Changes in Pressure at Constant Temperature

Heat must be transferred from the condenser at a rate of 2320 kW to achieve the required cooling and condensation. 

Before we leave this section, let us consider what we just did from a different perspective. The process for which we need to calculate SH — Q) may be depicted as shown below  

To calculate AH, in effect we constructed the following process path  

The total enthalpy change for the first step, AHa, is the negative of AH for the process in which acetone and nitrogen go from the reference conditions to the inlet conditions, or 

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The above equilibrium reaction is unaffected by volume change since the total number of coefficients of gases on each side of the reaction equation are equal. Pressure of a gas is inversely proportional to volume. When the volume of a gas increases, the pressure of the gas decreases. When the volume of a gas decreases, the pressure of the gas increases. Thus, change in pressure at constant temperature affects the equilibrium reaction conversely in respect to volume change. 

The first step is a change in pressure at constant temperature, the type of process described in Section 8.2. We saw in that section that... 

Changes in pressure at constant temperature. For a species undergoing an isothermal pressure change, AP,... 

For this change in pressure at constant temperature, the enthalpy change is very nearly zero (Section 7.15) and exactly zero if only ideal gases are involved. Thus for practical purposes the AH in Eq. (7.72) is equal to the AH for the constant-pressure process, while the AU refers to the constant-volume transformation. To a good approximation, Eq. (7.72) can be interpreted as... 

The change in volume with change in pressure at constant temperature is given by... 

Further, for changes of state at constant composition, the forms of the Gibbs-Duhem equation in Table 6.3 can be related to derivatives in Table 6.2. For example, for a change in pressure at constant temperature and constant composition, (4.3.13) combines with (4.3.15) to yield... 

Bulk modulus n. The modulus of volume elasticity, i.e., the resistance of a solid or liquid to change in volume with change in pressure, at constant temperature. The thermodynamic definition is ... 

The effect of large changes in pressure at constant temperature on the viscosity of various hydrocarbons is shown in Figure 3. There we see that the logarithm of the viscosity of liquid hydrocarbons and hydrocarbon mixtures increases almost linearly with increasing pressure. Alternatively, viscosity can be considered to be a function of density rather than pressure, and this is used in several of the models discussed later. The kinematic viscosity shows similar trends with respect to these variables mentioned above, however its variation with temperature is significantly more linear than dynamic viscosity so that the former is somewhat easier to correlate than the latter. Consequently, some correlations have been developed exclusively for the kinematic viscosity, as will be discussed later. 

We can represent a change in pressure with a vertical line on the phase diagram. For example, suppose we lower the pressure above a sample of water initially at 1.0 atm and 25 °C. We represent the change in pressure at constant temperature as movement along the line marked B in Figure 11.38. As the pressure drops, we move down the line and approach the vaporization curve. At the vaporization curve, the pressure stops dropping and vaporization occurs until the liquid is completely converted to vapor. Crossing the vaporization curve requires the complete transition from liquid to gas. Only after the liquid has all vaporized can the pressure continue to drop. 

Applied to a two-phase system, this says that the change in pressure with temperature is equal to the change in entropy at constant temperature as the total volume of the system (a + P) is increased, which can only take place if some a is converted to P ... 

Joule-Thompson Coefficient for Real Gases. This expresses the change in temperature with respect to change in pressure at constant enthalpy ... 

Remember that AG is a measure of the overall change in entropy at constant temperature and pressure, AS is the change in entropy of the system and —AH/T is the change in entropy of the surroundings (Section 7.9). 

This relationship identifies the surface energy as the increment of the Gibbs free energy per unit change in area at constant temperature, pressure, and number of moles. The path-dependent variable dWs in Eq. (2.60) has been replaced by a state variable, namely, the Gibbs free energy. The energy interpretation of y has been carried to the point where it has been identified with a specific thermodynamic function. As a result, many of the relationships that apply to G also apply to y ... 

Before we develop the mass action law, it is necessary to investigate how the Gibbs free energy changes with pressure at constant temperature, because in chemical equilibria, the partial pressures of gases can differ from 1 atm. 

Figure 12-10 A molecular interpretation of Boyle s Law— the change in pressure of a gas with changes in volume (at constant temperature). The entire apparatus is enclosed in a vacuum. 

From a thermodynamic point of view, phase diagrams may be constructed by changing the temperature (ii), pressure (12). or composition of a material. The present experiments are concerned with changes in composition at constant temperature and pressure, leading to a ternary phase diagram with polymer network I at one corner, monomer II at the second corner, and polymer network II at the third corner. According to classical concepts, at first there should be a mutual solution of monomer II in network I, followed by the binodal (nucleation and growth kinetics) and finally the spinodal (spinodal decomposition kinetics). 

Thus, for changes in composition at constant temperature and pressure, we have... 

In various applications, we will need expressions for the effect of changing the pressure at constant temperature on the internal energy, enthalpy, entropy, and Gibbs energy of a phase. We obtain the expressions by integrating expressions found in Table 7.1. For example. At/ is given by / (dU/dp)r P- The results are listed in the second column of Table 7.4. 

The basis for this procedure for evaluating the concentration of absorbed species at reaction conditions rests upon being able to measure adsorption while a much slower reaction step takes place. If the study is to go beyond the adsorption step, the reaction must be of the type that produces a change in pressure at constant volume and temperature. Figure 4 shows portions of a typical adsorption reaction history for the catalytic dehydration of t-butanol on Alumina lOOS which has been treated or "conditioned" with water (6). The reaction which is endothermic produces one mole of isobutylene and a mole of water for each mole of t-butanoL The steep decrease in pressxore during the first second (approximately) was caused by adsorption, then the slow rise resulted from the reaction. The ratio of adsorption rate to reaction rate for this case was about 1700. The temperature rose during the first three seconds as a result of the heat of adsorption then fell because of the endothermic reaction and heat loss to the reactor. The temperature lag may be due in part to the slower response of the thermocouple. The amount of t-butanol which was measured by the drop in pressure from the initial value to the minimum is considered to be the adsorption at reaction conditions. 

A sample of pure water is confined in a cylinder by a freely moving piston surmounted by weights to establish the confining pressure. Sketched here are conditions at the points labeled P, Q, and R in Figure 12-30. The transition from point Pto Q is accomplished by changing the pressure at constant temperature (isothermal). The transition from point Pto R is accomplished by changing the temperature at constant pressure (isobaric). 

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