Equations, mathematical Clapeyron

The Clausius-Clapeyron equation, one of the most famous in physical chemistry, is most applicable for this discussion. The equation states that the partial differential with respect to absolute temperature of the logarithm of a pure liquid s vapor pressure is inversely related to the liquid s absolute temperature. We again consider the liquid TCE and state the Clausius-Clapeyron equation mathematically (29) ... 

Numerous mathematical formulas relating the temperature and pressure of the gas phase in equilibrium with the condensed phase have been proposed. The Antoine equation (Eq. 1) gives good correlation with experimental values. Equation 2 is simpler and is often suitable over restricted temperature ranges. In these equations, and the derived differential coefficients for use in the Hag-genmacher and Clausius-Clapeyron equations, the p term is the vapor pressure of the compound in pounds per square inch (psi), the t term is the temperature in degrees Celsius, and the T term is the absolute temperature in kelvins (r°C -I- 273.15). 

It is of interest to consider the variation of vapor pressure with temperature. The vapor pressure of a liquid is constant at a given temperature. It increases with increasing temperature upto the critical temperature of the liquid. The liquid is completely in the vapor state above the critical temperature. The variation of the vapor pressure with temperature can be expressed mathematically by the Clapeyron-Clausius equation. Clausius modified the Clapeyron equation in the following manner by assuming that the vapor behaves like an ideal gas. 

As indicated by the plots in Figure 10.13a, the vapor pressure of a liquid rises with temperature in a nonlinear way. A linear relationship is found, however, when the logarithm of the vapor pressure, In Pvap, is plotted against the inverse of the Kelvin temperature, 1 /T. Table 10.8 gives the appropriate data for water, and Figure 10.13b shows the plot. As noted in Section 9.2, a linear graph is characteristic of mathematical equations of the form y = mx + b. In the present instance, y = lnPvap, x = 1/T, m is the slope of the line (- AHvap/R), and b is the y-intercept (a constant, C). Thus, the data fit an expression known as the Clausius-Clapeyron equation. ... 

Thermodynamics and kinetics can surely be counted—along with transport phenomena, chemistry, unit operations, and advanced mathematics—as subjects that form the foundation of Chemical Engineering education and practice. Thermodynamics is of course a very old subject. For example, it was the same Rudolf Clausius, who in 1865 coined two immortal sentences (1) "The energy of the universe is constant" and (2) "The entropy of the universe tends to a maximum," that developed the famous Clausius-Clapeyron equation, one of the most basic physico-chemical relationships. Classical thermodynamics was largely complete in the 19th century, before even the basic structure of the atom was understood. 

In some cases, a variable obeys a mathematical relation that can be linearized. This means finding new variables such that the curve in a graph of our data is expected to be a line instead of some other curve. In our vapor pressure example, these variables are found by manipulation of the Clapeyron equation. We assume that the volume of the liquid is negligible compared to that of the gas, and that the gas is ideal ... 

This is the form of the Antoine equation. It is an empirical equation used to fit experimental data. Its mathematical form, however, is not arbitrary but represents an empirical modification of the Clausius-Clapeyron equation, which turns out to be fairly accurate over an extended range, compared to the equation it is based on. Even so, the Antoine equation cannot accurately fit the entire subcritical region of a fluid. 

The general relationship between the amount of gas (volume, V) adsorbed by a solid at a constant temperature (T) and as a function of the gas pressure (P) is defined as its adsorption isotherm. It is also possible to study adsorption in terms of V and T at constant pressure, termed isobars, and in terms of T and F at constant volume, termed isosteres. The experimentally most accessible quantity is the isotherm, although the isosteres are sometimes used to determine heats of adsorption using the Clausius-Clapeyron equation. In addition to the observations on adsorption phenomena noted above, it was also noted that the shape of the adsorption isotherm changed with temperature. The problem for the physical chemist early in the twentieth century was to correlate experimental facts with molecular models for the processes involved and relate them aU mathematically. 

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