A certain gas obeys the van der Waals equation of state. Show that for just one
ID: 979094 • Letter: A
Question
A certain gas obeys the van der Waals equation of state. Show that for just one critical state specified by T_c and V_c that both Write the equation of state giving p = P/P_c in terms of t = T/T_c and v = V/V_c. Make plots of p as a function of v for t = 1/2, t = 1, and t = 2. Derive an expression for the isothermal compressibility, kt of a van der Waals gas in the reduced units. Make plots of kt vs. v at t = 1/2, t = 1, and t = 2. Describe the physical state that results when there are three allowed values of v for a given p and t. For such a case, what is the sign of the isothermal compressibility at each of the 3 volumes. Are all 3 volumes physically acceptable and why (or why not)?Explanation / Answer
Although no gas is truly ideal, many gasses follow the ideal gas law very closely at sufficiently low pressures. The ideal gas law was originally determined empirically and is simply
Real Gas - Real gas laws try to predict the true behavior of a gas better than the ideal gas law by putting in terms to describe attractions and repulsions between molecules. These laws have been determined empirically or based on a conceptual model of molecular interactions or from statistical mechanics.
A well known real gas law is the van der Waals equation
where a and b are either determined empirically for each individual compound or estimated from the relations.
The first parameter, a, is dependent upon the attractive forces between molecules while the second parameter, b, is dependent upon repulsive forces.
Another two parameter real gas equation is the Redlich-Kwong equation. It is almost always more accurate than the van der Waals equation and often more accurate than some equations with more than two parameters. The Redlich-Kwong equation is
where a and b are not identical to the a and b in the van der Waals equation.
Equations of state in terms of reduced variables give reasonable results without any empirically determined constants for a specific substance. However, these are not generally as accurate as equations using empirical constants. One such equation is
where reduced pressure and temperature are the unitless quantities obtained by dividing the value by the critical value. In the case of reduced volume, molar volume is divided by critical molar volume.
A two parameter equation which is no longer used much is the Berthelot equation
A somewhat more accurate modified Berthelot equation is
The Dieterici equation is another two parameter equation which has been seldom used in recent years.
The Clausius equation is a simple three parameter equation of state.
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