Variation of pressure with volume

Summaries of the properties of gases, particularly the variation of pressure with volume and temperature, are known as the gas laws. The first reliable measurements of the properties of gases were made by the Anglo-Irish scientist Robert Boyle in 1662 when he examined the effect of pressure on volume. A century and a half later, a new pastime, hot-air ballooning, motivated two French scientists, Jacques Charles and Joseph-Louis Gay-Lussac, to formulate additional gas laws. Charles and... 

Figure 10.4 Variation of pressure with volume for the van der Waals equation of state for a T >Tc (dotted line) bT=Tc (solid line) and cT

The variation of entropy with volume and pressure under conditions of constant temperature is determined by using Equation 2-130 ... 

From equation 2.43 in Volume 1, the variation of pressure with height is given by ... 

Vc crystalline Va, amorphous). The densities of the pure crystalline (pc) and pure amorphous (pa) polymer must be known at the temperature and pressure used to measure p. The value of pc can be obtained from the unit cell dimensions when the crystal structure is known. The value of pa can be obtained directly for polymers that can be quenched without crystallization, polyfethylene terephtha-late) is one example. However, for most semi-crystalline polymers the value of pa is extrapolated from the variation of the specific volume of the melt with temperature [16,63]. 

The contraction of solids on heating seems anomalous because it offends the intuitive concept that atoms will need more room to move as the vibrational amplitudes of the atoms increase. However, this argument is incomplete. Figure 11.9 plots schematically the variation of A with V at two temperatures, for both positive and negative thermal expansion. The volumes marked explicitly on the E-axis give the minima of each A vs. V isotherm. These are the equilibrium volumes at temperatures T and T2 respectively (J2 > 7j) and zero pressure. 

In high-pressure biochemistry the buffer system should be chosen with care. As should be clear from Section IV,A, ionizations are followed by negative volume changes and therefore increase with pressure. The variation of pH with pressure depends on the volume of ionization AVZ, which can differ as much as —30 cm3 mol-1 from one buffer to another. 

Eontanella and co-workers studied the effect of high pressure variation on the conductivity as well as the H, H, and O NMR spectra of acid form Nafionl 17 membranes that were exposed to various humidities. Variation of pressure allows for a determination of activation volume, A V, presumably associated with ionic and molecular motions. Conductivities (a) were obtained from complex electrical impedance diagrams and sample geometry, and A V was determined from the slope of linear isothermal In a versus p graphs based on the equation A E = —kJ d In a/d/j] t, where p is the applied pressure. At room temperature, A Ewas found to be 2.9 cm mol for a sample conditioned in atmosphere and was 6.9 cm mol for a sample that was conditioned in 25% relative humidity, where the latter contained the lesser amount of water. 

Figure 18. Variation of the specific volume at the glass transition temperature Tg with the glass transition temperature Tg as calculated from the LCT for constant pressure P = 1 atm 0.101325 MPa) F-F and F-S polymer fluids. Both and Tg are normalized by the corresponding high molar mass limits (v or T ). (Used with permission from J. Dudowicz, K. F. Freed, and J. F. Douglas, Journal of Physical Chemistry B 109, 21285 (2005). Copyright 2005 American Chemical Society.)...

In Section 2.1.1 we saw that, for an ideal gas, the numerical values of the pressure, p, and volume, v, are related according to/>°c 1/v, or p — c/v, where c — nrt (a constant). We can now explore how well the ideal gas law works for a real gas by considering experimental data1 for 1 mol of C02 at T=313 K. The ideal gas law suggests that pressure is inversely proportional to the volume and so in the first two rows of Table 2.3 we present the variation of p with 1/v for the experimental data (note that the working units for the pressure and volume in this case are atm and dm3, respectively). In the third row, we show values for l/vB, obtained using the ideal gas equation, where, in this case, the constant of... 

FIGURE 1.1 Variation of pressure-volume product with pressure. 

In the derivation of Eq. (1.13), we have neglected the variation of aY with temperature. In case the thermal expansion is constant, so that a0 = aT7, and the specific heat is independent of volume or pressure, Eq. (1.13) takes the simple form... 

Volume variations with conversion are large for constant-pressure gas-phase reactions with change in mole number. Here, as a rule, operation at constant volume poses no difficulties. Liquid-phase reactions may also entail volume contraction or expansion. However, these are not related to changes in mole number and can be predicted only if information on partial molar volumes is at hand. Because liquids are essentially incompressible, even at elevated temperature, it is unsafe to conduct liquid-phase reactions without a gas cap in a closed reactor. Some variation of liquid-phase volume with conversion therefore is apt to occur. Fortunately, the variation at constant temperature is usually so small that it can be neglected in the evaluation or accounted for by a minor correction. 

Reconsider the balloon discussed in Prob. 14-51. Assuming the volume to remain constant and disregarding the diffusion of air into the balloon, obtain a relation for the variation of pressure in the balloon with time. Using the results obtained and the iiuinetical values given in the problem, determine how long it will take for the pressure inside the balloon to drop to lOOkPa. 

Heat capacities are also useful in determining the variation of temperature with pressure or volume in isentropic processes. For this purpose introduce Eq. (1.3.8) to write... 

The magnitude of the variation of with pressure is thus determined by the variation of a with temperature. Since docjdT is always positive, the heat capacity at constant pressure decreases with increased pressure, but the effect becomes small at high temperatures since docjdT falls off more rapidly than T increases (c/. fig. 12.1). A similar formula is readily obtained for the effect of volume on c. ... 

A schematic plot of the variation of the pressure with volume, as predicted by the van der Waals equation of state, at various temperatures is given in Fig. 10.2. At temperatures above the critical temperature, the pressure-volume variation is monotonic and qualitatively similar to that of an ideal gas (see dotted-line). At temperatures below tire critical temperature, the pressure-volume curve begins to oscillate, exhibiting a van der Waals" loop (see dashed-line). This behavior is unphysical, but represents tire vapor-liquid transition, and should be replaced by the solid line. The precise location of the solid line is given by the Maxwell construction. 

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