Question

What is the physical significance of the two constants that appear in the van der Waals equation of state? On what basis are they determined?

Short answer

Answer: The van der Waals constants 'a' and 'b' account for the non-ideal behavior of real gases. The constant 'a' represents the strength of the molecular attraction, while 'b' represents the effective size of the gas molecules. These constants are determined experimentally for each specific gas by measuring their critical temperature, pressure, and volume, which depend on the molecular properties, such as size, shape, and attractive forces between the molecules. Using the van der Waals equation of state with these constants provides a more accurate description of gas behavior than the ideal gas law.

Step-by-step solution

Introduce the van der Waals equation of state

The van der Waals equation of state for a real gas can be written as: $$\left(P + a\frac{n^2}{V^2}\right)(V - nb) = nRT$$ Here, P is the pressure, V is the volume, T is the temperature, n is the number of moles of gas, R is the universal gas constant, and 'a' and 'b' are the van der Waals constants.

Explain the role of 'a' in the equation

The constant 'a' accounts for the attractive forces between the gas molecules. As the molecules come closer together (smaller volume), the attractive forces become stronger, causing a decrease in the effective pressure acting on the gas. This means that the observed pressure (P) will be less than the pressure predicted by the ideal gas law. Thus, the term \(a\frac{n^2}{V^2}\) compensates for this discrepancy by adding it to the observed pressure.

Explain the role of 'b' in the equation

The constant 'b' represents the effective size or excluded volume of the gas molecules. This takes into consideration the fact that gas molecules have a finite size and cannot be compressed into a volume smaller than the sum of their own volumes. The term 'nb' subtracts the occupied volume from the total available volume, resulting in the effective volume (V - nb) in which the gas molecules can move.

Describe how the constants 'a' and 'b' are determined

The van der Waals constants 'a' and 'b' are determined experimentally for each specific gas. They depend on the molecular properties of the gas, such as the size and shape of the molecules, and the strength of the attractive forces between them. Various methods have been developed to obtain these constants from experimental data, such as measuring the critical temperature, pressure, and volume of a gas. Once these constants have been found for a particular gas, they can be used to describe the behavior of the gas more accurately than the ideal gas law.