Structure of Liquids

Structure of Liquids

The structure of liquids is less well established than that of gases or solids.
To understand the structure of liquids, we will consider the theories of liquid state and other various approaches.

1.   Vacancy Theory of Liquids

A Liquid is generally less dense than the corresponding solid;
\(\implies\) intermolecular space in a liquid is more than that in a solid.
In 1961, Ering and Ree proposed this theory.
They suggested that the intermolecular spaces in a liquid is not randomly distributed. Instead these spaces contains ‘holes’ or vacancies of molecular-size.

Ering and Ree Theory
Thus, the liquid is considered as a random collection of molecules and these holes as shown in the fig.
In this theory, it was assumed that the molecules around a given hole can easily jump into it and thus they show ‘gas-like’ behaviour.
While the remaining molecules which can’t jump inside the hole show ‘solid-like’ behaviour.
Considering, Vl = Molar volume of liquids
Vs = Molar volume of solids ;
it is shown that :
mole fraction of gas-like molecules,
\(x_g = \frac{V_l – V_s}{V_l}\)

and the mole fraction of remaining solid-like molecules,
\(x_s = \frac{V_s}{V_l}\)

Ering & Ree calculated the boiling point, melting point, critical constants and other typical thermodynamic properties of argon(Ar).
And the theoretical values for Ar were found to be in close agreement with the experimental values.

2.   Free Volume Theory

As we know, the order of intermolecular spaces (space between two molecules or atom) in states of matter is : Solid<< Liquid< Gas
Hence the gas molecules move over considerable distance before colliding with each other;
while liquids molecules move on infinitesimally small distance before colliding with one another because the intermolecular spaces in liquids are smaller.

Free volume theory assumes that each molecule in a liquid is tightly surrounded by almost 10 to 12 neighbouring molecules forming a spherical cage or cell around the molecule in center as shown in the fig.
This spherical cage can be approximated to a spherical box of radius \(r_f\) which is only slightly bigger than the radius \(r_m\) of enclosed molecule in centre as shown in the fig.free volume theory
It is evident that the caged molecule at the centre can move within a small spherical cage of radius \(r_f\). This available volume  for 1 mole of liquid molecules is known as free Volume.
Thermodinamically, it can be shown that the magnitude of free volume \(\approx\ 0.37 cm^3\)
$$= 0.37×10^{24} {Å}^3$$
\(\implies\) free volume per molecule = $$ \frac{0.37×10^{24} {Å}^3}{N_A}$$
$$=  \frac{0.37×10^{24} {Å}^3}{6.022× 10^{23}}$$
$$=  0.61\,{Å}^3$$
\(Also\, free\, volume\, per\, molecule = \frac{4}{3} \pi r^3 \)
$$Hence, \frac{4}{3} \pi r^3 =0.61\,{Å}^3$$
$$\implies r_f=0.54\,{Å}$$

Note :
The radius rf represents the average distance  that a molecule traverse betweens collisions with the walls of the spherical cage.

Qus. Calculate the number of collisions per second that a molecule make with the walls of the container.
(given that rf=0.54 Å and average speed of molecules of liquid is 3×10⁴ cm s-1

\(\implies\)
As we know, collision frequency = \(\frac{\bar{a}}{r_f}\)
\(=\frac{3×10^4 \,cm \,s^{-1}}{0.54×10^{-8} \,cm}  \)

= 5.6 × 1012 Collisions per second

3.   X-ray diffraction studies

The x-ray diffraction studies of solids show a series of bands showing well defined maxima. Because the atoms in crystalline solids are arranged in regular pattern i.e, solids possess long range order.

The x-ray diffraction studies of gases show a continuous scattering of x-rays without showing any maxima or minima. Because gas molecules possess no order or there is complete randomness or disorder in gases due to the random movement of molecules in all directions.

The x-ray diffraction pattern of liquids show only 1 or 2 well defined maxima which diminish with distance. Because there is partially ordered structure in liquids and hence only short range order exists in liquids.
In this way liquids occupy an intermediate place between the complete order of solids and complete disorder of gaseous state.
The x-ray diffraction pattern of a liquid represents the average distribution of liquid molecules relative to one another.

X-ray diffraction pattern of liquid mercury
Intensity of scattered radiation \(I(\theta)\) is a function of scattering angle \(\theta\) . It is given by the relation:

$$I(\theta) \propto \int\limits_0^\infty p(r) \frac{sin\,kr\, dr}{kr} $$
where \(k= \frac{4 \pi}{\lambda} sin\frac{ \theta} {2}\)
p(r) dr = probability of finding a particle between r and r+dr
The intensity of scattered X-rays decreases with increase in the value of θ as shown in fig.
x-ray diffraction pattern of liquids

It is more convenient to study the location of particles in a liquid in terms of Radial distribution function .
Radial distribution function of liquid mercury at 293 K is as shown in the fig :Radial distribution function for liquid mercury
It is clear from this graph that :
• There is only one sharp maximum.
• But as we move away from centrally chosen atom then the maxima are diminished. Hence there is short range order in liquids.
• i.e., liquids have no long range order, the Radial distribution function becomes constant after a few molecular diameters.
• For small values of r , the radial distribution function for a liquid is zero. This is due to the small range intermolecular repulsions.
• The first maxima appears at about 10 to 11 nearest neighbours that surround each molecule.

The radial distribution function characterizes the average structure of a liquid (i.e., it characterizes the average distribution of its molecules relative to one another.)

Incompressibility

As the molecules in liquids are randomly arranged like those of a gas.
Also there are strong forces of attraction and compact structure in liquids resembling that of a solid. i.e., Liquids are incompressible
Liquids represent an intermediate state between order and disorder.

Change in molar volume on fusion and vaporisation

When a solid melts, its molar volume increases by about 1%
whereas when a liquid is converted into vapours at its boiling point, the increase in volume is about 100-1000 times .
A small change in volume during fusion indicates that there is some disorder when the solid melts.
But when a liquid changes into vapour , the large increase in volume indicates a large disorder from liquid to gaseous state.
It reveals that liquids do not have perfect order as in solids but they are not so disordered like gases.

Enthalpy of fusion and vaporisation

Enthalpy of fusion \( \Delta H_{f}\) :
It is defined as the amount of heat required to melt one mole of solid at the melting point.

Enthalpy of vaporisation \(\Delta H_{v}\) :
It is defined as the heat required to convert one mole of a liquid into vapour at its boiling point.

Enthalpy of fusion is found to be much lesser than the enthalpy of vapourisation.
i.e., \(\Delta H_f\) <<< \(\Delta H_{v}\)

For e.g.
$${(\Delta H_f)}_{ice} = 6.1 KJ mol^{-1}$$
$${(\Delta H_v)}_{water} = 40.7 KJ mol^{-1}$$
i.e., heat of fusion of ice is much lesser than the heat of vaporisation of water

The lower value of enthalpy of fusion suggests that some disorder is introduced when a solid is melted at its melting point.
The heat of vaporisation is a measure of the strength of the intermolecular forces holding molecules in the liquid state. Thus, a large value of enthalpy of vaporisation means more heat is absorbed in changing the liquid into vapour.
In other words, large disorder is caused in going from liquid to vapour state. fusion and vaporisation

\(\implies\) the constituent particles of liquid are not as perfectly ordered like in solids but they are not so disordered like those in a gas.

Entropy of fusion and Entropy of vaporisation

Entropy of fusion \(\Delta S_f\) :
• It is the entropy change accompanying melting of one mole of solid at the melting point

Entropy of vaporisation \(\Delta S_v\) :
• It is the change in entropy involving vaporisation of 1 mole of liquid at its boiling point.

Entropy of fusion is found to be lesser than the entropy of vaporisation.
i.e., \(\Delta S_f\) < \(\Delta S_v\)

For example
$${(\Delta S_f)}_{ice} = 22.1 J K^{-1}$$ at 273 K
$${(\Delta S_v)}_{water} = 109 J K^{-1}$$ at 373K
i.e., entropy of fusion of ice is less than the entropy of vaporisation of water.
As entropy is a direct measure of disorder.
\(\implies\) lesser disorders introduced when a solid changes to liquid than when a liquid converts into a gas.
\(\implies\) liquid state is an intermediate state of perfect order of solids and complete disorder of gases.

Structural differences between solids, Liquids and Gases :

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