The Perfect-gas Equation

It is interesting that it was not until the early years of the seventeenth century that the word gas was used. This word was invented by a Belgian physician, . B. van Helmont (1577-1644), to fill the need caused 

Gases differ remarkably from liquids and solids in that the volume of a sample of gas depends in a striking way on the temperature of the gas and the applied pressure. The volume of a sample of liquid water, say 1 kg of water, remains essentially the same when the temperature and pressure are changed somewhat. Increasing the pressure from I atm to 2 atm causes the volume of a sample of liquid water to decrease by less than 0.01% and increasing the temperature from 0°C to 100°C causes the volume to increase by only 2%. On the other hand, the volume of a sample of air is cut in half when the pressure is increased from 1 atm to 2 atm, and it increases by 36.6% when the temperature is changed from 0°C to 100°C. 

We can understand why these interesting phenomena attracted the attention of scientists during the early years of development of modern chemistry through the application of quantitative experimental methods of investigation of nature, and why many physicists and chemists during the past century have devoted themselves to the problem of developing a sound theory to explain the behavior of gases. A part of this theory is presented in Appendix III. 

In addition to the desire to understand this part of the physical world, there is another reason, a practical one, for studying the gas laws. This reason is concerned with the measurement of gases. The most convenient way to determine the amount of material in a sample of a solid is to weigh it on a balance. This can also be done conveniently for liquids or we may measure the volume of a sample of a liquid, and, if we want to know its weight, multiply the volume by its density, as found by a previous experiment. The method of weighing is, however, not conveniently used for gases, because their densities are very small volume measurements can be made much more accurately and easily by the use of containers of 

It has been found by experiment that at low density all ordinary gases behave in nearly the same way. The nature of this behavior is described by the perfect-gas laws (often referred to briefly as the gas laws). 

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Where W is weight and Rj is a specific constant for the gas involved. This is the perfect gas equation. Going one step further, by making W, in pounds, equal to the molecular weight of the gas (one mole), the formula becomes ... 

The perfect-gas equation of state for multicomponent mixtures depends on the species composition. Representing the composition as either mass fraction Yk or mole fraction Xk leads to... 

Finally, the perfect-gas equation of state in the nondimensional variables is... 

The columns of cells below row 16 contain the values of the dependent variables at the node points. They will all be iterated until a final solution is achieved. The formula in each cell represents an appropriate form of the difference equations. Each column represents an equation. Column B represents the continuity equation, column C represents the radial momentum equation, column D represents the circumferential momentum equation, and column E represents the thermal energy equation. Column F represents the perfect-gas equation of state, from which the nondimensional density is evaluated. The difference equations involve interactions within a column and between columns. Within a column the finite-difference formulas involve the relationships with nearest-neighbor cells. For example, the temperature in some cell j depends on the temperatures in cells j — 1 and j + 1, that is, the cells one row above and one row below the target cell. Also, because the system is coupled, there is interaction with other columns. For example, the density, column F, appears in all other equations. The axial velocity, column B, also appears in all other equations. 

A similar example is the introduction of the universal gas constant, R, which ensures that in the perfect gas equation of state p V = n R T the already fixed secondary... 

In every case yet worked out, the adsorbed films of soluble substances are of the gaseous type there are always corrections, except in very dilute solutions, to the perfect gas equation due to the area actually occupied by the adsorbed molecules and in many cases also there are corrections due to the lateral adhesion between them. 

The proportionality of surface tension lowering to the concentration thus indicates that the adsorbed film is gaseous, with negligible corrections to the perfect gas equation. 

In the following sections of this chapter these three laws are formulated and applied in the solution of some problems. It is also shown that they can be combined into a single equation, which is called the perfect-gas equation. 

If the number of moles in a sample of gas, n, remains constant and the temperature T remains constant, the perfect-gas equation simplifies to... 

The volume occupied by one mole of gas at standard conditions is seen from the perfect-gas equation to be just the product of R and the temperature 0° C on the absolute scale. The value of R can hence be found by dividing 22.4 by 273 ... 

Some of the ways in which the perfect-gas equation can be used in the solution of chemical problems are discussed in the following paragraphs. 

Real gases differ in their behavior from that represented by the perfect gas equation for two reasons. First, the molecules have finite size, so that each molecule prevents others from making use of a part of the volume ot the gas container. This causes the volume of a gas to be larger than that calculated for ideal behavior. Second, the molecules even when some distance apart do not move independently of one another, but attract one another slightly. This tends to cause the volume of a gas to be smaller than the calculated volume. 

It is striking that the equation is identical in form with the perfect-gas equation van t Hoff emphasized the similarity of a dissolved substance and a gas. 

It is important to stress that there is a fundamental difference between the van t Hoff formula and the perfect gas equation... 

The mostly measurable input data on the density po(T) at ambient pressure have been used below to develop the predictable model. For the vapor pressure at low temperatures it is proposed to use the perfect-gas equation ... 

That is, the osmotic pressure obeys the generalised gas law, which includes Boyle s Law, Gay-Lussac s, and the Avogadro Hypothesis It will be observed that the R is numerically identical with the R of the perfect gas equation That is, the osmotic pressure of a dilute solution is related to the molecular volume, or the inverse of this the molecular concentration quantitatively as the pressure of a perfect gas is related to the volume For substances in dilute solution R = i 985 calones per gram-mole It may be noted that in the particular case in which V - v the osmotic pressure is identical with the gas pressure... 

The state of a perfect gas can be defined by specifying P, V, and T. As PV = nRT for a fixed mass of gas we need specify only two of P, V, and T since this will be sufficient to fix the remaining variable. Indeed such is the case for any pure substance (or mixture of fixed composition) even though it may not follow the perfect gas equation. We may write T f(P, V). Such an equation which links P, Ff and T is called an equation of state. 

We have assumed earlier that this is so, in fact our definition of a perfect gas included this condition. Now we have shown it to be a consequence of the perfect-gas equation of state. Closely similar arguments show that for a... 

El. 9(b) Use the perfect gas equation to compute the amount then convert to mass. 

Inserting the constant of proportionality, R, yields the perfect gas equation RnT... 

D1.5 The van der Waals equation corrects the perfect gas equation for both attractive and repulsive interactions between the molecules in a real gas. See Justification 1.1 for a fiiller explanation. 

P2.1 The temperatures are readily obtained from the perfect gas equation, T =... 

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