6 Vapour Pressure
6.4 VAPOUR PRESSURE PLOTS
6.4.1 Equal-Temperature Reference-Substance Plots
For n-heptane at 325 K, the vapour pressure is
S
2911.32
ln P =− =3.0134 325−56.56
Therefore, PS = 20.36 kPa.
(b) The normal boiling point is the temperature at which the vapour pressure is 101.3 kPa. Substitute PS = 101.3 kPa in the Antoine equation.
2911.32ln 101.3=− −56.56
Solving this equation, T = 371.62 K.
6.4 VAPOUR PRESSURE PLOTS
Vapour pressure, as we have seen, is a strong function of temperature. Generally, a large number of experimental data would be necessary to establish a functional relationship between vapour pressure and temperature graphically. Frequently in chemical engineering calculations, it becomes necessary to interpolate or extrapolate vapour pressure data from a very few experimental observations, these being easily carried out if a straight line results when the vapour pressure is plotted against temperature. The Clapeyron equation, i.e. Eq. (6.2) provides a relationship which indicates that a plot of the logarithm of the vapour pressure against the reciprocal of temperature is a straight line. It is observed that this relationship predicts the dependence of vapour pressure on temperature accurately when the pressure is low. But, there is a considerable deviation from the linear behaviour at higher pressures. The
simplifications that were used in arriving at this relationship are no longer valid if the pressure is high. At high pressures, the molar volume of the liquid is comparable with that of the vapour and cannot be neglected. Also, the vapour deviates from ideal behaviour and the latent heat of vaporization is not constant. The reference substance plots for vapour pressures as useful in such cases, as these provide the vapour pressure as linear functions of temperature even at higher pressures. Two such plots are in common use: The equal-pressure reference-substance plots and the equal-temperature reference-substance plots.
6.4.1 Equal-Temperature Reference-Substance Plots
In equal-temperature reference-substance plots, the logarithm of the vapour pressure of a substance is plotted against the logarithm of the vapour pressure of a reference substance, both at the same
temperature. Since the vapour pressure versus temperature data for water are readily available, water is usually chosen as the reference substance.
The Clapeyron equation (Eq. 6.6) may be written as
S =− l⎛⎞1
dP dRT ⎜⎟⎝⎠(6.9)
where PS is the vapour pressure of the substance. The vapour pressure of the reference substance ( P
S
R) at the same temperature may be written as
S
=−
l
⎛⎞1 dP dRT
R ⎜⎟⎝⎠(6.10)
Dividing Eq. (6.9) by Eq. (6.10), we get
SdP Rd T)(6.11)dPS =l
l(/)(1/)R R
Equation (6.11) leads to ln
S
l
PPS constant (6.12)l =+
If the ratio l/lR is constant, this equation predicts a linear behaviour when the logarithm of vapour pressure of a substance is plotted against the logarithm of the vapour pressure of the reference
substance. Though the latent heat of vaporisation may vary with temperature, it is a reasonably good approximation to assume that land lR both vary to the same extent that their ratio remains constant.
Thus, Eq. (6.12) provides a better means for predicting the vapour pressure variation with temperature than Eq. (6.1)
The equal-temperature reference-substance plots are known as the Cox chart because these are prepared based on the method suggested originally by E.R. Cox (Ind. Eng. Chem., 15, 592, 1923).
Figure 6.2 gives the Cox chart in which the vapour pressure of different substances are given in mm Hg. and temperature is given in °C. In these plots, the logarithm of the vapour
100,000 10000 1000 600 400 200 100 60 40 20 10 6 4 2
1 0 10 20 30 50 75 100 150 200 250 300 400 500 600 700 1000 Temperature, °C Figure 6.2 The Cox chart.
pressure of the substance is plotted as the ordinate against the logarithm of the vapour pressure of the reference at the same temperature as the abscissa. For convenience, the abscissa scale is converted to read temperature instead of vapour pressure. This non-linear temperature axis is created as follows.
Refer Figure 6.3. The ordinate reads the vapour pressure on the logarithmic scale. A straight line with positive slope is drawn, which represents the vapour pressure of the reference substance, say water.
Take the vapour pressure of water at a desired temperature on the y-axis, and move horizontally to the straight line representing the vapour pressure. For example, the vapour pressure of water at 75°C is 289.2 mm Hg (38.55 kPa). From 289.2 on the y-axis, move horizontally to point C on the straight line AB. From point C move vertically downwards to point D on the x-axis. Point D is marked to read 75°C (348 K). The temperature 50°C (323 K) can be marked by taking the vapour pressure 92.6 mm Hg (12.34 kPa) on the vertical axis, moving horizontally to the line AB and then vertically to the x-axis to locate 50°C (323 K). Similarly, temperatures 10°C, 25°C, 100°C, 150°C, etc., can be marked off by taking the vapour pressures of water at the respective temperatures. Figure 6.3 illustrates the
construction of the Cox chart.
B 10,000 3571.2 760.01000 289.2 C 92.6 100 23.8 9.2 10 A
D 10 10 25 50 75 100 150 Temperature, °C
Figure 6.3 Construction of Cox chart.
The vapour pressures of other substances plotted on this special coordinate scales will yield straight lines over a considerable range of temperature. These lines can be drawn from the vapour pressure of a substance at just two temperatures. Another feature of the Cox chart is that the lines of vapour
pressures of the members of a homologous series of compounds or members of a group of closely related compounds converge at a point, called the infinite point which is characteristic of that group.
This feature makes it possible to establish the complete vapour pressure–temperature relationship of a member of the group with only one experimental value and the infinite point.
It has been shown that the temperature scale on the Cox chart with water as the reference substance is nearly equivalent to the function (T+C)–1 where C is approximately equal to
–43 K. Thus the Cox chart resembles the Antoine vapour pressure equation:
lnSPAB (6.13)=− −43
EXAMPLE 6.5 Construct the Cox chart for the vapour pressure of acetone given that the vapour
pressures are 8.52 kPa and 194.9 kPa at 273 K and 353 K respectively. The vapour pressure of water is obtained from steam tables as given below:
T (K) 273 293 313 323 333 353 373 PS (kPa) 0.61 2.33 7.37 12.34 19.90 47.35 101.3 Compare the vapour pressure at 323 K given by the Cox chart with the experimental value of 73.94 kPa.
Solution Pressures are marked on the logarithmic scale on the vertical axis. A straight line AB is
drawn with a convenient slope as shown in Figure 6.4. The temperature scale is established by the procedure explained earlier. For example, consider the second data point. At 293 K the vapour
pressure is 2.33 kPa on the vertical axis, move horizontally to line AB and mark point C. From point C move vertically downwards and mark T = 293 K on the horizontal axis. Thus all temperatures can be marked. Points D and E are located such that they give the vapour pressures of acetone at 273 K and 353 K respectively. Line DE is drawn which can be used for interpolating and extrapolating the vapour pressure data of acetone at any temperature. Read the vapour pressure of acetone at 323 K from the graph. Vapour pressure of acetone at 323 K = 74.5 kPa.
194.9E 200 101.3B 100 74.50 47.35
19.90 12.34D 108.52 7.37
2.33C 1.0 0.61 A
273 293 313 323 333 353 3730.1 Temperature, K
Figure 6.4 Construction of the Cox chart for Example 6.5.
Alternatively, the vapour pressure of acetone at 323 K can be determined analytically since the equal-temperature reference-substance plot can be assumed linear. That is, the vapour pressure of water is plotted against the vapour pressure of acetone at the same temperature, the resulting curve should be a straight line. We have the following vapour pressure data:
At 273 K, the vapour pressure of water = 0.61 kPa. The vapour pressure of acetone = 8.52 kPa. At 353 K, the vapour pressure of water = 47.35 kPa. The vapour pressure of acetone = 194.9 kPa. The vapour pressure of water at 323 K is 12.34 kPa.
The vapour pressure of acetone at 323 K (PS) is now obtained by linear interpolation, knowing that the vapour pressure of water at this temperature is 12.34 kPa.
log194.9log 8.52 log PS−− log 8.52
log 47.35=−−log 0.61
2.28980.9304 log PS−−0.9304
1.6753=++0.2147
log PS = 1.8698. Therefore, the vapour pressure of acetone at 323 K, PS = 74.09 kPa. The vapour pressure given by the equal-temperature reference-substance method is greater than the experimental value of 73.94 kPa by 0.15 kPa. Therefore,
0.15 ×=0.20%percent deviation = 73.94