(a)
Interpretation:
The van der Waals equation of state has to be expressed as the power series of
Concept Introduction:
van der Waals equation:
van der Waals equation represents the real gas equation. Real gas molecules have their own volume and there is force of attraction and repulsion constantly working between the real gas molecules unlike the ideal gas molecules. Hence ideal gas equation is subjected to modify with pressure and volume correction and thus van der Waals equation has been formed for real gases.
(a)
Explanation of Solution
A mathematical function of the form
Now the van der Waals equation of state has to be expressed as an expansion series of
Hence van der Waals equation can be represented as,
Now the form needed for expansion is
Now according to the series expansion,
Hence here
So the expansion form can be written as,
The equation can be rearranged as,
Thus the van der Waals equation can be represented as the power series of
(b)
Interpretation:
An expression of the Boyle temperature has to be derived in the terms of van der Waals constants
Concept Introduction:
van der Waals equation:
van der Waals equation represents the real gas equation. Real gas molecules have their own volume and there is force of attraction and repulsion constantly working between the real gas molecules unlike the ideal gas molecules. Hence ideal gas equation is subjected to modify with pressure and volume correction and thus van der Waals equation has been formed for real gases.
Virial equation:
General equation of states for real gases is virial equation which is proposed by Kammerlingh-Onnes. He proposed the equation as,
Here,
The 1st virial coefficient is
Boyle temperature:
The temperature at which real gas starts behaving ideally is called the Boyle temperature.
At this temperature the 2nd virial coefficient becomes zero and 3rd, 4th and higher virial coefficients become insignificant. The virial equation for real gas becomes ideal gas equation
(b)
Explanation of Solution
At Boyle temperature the 2nd virial coefficient becomes zero.
Now, from part (a) it has been obtained that,
Virial equation is,
Thus comparing the above two equations the 2nd virial coefficient is,
At Boyle temperature
Thus,
Thus the expression for Boyle temperature is
(c)
Interpretation:
Boyle temperature for carbon dioxide has to be calculated for given van der Waals constant values.
Concept Introduction:
van der Waals equation:
van der Waals equation represents the real gas equation. Real gas molecules have their own volume and there is force of attraction and repulsion constantly working between the real gas molecules unlike the ideal gas molecules. Hence ideal gas equation is subjected to modify with pressure and volume correction and thus van der Waals equation has been formed for real gases.
Virial equation:
General equation of states for real gases is virial equation which is proposed by Kammerlingh-Onnes. He proposed the equation as,
Here,
The 1st virial coefficient is
Boyle temperature:
The temperature at which real gas starts behaving ideally is called the Boyle temperature.
At this temperature the 2nd virial coefficient becomes zero and 3rd, 4th and higher virial coefficients become insignificant. The virial equation for real gas becomes ideal gas equation
(c)
Answer to Problem 1.3PR
The Boyle temperature for carbon dioxide is
Explanation of Solution
From the above part (b) the expression for Boyle temperature has been obtained as,
Given that the values of van der Waals constants are,
Thus the value of Boyle temperature for carbon dioxide is,
Hence the Boyle temperature for carbon dioxide is
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