Concept explainers
Interpretation: The values of the isothermal compressibility and the expansion coefficient of a van der Waals gas have to be calculated. Also, it has to be shown that
Concept introduction: The van der Waals equation defined the behavior of real gases. This includes the forces of attraction and repulsion in the real gases. This is represented by the formula given below as,
Answer to Problem 2D.5P
The expansion coefficient of a van der Waals gas is,
The isothermal compressibility of a van der Waals gas is,
It is proved that
Explanation of Solution
The expression for one mole of a van der Waals gas is,
The above expression is also written as,
Differentiate the above equation as follows,
At constant pressure,
Now, the value of the expansion coefficient of a van der Waals gas is,
Substitute the value of
As, it is known that
Hence, the expansion coefficient of a van der Waals gas is,
Now, At constant temperature,
The above expression is simplified as,
As, it is known that
Now, the value of isothermal compressibility is given by the expression,
Substitute the value of
Hence, the isothermal compressibility of a van der Waals gas is,
According to Euler’s chain relation, the partial derivates are expressed as,
Since,
And,
Substitute the values of
At constant volume,
Now, substitute the value of
Hence, it is proved that
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Chapter 2 Solutions
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