--%>

Define Virial Equation

The constant of vander Waal's equation can be related to the coefficients of the virial equation. 

Vander Waal's equation provides a good overall description of the real gas PVT behaviour. Now let us focus on the description that this equation gives to the onset of nonideal behaviour. This stage is shown most clearly on displays of Z = PV/(RT) versus P. the first deviations from the ideal gas value of Z = 1 show up as straight line sections in Z-versus-P plots. These initial stages of non ideal behaviour are described by the simple virial expressions Z = 1 + BPP or Z = 1 + BV/V. we begin, therefore, by rearranging van der Waal's equation to a form that can be compared with the virial equations.

Multiplication of van der Waal's equation, in the form and for 1 mol by V/(RT) converts this equation to 

PV/RT = V/(V - b) - a/RTV = 1/(1 - b/V) - a/RTV


We develop an equation with the form of the virial equation with volume terms by recognizing that the 1/(1 - b/V) term can be expanded by using the binomial expansion (1 - x)-1 = 1 + x + x2 + .... If only the first three terms of the series are exhibited, we develop to

54_virial equation.png 

Comparison with virial equations, shows that van der Waals' equation implies the definition

BV = b - a/RT 

Experimental values for BV for neon are plotted, notice that the temperature dependence of this second virial coefficient is generally consistent with that suggested values of BV are negative at low temperatures where the second term dominates, and these values increase and become positive at higher temperatures where the first term dominates. The curve is drawn on the basis with a and b values adjusted to give a good fit to the experimental results. Fitting second virial coefficient data provides, as this example illustrates, another way for assigning values to van der Waals' a and b parameters.

Van der Waals' excluded volume and molecular diameters

The excluded volume b, introduced by van der Waals' as an empirical correction term, can be related to the size of the gas molecules. To do so, we assume the excluded volume is the result pairwise coming together of molecules. This assumption is justified when b values are obtained from second virial coefficient data. Fitting values for the empirical constants of van der Waals' equation. 

So that we need to deal with a single molecular size parameter, we treat molecules as spherical particles. The diameter of a molecule is d. the volume of a molecule is 4/3 ∏ (d/2)3.
The volume in which a pair of molecules cannot move because of each other's presence is indicated by the lightly shaded region. The radius of this excluded volume sphere is equal to the molecular diameter d. the volume excluded to the pair of molecules is 4/3∏d3. We thus obtain
Excluded volume per molecule = ½ (4/3 ∏d3)

= 4[4/3∏(d/2)3]


The expression in brackets is the volume of a molecule. Thus the excluded volume per molecule is 4 times the actual volume of the molecule.

Van der Waals' b term is the excluded volume per mole of molecules. Thus we have, with N representing Avogadro's number


B = 4 N [4/3∏ (d/2)3] = 4 N (volume of molecule)

Molecular size and Lennard-Jones Intermolecular attraction term based on second virial coefficient data:

Gas Excluded volume b, L mol-1 Molecular diam. D, pm εLJ, J × 10-21
He 0.021 255 0.14
Ne 0.026 274 0.49
Ar 0.050 341 1.68
Kr 0.058 358 2.49
Xe 0.084 405 3.11
H2 0.031 291 0.52
N2 0.061 364 1.28
O2 0.058 358 1.59
CH4 0.069 380 1.96
C(CH3)4 0.510 739 3.22

 

   Related Questions in Chemistry

  • Q : Simulate the column in HYSYS The

    The objective of this work is to separate a binary mixture and to cool down the bottom product for storage. (Check table below to see which mixture you are asked to study). 100 kmol of feed containing 10 mol percent of the lighter component enters a continuous distillation column at the m

  • Q : Calculating total vapour pressure

    Select the right answer of the question. The vapour pressure of two liquids P and Q are 80 and 600 torr, respectively. The total vapour pressure of solution obtained by mixing 3 mole of P and 2 mole of Q would be: (a) 140 torr (b) 20 torr (c) 68 torr (d) 72 torr

  • Q : Describe characteristics of halides and

    Halides characteristics

  • Q : Mole fraction and Molality Select the

    Select the right answer of the following question.What does not change on changing temperature : (a) Mole fraction (b) Normality (c) Molality (d) None of these

  • Q : Question based on vapour pressure and

    Give me answer of this question. The vapour pressure of water at 20degreeC is 17.54 mm. When 20g of a non-ionic, substance is dissolved in 100g of water, the vapour pressure is lowered by 0.30 mm. What is the molecular weight of the substances: (a) 210.2 (b) 206.88

  • Q : Reaction of calcium carbonate Give me

    Give me answer of this question. What is the volume of 0.1NHcl required to react completely with 1.0g of pure calcium carbonate : (Ca= 40, C= 12 and o = 16 ) (a)150cm3 (b)250cm3 (c)200cm3 (d)100cm3

    Q : Latent heat of vaporization Normal

    Normal butane (C4H10) is stored as a compressed liquid at 90°C and 1400 kPa. In order to use the butane in a low-pressure gas-phase process, it is throttled to 150 kPa and passed through a vaporizer. The butane emerges from the vaporizer as a

  • Q : Partial vapour pressure of volatile

    Choose the right answer from following. For a solution of volatile liquids the partial vapour pressure of each component in solution is directly proportional to: (a) Molarity (b) Mole fraction (c) Molality (d) Normality

  • Q : What are electromotive force in

    The main objective of this particular aspect of Physical Chemistry is to examine the relation between free energies and the mechanical energy of electromotive force of electrochemical cells. The ionic components of aqueous solutions can be treated on the basis of the

  • Q : Nuclear Magnetic Resonance The nuclear

    The nuclear states produced by a magnetic field are studied in nuclear magnetic resonance spectroscopy.The frequency of the radiation that corresponds to the nuclear magnetic energy level spacings and the weakness of the radiation absorption that must be e