A \"generic\" phase diagram for a typical pure chemical substance is given in Fi

Question

A "generic" phase diagram for a typical pure chemical substance is given in Fig 5.12 of Atkins. What is the physical significance of the critical point in the phase diagram? The van der Waals equation is the simplest equation of state that can be used to describe critical behavior for pure substances. Use the values for the van der Waals a and b coefficients for carbon dioxide (Table 1.5 of Atkins) to estimate the values for the critical constants for CO2 (pc, Vc, and Tc). Compare your values to the experimental values (pc = 72.85 atm, Vc = 94.0 cmVmol, and Tc = 304.2 K), and briefly discuss the agreement between the experimental values and those predicted using the van der Waals coefficients.

Explanation / Answer

a) The significance of the critical point is most easily seen by considering the critical temperature, TC. For a substance initially in the gas phase, the substance can be converted from gas to condensed phase (solid or liquid) by an isothermal reversible compression if T < TC. If T > TC an isothermal reversible compression will never result in a phase transition (that is, the properties of the substance will change continuously from those of a gas at low pressure to those of a liquid at high pressure, but without a phase transition ever taking place). pC and VC represent the values for pressure and temperature at the critical temperature on the liquid-gas phase boundary.

b) For CO2, a = 3.610 L2.atm/mol2, and b = 0.0429 L/mol. So (eq 1.24)

VC = 3 (0.0429 L/mol) = 0.1287 L/mol = 128.7 cm3/mol (experimental value is 94.0 cm3/mol)

TC = 8a = 8 (3.610 L2.atm/mol2)

27Rb 27 (0.08206 L.atm/mol.K)(0.0429 L/mol)

= 303.8 K (experimental value is 304.2 K)

pC = a = (3.610 L2.atm/mol2) = 72.65 atm (experimental value is 72.85 atm)

27b2 27 (0.0429 L/mol)2

The values for the critical temperature and pressure calculated using the van der Waals coefficients are within a few tenths of a percent of the experimental value. The value for VC calculated from the van der Waals coefficients is 37% higher than the experimental value. The van der Waals equation typically does a good job in predicting TC and pC, but usually overestimates the value for VC.

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