportion to equal 1, we would have to adjust the units of the measured quantities for each new equation.

 

The letter “n” in the equation is called the molar mass. It is a variable that changes from type of gas to type of gas and it too is calculated from the ideal gas law to make the equation equal one rather than another number.

 

          "The molar mass of the unknown gas can be found using the ideal gas law,              provided the temperature and pressure of the gas are also known."

 

Since n varies from experiment to experiment, R is superfluous and could be included in the calculation of n. If we introduced a new variable, say m = nR then the formula would be simplified into

 

PV = mT or PV/mT = 1

 

The experiment would be performed like this. Fill a hollow cube whose volume, V, can be easily calculated, with gas, measure the pressure, P. inside with a manometer and the temperature, T, with a thermometer. Set m = PV/T. Then if we change P, V, or T without adding any gas, the equation will still hold. Proportions still hold as long as the numbers in the numerator vary when those in the denominator do to make them so.

 

Since m is a constant for a type of gas—hydrogen for example, once m is calculated for hydrogen it can be used in other experiments upon volumes of hydrogen elsewhere. Thus the experiment can be repeated. That m is constant for a type of gas must be experimentally determined.

 

Since V is a measurement of volume, any other enclosed space with the same volume as the cube can be used. The equation predicts that if you heat a gas in a closed space the pressure will increase with the temperature. Or if you compress a gas, making the volume smaller, the temperature and pressure will increase. This result is useful in, for example, manufacturing the compressor for a refrigerator.