We thus find the pressure values, p1, p2, p3, p4, p5, etc., and the corresponding volumes, v1, v2, v3, v4, v5, etc., and we may then plot these values so as to make a graph.
In this figure the values represented along the horizontal axis are pressure-values, and those represented along the vertical axis are volume-values. We have so made the experiment that we can make the pressure-values whatever we choose—let us call them the values of the independent variable or argument. For each value of the pressure, or argument, there is a corresponding value of the volume, which depends on the pressure—let us call these values of the volume values of the dependent variable or function.
We can make arbitrary values of the pressure, but whenever we do this the corresponding values of the volume are fixed. We say, then, that the volume is a function of the pressure. In general, when we choose one value of an independent variable, or argument, there can be only one, or a small number, of values of the dependent variable, or function. If there are two or more values of the function for one value of the argument each of these is necessarily determined by the value which we choose to assign to the argument. There is a strict functionality between the two series of variables. In the experiment we have chosen this functionality is expressed by the equation pv = k(1 + at), where p is the pressure, v the volume, k and a constants, and t is the temperature at which the experiment is carried out. In a number of experiments like that which we have mentioned, k, a, and t are the same throughout, and this is why we call them constants. We give p any value we like, and then v can be calculated from the equation.
RATE OF VARIATION
If we know the equation pv = k(1 + at), we can find how much the volume changes when the pressure changes, that is, the rate of variation of v with respect to p. But even if we don’t know that this equation applies, we can still find the rate of variation from our experiments. We see from the graph that, when the pressure increases from p1 to p2, the volume decreases from v1 to v2 but that if the pressure is again increased to p3, that is, by a similar amount to the increase of pressure from p1 to p2, the volume decreases from v2 to v3. Now we find, by measurements made on the graph, that the decrease v1 to v2 is greater than the decrease v2 to v3, and the latter decrease is greater again than the decrease from v3 to v4.
Fig. 28. Evidently the rate of variation of volume is not like the rate of variation of pressure, that is, the same throughout, and when we look at the graph we see that the rate of variation is greatest where the slope of the curve is steepest. The latter is steepest near the point a, less steep near the point b, and still less steep near the point c. Now any small part of the curve is indistinguishable from a straight line. Let us draw a straight line ee1, which appears to coincide with a small part of the curve near a, and similar straight lines ff1, and gg1, which also appear to coincide with small parts of the curve near b and c. Then the steepness of the curve will be proportional to the angles which these straight lines make with the axis op, and these angles are measured by their tangents, that is, by the ratio oe1/oe, which is the tangent that e1e makes with op, the ratio of1/of, and the ratio og1/og.
The point a on the curve corresponds with a pressure a1 and a volume a11. The point b corresponds with a pressure b1 and a volume b11, and c with a pressure c1 and a volume c11. The average rate of variation of the volume of the gas, as the pressure changes from a to c, is therefore proportional to the sum of the tangents oe1/oe and og1/og, divided by 2.