Boyle curve

This curve describes any inverse relationship. The commonest value for the constant, k, in anaesthetics is 1, which gives rise to a curve known as a rectangular hyperbola. The line never crosses the x or the y axis and is described as asymptotic to them (see definition below). Boyle s law is a good example (volume = 1/pressure). This curve looks very similar to an exponential decline but they are entirely different in mathematical terms so be sure about which one you are describing. 

The Boyle temperature is that temperature, for a given gas. at which Boyle s law is most closely obeyed ill the lower pressure range. At tills temperature, the minimum point (of inflection) in the pV-T curve falls on the pV axis. See Compression (Gas) and Ideal Gas Law. 

The validity of Boyle s law can be demonstrated by making a simple series of pressure-volume measurements on a gas sample (Table 9.2) and plotting them as in Figure 9.6. When V is plotted versus P, as in Figure 9.6a, the result is a curve in the form of a hyperbola. When V is plotted versus 1/P, as in Figure 9.6b, the result is a straight line. Such graphical behavior is characteristic of mathematical equations of the form y = mx + b. In this case, y = V,m = the slope of the line (the constant k in the present instance), x = 1 /P, and b = the y-intercept (a constant 0 in the present instance). (See Appendix A.3 for a review of linear equations.)... 

No actual gas follows the ideal gas equation exactly. Only at low pressures are the differences between the properties of a real gas and those of an ideal gas sufficiently small that they can be neglected. For precision work the differences should never be neglected. Even at pressures near 1 bar these differences may amount to several percent. Probably the best way to illustrate the deviations of real gases from the ideal gas law is to consider how the quantity PV/RT, called the compressibility factor, Z, for 1 mole of gas depends upon the pressure at various temperatures. This is shown in Figure 7.1, where the abscissa is actually the reduced pressure and the curves are for various reduced temperatures [9]. The behavior of the ideal gas is represented by the line where PV/RT = 1. For real gases at sufficiently low temperatures, the PV product is less than ideal at low pressures and, as the pressure increases, passes through a minimum, and finally becomes greater than ideal. At one temperature, called the Boyle temperature, this minimum... 

Above the Boyle temperature the deviations from ideal behavior are always positive and increase with the pressure. The initial slope of the PV curves at which P = 0 is negative at low temperatures, passes through zero at the Boyle temperature, and then becomes positive. A maximum in the initial slope as a function of temperature has been observed for both hydrogen and helium, and it is presumed that all gases would exhibit such a maximum if heated to sufficiently high temperatures. This behavior of the initial slope with temperature is illustrated in Figure 7.2 [10]. 

Much of the data supporting Westheimer s hypothesis was critically examined by Bordwell and Boyle (1975) and in their estimation found wanting. They redefined the maximum of the log hAd versus ApK curve of Bell and Goodall by taking into account secondary KIE s and data from other systems. The consequence of this treatment was that the slopes on either side of the maximum were much shallower. Bordwell concluded that either the ratio Ah/ d is relatively insensitive to the symmetry of the transition state or that symmetry does not change over wide ranges of ApA. 

It is seen that as the temperature is raised, the dip in the curve becomes smaller and smaller. At 50 C, the curve seems to remain almost horizontal for an appreciable range of pressure varying between 0 and about 100 atmosphere showing thereby that the compressibility factor Z becomes almost equal to unity under these conditions. In other words, the product PVremains constant and hence Boyle s law is obeyed within this range ofpressure at 50 C. 

As the pressure increases on a gas, the volume of the gas will decrease. The graph that demonstrates Boyle s Law is curved as indicated by choice E. 

Boyle s law states that the volume of a given amount of gas held at a constant temperature varies inversely with the pressure. Look at the graph in Figure 14-2 in which pressure versus volume is plotted for a gas. The plot of an inversely proportional relationship results in a downward curve. If you choose any two points along the curve and multiply the pressure times the volume at each point, how do your two answers compare Note that the product of the pressure and the volume for each of points 1, 2, and 3 is 10 atm-L. From the graph, what would the volume be if the pressure is 2.5 atm What would the pressure be if the volume is 2 L ... 

As the vapour pressure curve shows, the boiling point of a compound can be lowered by reducing the total pressure this also has consequences on equipment design. According to the law of Boyle and Mariotte, p V = constant, the volume of gases increases under reduced pressure. These higher values have to be taken into account when calculating evaporation and condensation areas. 

When the volume of a gas is plotted against its pressure at constant temperature, the resulting curve is one branch of a hyperbola. Figure 12-4b is a graphic illustration of this inverse relationship. When volume is plotted versus the reciprocal of the pressure, /P, a straight line results (Figure 12-4c). In 1662, Boyle summarized the results of his experiments on various samples of gases in an alternative statement of Boyle s Law ... 

At the Boyle temperature the Z versus p curve is tangent to the curve for the ideal gas at p = 0 and rises above the ideal gas curve only very slowly. In Eq. (3.8) the second term drops out at 7, and the remaining terms are small until the pressure becomes very high. Thus at the Boyle temperature the real gas behaves ideally over a wide range of pressures, because the effects of size and of intermolecular forces roughly compensate. This is also shown in Fig. 3.4. The Boyle temperatures for several different gases are given in Table 3.2. 

The data in Table 3.2 make the curves in Fig. 3.2 comprehensible. All of them are drawn at 0 °C. Thus hydrogen is above its Boyle temperature and so always has Z-values greater than unity. The other gases are below their Boyle temperatures and so have Z-values less than unity in the low-pressure range. 

It is convenient to represent the data in Table 5.1 by using two different plots. The first type of plot, P versus V, forms a curve called a hyperbola shown in Fig. 5.5(a). Looking at this plot, note that as the volume drops by about half (from 58.8 to 29.1), the pressure doubles (from 24.0 to 48.0). In other words, there is an inverse relationship between pressure and volume. The second type of plot can be obtained by rearranging Boyle s law to give... 

Boyle s law states that the volume of a fixed amount of gas held at a constant temperature varies inversely with the pressure. Look at the graph in Figure 13.1, in which pressure versus volume is plotted for a gas. The plot of an inversely proportional relationship results in a downward curve. 

In Fig. 9 the pressure-volume relations of a gas-liquid system are represented. A corresponds to a dilute unsaturated vapour. On compression at constant temperature the pressure and volume change more or less in accordance with Boyle s law and the curve AB is followed. Imagine the vapour to be tested at various points by being placed in contact with a continuous surface of its liquid. Up to B, the saturation point, it would take up liquid which would evaporate into it. At B there would be equilibrium, and if in presence of the liquid the pressure were infinitesimally raised, complete condensation would occur at constant pressure the line BC would be followed to the point C. If pressure were raised further, the compression curve of the liquid, CD, would be traversed. The only variable... 

FIG. 4. Boyle s law, which established the inverse relationship of pressure and the volume of a gas at constant temperature, derived from the experiment illustrated below. Mercury dropped in long arm of tube drove the trapped air into short arm. Doubling the column of mercury halved the column of air. Relationship is plotted in the curve above, a section of one branch of the hyperbola. 

At low temperature the curve of pV vs. p for gases has a minimum. At higher temperature the Boyle temperature, the point of zero slope occurs at p = 0. At this point the effect of the excluded volume of the gas is compensated by the interaction. This is similar to the d-temper-ature of polymer solutions (interaction compensates excluded volume). 

The extrapolation behaviour of empirical multi-parameter equations of state has been summarized by Span and Wagner. " Aside from the representation of shock tube data for the Hugoniot curve at very high temperatures and pressures, an assessment of the extrapolation behaviour of an equation of state can also be based on the so called ideal curves that were first discussed by Brown. While reference equations of state generally result in reasonable estimates for the Boyle, ideal, and Joule-Thomson inversion curves, the prediction of reasonable Joule inversion curves is still a challenge. Equations may result in unreasonable estimates of Boyle, ideal and Joule-Thomson plots especially when the equations are based on limited experimental data. 

Plotting the values of volume versus pressure for a gas at constant temperature gives a curve like that in Figure 2.2. The general volume-pressure relationship that is illustrated is called Boyle s law. Boyle s law states that the volume of a fixed mass of gas varies inversely with the pressure at constant temperature. 

We find that for a perfect gas, the curve is horizontal (Boyle-Mariotte law) whereas the curve for a real gas differs, becoming more horizontal the lower the temperature and the higher the pressure. 

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