Temperature, Boyle

Figure A2.1.7 shows schematically the variation o B = B with temperature. It starts strongly negative (tiieoretically at minus infinity for zero temperature, but of course iimneasiirable) and decreases in magnitude until it changes sign at the Boyle temperature (B = 0, where the gas is more nearly ideal to higher pressures). The slope dB/dT remains...

It is widely believed that gases are virtually ideal at a pressure of one atmosphere. This is more nearly tnie at relatively high temperatures, but at the nonnal boiling point (roughly 20% of the Boyle temperature), typical gases have values of pV/nRT that are 5 to 15% lower than tlie ideal value of unity. 

The temperature at which 2(7) is zero is the Boyle temperature Jg. The excess Hehuholtz free energy follows from the tlrenuodynamic relation... 

The first seven virial coefficients of hard spheres are positive and no Boyle temperature exists for hard spheres. 

Theta conditions in dilute polymer solutions are similar to tire state of van der Waals gases near tire Boyle temperature. At this temperature, excluded-volume effects and van der Waals attraction compensate each other, so tliat tire second virial coefficient of tire expansion of tire pressure as a function of tire concentration vanishes. On dealing witli solutions, tire quantity of interest becomes tire osmotic pressure IT ratlier tlian tire pressure. Its virial expansion may be written as... 

In Chap. 1 we referred to these as 0 conditions, and we shall examine the significance of this term presently. Note that 0 conditions for a polymer solution are analogous to the Boyle temperature of a gas Each behaves ideally under its respective conditions. 

It is interesting to note that for a van der Waals gas, the second virial coefficient equals b - a/RT, and this equals zero at the Boyle temperature. This shows that the excluded volume (the van der Waals b term) and the intermolecular attractions (the a term) cancel out at the Boyle temperature. This kind of compensation is also typical of 0 conditions. 

Following Flory (1969), a 0 solvent is a thermodynamically poor solvent where the effect of the physically occupied volume of the chain is exactly compensated by mutual attractions of the chain segments. Consequently, the excluded volume effect becomes vanishingly small, and the chains should behave as predicted by mathematical models based on chains of zero volume. Chain dimensions under 0 conditions are referred to as unperturbed. The analogy between the temperature 0 and the Boyle temperature of a gas should be appreciated. 

When we make a plot of Z versus 1/V, we expect that the value of Z = 1 when 1/y = 0 or at infinite volume. The first virial coefficient B T) is the slope of the line at 1/y = 0. B T) is usually negative at very low temperatures due to the attractive forces. But when the temperature increases, B T) will turn positive, and the temperature at which B becomes zero is called the Boyle temperature, since that is the temperature where Boyle s law applies exactly (at infinite volume). 

The Boyle temperature T is defined as the temperature for which the second virial coefficient is zero, i.e. TR = 1. The ratio T /Tc = 27/8 is well known for a van der Waals fluid. In this transition, two opposing contributions to the free energy, i.e. translational entropy of the fluid molecules and the van der Waals attractive interaction, are balanced. 

The leading correction to ideality arises from the second virial coefficient B(T), whose qualitative T dependence is shown in Fig. 2.6 (approximating the experimental data for C02). As shown in the figure, B(T) rises from strongly negative values near the low-T condensation limit to weakly positive values at very high T. At an intermediate T known as the Boyle temperature rBoyle, the second virial coefficient vanishes ... 

Figure 2.6 Representative T dependence of the second virial coefficient B(T), showing the strong negative deviations from ideality at small T, the weak positive deviations at high Tand the Boyle temperature (TBoyie — 750K for C02) where B(TBoylc) vanishes.

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. 

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]. 

The temperature at which B( T) = 0 is the temperature at which a gas behaves most nearly like an ideal gas, and is called the Boyle temperature Tg. From Figure 7.9 we can see that... 

Example 4. Find an expression for the Boyle temperature of a van der Waals gas. 

Find the Boyle temperature for a gas obeying the Berthelot equation. 

If the temperature is less than Boyles temperature, the value of Z first decreases and then reaches a minimum value and finally as the pressure is gradually increased, the value of Z starts increasing. Different gases have different Boyle temperatures. For hydrogen and helium, Boyle temperatures are -80°C and -240°C, respectively. It means that at -80°C, hydrogen obeys Boyles law within a maximum range of pressure. 

In general, the more easily liquefiable gases have a.Boyle temperature. 

This temperature is called the Boyle point or Boyle temperature TBoyle. Below this temperature, the value of Z at first decreases, approaches a minimum and then increases as the pressure is increased continuously. Above 50 C, the value of Z shows a continuous rise with increase in pressure. 

The Boyle temperature is different for different gases. For example, the Boyle temperature for hydrogen is -165 C and for helium it is -240 C. Thus, at -165 C, hydrogen gas obeys Boyle s law for an appreciable range of pressure. However, at any temperature below -165 C, the plot of Z vs P first shows a fall and then a rise as pressure is increased continuously. At a temperature above -165 C, however, Z shows a continuous rise with increase in pressure. 

The Boyle Temperature. As already mentioned, the temperature at which a real gas obeys Boyle s law, is known as the Boyle temperature TB It is given by the expression... 

Since at the Boyle temperature, the second virial coefficient is zero, that is,... 

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