States of Matter (Gases and Liquids)

Study Notes

 

Intermolecular Forces

Attractive and repulsive forces between interacting particles (atoms and molecules)

Do not include ionic and covalent bonds

Attractive intermolecular forces are known as van der Waals forces.

Dispersion Forces or London Forces

Forces of attraction between two temporary dipoles

Occur between atoms or non-polar molecules

Atoms or non-polar molecules have no dipole moment as they are electrically symmetrical. However, for some reason, momentarily an atom or non-polar molecule becomes electrically unsymmetrical. As a result, instantaneous dipole is developed on that atom for a very short time. This instantaneous dipole induces dipole on another atom close to it.

E ∝ , where E = Interaction energy

r = Distance between two interacting particles

Occur only at short distances (~500 pm)

Depend on the polarisability of the particle

Dipole-Dipole Forces

Forces between molecules possessing permanent dipole

London force < Dipole-dipole force < Ion-ion force

Increase with decrease in distance

For stationary polar molecules, Interaction energy

E ∝ 

And for rotating polar molecules,

E ∝ 

r = Distance between the polar molecules

For polar molecules,

Total intermolecular forces = Dipole-dipole forces + London forces

Dipole-dipole interaction between two HCl molecules

Hydrogen Bonding

Special case of dipole-dipole interaction

Force between hydrogen attached to an electronegative atom of one molecule and an electronegative atom of different molecule

Limited to electronegative atoms N, O, and F (sometimes Cl also)

Formation of hydrogen bond:

Dipole-Induced Dipole forces

Forces between polar molecules and non-polar molecules

Occur because polar molecule induces dipole on non-polar molecule

Depends upon the dipole moment present in the permanent dipole and the polarisability of the non-polar molecule.

E ∝  , where E = Interaction energy

r = Distance between the two molecules

Intermolecular forces between polar and non-polar molecules

= London forces + Dipole-induced dipole forces

Repulsive Forces

Increase with decrease in distance

Due to this reason, solids are harder to compress than liquids, and liquids are harder to compress than gases.

Thermal Energy

Energy of a body resulting from atomic and molecular motion

Measure of average kinetic energy of particles of matter

Directly proportional to temperature

Intermolecular Forces vs. Thermal Interaction

Intermolecular forces −−− hold molecules together

Thermal interactions −−− keep molecules apart

Intermolecular forces and thermal energy balance each other to different extent resulting in the three states of matter (solid, liquid, and gas).

Gas Laws

Boyle’s Law

Relation between pressure (p) and volume (V)

Statement − At constant temperature, the pressure of a fixed amount (number of moles, n) of a gas is inversely proportional to its volume.

Mathematically,

    (at constant T and n)

    (where k₁ is proportionality constant)

 

From the above equation, it is found that at constant temperature, the product of pressure and volume of a fixed amount of a gas is constant.

The value of k₁ depends upon

amount of the gas

temperature of the gas

units of p and V

Graphical representation of constant temperature

Each line is called isotherm (at constant temperature plot).

If at constant temperature,

V₁ = Volume of a gas at pressure p₁

V₂ = Volume of the same gas at pressure p₂

Then,

p₁V₁ = p₂V₂ = Constant

 

We know that, relationship between density (d) and pressure (p):

 

Where, m = Mass of a gas

V = Volume of the gas

 

 

 

From the above equation, it is known that density is proportional to the pressure of a fixed amount of a gas.

Charles’ Law

Relation between temperature (T) and volume (V)

Statement − At constant pressure, the volume of a fixed amount of a gas is directly proportional to its absolute temperature.

Mathematically,

 

   , where k₂ = Proportionality constant

The value of k₂ depends upon

pressure of the gas

amount of the gas

unit of volume

Graphical representation

Straight lines, interception on zero volume at 273.15 °C

Each line is called isobar (constant pressure plot).

Derivation

For each degree rise in temperature, volume of a gas increases by  of the original volume of the gas at 0°C.

Suppose, V₀ = Volume of a gas at 0 °C

V_t = Volume of the same gas at t °C

Then,

 

 

           ……… (i)

According to Kelvin temperature scale (also called absolute temperature scale or thermodynamic scale),

T = 273.15 + t

T₀ = 273.15

From equation (i), we obtain

 

 

Or, we can write

 

 

 

 

Gay-Lussac’s Law

Relation between pressure and temperature

Statement − At constant volume, the pressure of a fixed amount of a gas is directly proportional to the temperature.

Mathematically,

p ∝ T

Þ  = constant = k₃

Graphical representation at constant volume

Each line is called isochore (constant volume plot).

If at constant volume,

p₁ = Pressure of a gas at T₁

p₂ = Pressure of the same gas at T₂

Then,

 

Avogadro Law

Relation between volume (V) and amount of substance (number of moles n)

Statement − Under the same conditions of temperature and pressure, equal volumes of all gases contain equal number of molecules.

That is, at constant temperature and pressure, the volume of a gas depends upon the amount (number of mole n) of the gas.

Mathematically,

V ∝ n

V = k₄n

Where, k₄ = Proportionality constant

Avogadro constant = Number of molecules present in one mole of a gas = 6.022 × 10²³

At STP (273.15 K and 1 bar), molar volume of an ideal gas is 22.7 L mol⁻¹.

If m = Mass of a gas

M = Molar mass of the gas

Then,

 

Therefore,   (Since V = k₄ n)

 

      

From the above equation, it can be concluded that at a given temperature and pressure, density of a gas is directly proportional to its molar mass.

 

Ideal Gas

The gas which strictly follows Boyle’s law, Charles’ law and Avogadro law

The intermolecular forces are assumed to be absent between the molecules of an ideal gas.

Under a certain specific condition (when the intermolecular forces are negligible), real gases follow the above laws.

Ideal Gas Equation

Equation obtained by the combination of Boyle’s law, Charles’ law and Avogadro law

Boyle’s law: V ∝   …(At constant T and n)

Charles’ law: V ∝ T …(At constant p and n)

Avogadro law: V ∝ n …(At constant p and T)

By combining the above three laws, we have

 

 

pV = nRT             …….(i)

R = Proportionality constant, known as Universal Gas Constant

Equation (i) is called ideal gas equation.

At STP, for one mole of a gas, R = 8.314 J K⁻¹ mol⁻¹

Or, R = 8.20578 × 10⁻² L atm K⁻¹ mol⁻¹

It is also called the equation of state

Reason: It relates between four variables and describes the state of a gas.

Combined gas law:

If the temperature, volume and pressure of a fixed amount of a gas vary from T₁, V₁ and p₁ to T₂, V₂ and p₂, then we have

          ……… (ii)

          ……… (iii)

From equations (ii) and (iii), we have

           ……… (iv)

Equation (iv) is called combined gas law.

Relation between density and molar mass of a gaseous substance:

pV = nRT

 

 

      (Where, Density,  )

 

Dalton’s Law of Partial Pressures

Partial pressure: Pressure exerted by the individual gases in a mixture

Statement: At constant temperature, the total pressure exerted by a mixture of two or more non-reacting gases, enclosed in a definite volume, is equal to the sum of the partial pressures of the individual gases.

Mathematically,

p_total = p₁ + p₂ + p₃ + ……….. (At constant T and V)

Where, p_total = Total pressure exerted by the mixture

p₁ + p₂ + p₃, ... = Partial pressures of the individual gases

p_{dry gas} = p_total − Aqueous tension

Where, p_{dry gas} = Pressure of dry gas

p_total = Total pressure

Aqueous tension: Pressure exerted by saturated water vapour

Partial pressure in terms of mole fraction:

Suppose three gases are enclosed in a vessel of volume, V at temperature, T and exert partial pressures, p₁, p₂ and p₃ respectively. Then, we have

 

Where, n₁, n₂, n₃ = Number of moles of the gases

Now, p_total = p₁ + p₂ + p₃

 

 

By dividing p₁ by p_total, we have

 

  (as n = n₁ + n₂ + n₃)

 

         (x₁ is the mole fraction of the first gas)

General equation

p_i = x_i p_total

p_i = Partial pressure of i^th gas, x_i = Mole fraction of i^th gas

Kinetic Molecular Theory of Gases

The kinetic theory explains the macroscopic properties of gases, such as temperature, pressure and volume, considering their molecular composition and motion.

Postulates:

A gas consists of very small particles, atoms and molecules, which move in a random motion.

These particles have the same mass.

There is no force of attraction between these particles.

The volume of the particles is negligible compared to the total volume of the gas.

Collision of the gas particles takes place either with other particles or with the wall of the container.

The average kinetic energy of gas particles depends only on the temperature of the system. The total kinetic energy of gas particles remains constant.

Pressure of the gas arises due to the collision of its molecules with the wall of the container.

Average speed of molecules,

Mean of square of speeds,

Root mean square speed,

 

Maxwell Distribution of Molecular Velocity

Gas is a collection of tiny particles that are separated from each other by a large empty space; these particles move rapidly in a random motion in all directions. Because of this motion, molecules frequently collide with one another as well as with the wall of the container, thereby resulting in the change in their velocity and direction. Although the molecules in a given sample do not have the same velocity, the distribution of velocities remains constant at a particular temperature.

Important features of Maxwell's distribution curve:

The fraction of molecules having very high or very low molecular speeds is very small. Most number of molecules have speed near to the most probable speed, .

The total area under the curve remains constant at different temperatures; it is the measure of the total number of molecules possessing a particular velocity.

The distribution of molecules is also dependent on the molecular mass of a gas. At constant temperature, a gas with higher molecular mass shows a narrow distribution of speeds compared to that shown by a low-molecular-weight gas.

Maxwell Distribution of Molecular Velocity

The maximum in the curve represents speed possessed by maximum number of molecules. This speed is called most probable speed, .

The fraction of molecules with speeds greater than the minimum increases with the increase in speed, reaches to the maximum value and then starts to decrease.

At constant temperature, the fraction of molecules having a certain speed remains the same, even though the molecules change their speeds continuously because of collisions.

Behaviour of Real Gas

Real gases deviate from ideal gas behaviour

According to Boyle’s law, at constant temperature, pV of a gas is constant, and the pV vs p graph is a straight line parallel to the x-axis at all pressures. But real gases do not behave like this.

Reasons for deviation from the ideal gas behaviour − The two postulates of the kinetic theory which do not hold good:

There is no force of attraction and repulsion between the molecules of a gas.

Volume of the molecules of a gas is negligibly small in comparison to the space occupied by the gas.

van der Waals equation:

Real gas deviates from ideal gas behaviour as there are forces of attraction and repulsion between the molecules of a gas. At high pressure, while striking the walls of a container, the molecules of a gas are dragged back by other molecules due to the existing force of attraction; therefore, the pressure exerted by the gas is lower than the pressure exerted by an ideal gas.

 

                  Observed        Correction

                 pressure            term

Where,

n = Number of moles of real gas

V = Volume of the gas

a = van der Waals constant, whose value depends upon the nature of the gas

Real gas deviates from ideal gas behaviour as the volume occupied by the gas molecules becomes significant. This is because the movement of the gas particles is restricted to the volume (V − nb), instead of V. Here, nb is the approximate total volume of the molecules themselves, excluding the spaces between them.

This means, V_ideal = V_real − nb

Therefore, the ideal gas equation pV = nRT can be rewritten as

 

This equation is called Van der Waals equation.

Significance of ‘a’:

It is the measure of the magnitude of attractive forces between the molecules of a gas.

Unit: atm L² mol⁻² or bar L² mol⁻²

Larger the value of ‘a’, larger will be the intermolecular force of attraction.

It is independent of pressure and temperature.

Significance of ‘b’:

It is the measure of the effective size of gas molecules.

Unit: L mol⁻¹

Real gases behave like ideal gas under certain specific conditions when the intermolecular forces are negligible.

When pressure approaches zero, real gases behave like ideal gas.

When the volume of a gas is so large that the volume occupied by the molecules is negligible, the gas shows ideal gas behaviour.

Compressibility factor (Z):

It is the measure of deviation from ideal behaviour.

 

Or

 

For ideal gas, Z = 1

At low pressure, Z ≈ 1

At high pressure, Z > 1

At intermediate pressure, Z < 1

Boyle temperature or Boyle point:

The temperature at which a real gas shows ideal gas behaviour over an appreciable range of pressure

Depends upon the nature of the gas

Above Boyle point − positive deviation from ideal behaviour

Below Boyle point − negative deviation from ideal behaviour

At low pressure and high temperature, gases behave ideally.

Liquefaction of Gases

A gas can be liquefied by increasing its pressure and decreasing its temperature.

Critical temperature (T_c) − the temperature at which a gas can be liquefied

Critical pressure (p_c) − the pressure of a gas at critical temperature

Critical volume (V_c) − the volume of 1 mole of a gas at critical temperature

Isotherm of carbon dioxide is shown in the figure.

Critical temperature, critical pressure and critical volume are called critical constants.

Vapour Pressure of Liquids

Liquid State

Intermolecular forces are stronger than those in gaseous state

Liquids have definite volume.

Reason: Molecules do not separate due to intermolecular force of attraction.

Liquids can flow.

Reason: Molecules can freely move past one another.

Physical Property of Liquid − Vapour Pressure

Equilibrium vapour pressure or saturated vapour pressure: Vapour pressure in the state of equilibrium between liquid phase and vapour phase

Boiling point:

The temperature at which the vapour pressure of a liquid is equal to the external pressure

Normal boiling point − Boiling point at 1 atm pressure

Standard boiling point − Boiling point at 1 bar pressure

The standard boiling point of a liquid is slightly lower than its normal boiling point.

Reason − 1 atm pressure is slightly greater than 1 bar pressure.

Example − Water has a normal boiling point of 100°C (373 K) and a standard boiling point of 99.6 °C (372.6 K).

The boiling point of a liquid is lower at high altitudes than at sea level.

Reason − Atmospheric pressure is lower at high altitudes than at sea level.

This means that the boiling point of a liquid can be varied by changing the pressure over the liquid.

Surface Tension

Force acting per unit length perpendicular to the line drawn on the surface of liquid

Denoted by Greek letter γ (gamma). Unit = Nm⁻¹

Reason for surface tension − A molecule in the bulk of liquid does not experience any net force as it experiences equal intermolecular forces from all the sides. However, there are no intermolecular forces above a molecule on the surface of liquid. Therefore, a molecule on the surface of liquid experiences net attractive force towards the interior of the liquid.

As a result of surface tension, liquid tends to minimize their surface area.

Surface energy

Energy required to increase the surface area of the liquid by one unit. Unit = J m⁻²

Liquid droplets are spherical because sphere has minimum surface area and liquids tend to have minimum surface area due to surface tension.

Surface tension causes capillary action.

Dependence of surface tension:

The attractive forces between the molecules increase with increase in attractive force.

The intermolecular forces decrease with the increase in temperature because with the increase in temperature, kinetic energy of particles increases and hence, effectiveness of intermolecular attraction decreases.

Viscosity

Resistance of flow

Stronger the intermolecular forces, higher is the viscosity.

Laminar flow

Type of flow which involves a regular velocity gradation in passing from one layer to the next

For a given layer in a flowing liquid, the layer above it accelerates its flow while the layer below it retards its flow.

Velocity gradient   (change in velocity with distance)

Where, dz = Distance

du = Change in velocity

We can write,

F ∝ A

Where, F = Force required to maintain the flow of layers

A = Area of contact

And,  

Therefore,  

 

Where, η = Proportionality constant known as coefficient of viscosity

Viscosity coefficient

Force when velocity gradient is unit and the area of contact is unit area

SI unit = Ns m⁻²

1 Ns m⁻² = 1 Pa S = 1 kg m⁻¹ s⁻¹

CGS unit = poise

1 poise = 1 g cm⁻¹ s⁻¹ = 10⁻¹ kg m⁻¹ s⁻¹

Greater the viscosity, more slowly the liquid flows.

Glass is an extremely viscous liquid.

With increase in temperature, viscosity of liquids decreases.

Reason − With increase in temperature, kinetic energy of molecule increases and therefore, it is easier to overcome the intermolecular forces to slip past one another between the layers.

 

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