Chemistry

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System and Surrounding

Before studying the laws of thermodynamics and their applications, it is important to understand the meaning of a few terms used frequently in thermodynamics.

System:

The universe is divided into two parts, the system and its surroundings. The system is the part of universe which is under thermodynamic consideration. It is separated from the rest of the universe by real or imaginary boundaries.

Example:

The system may be water in a beaker, a balloon filled with air, an aqueous solution of glucose etc. On the basis of physical and chemical properties, systems can be divided into two types.

A system is called homogeneous if the physical state of all its constituents are the same. Example: a mixture of gases, completely miscible mixture of liquids etc.

A system is called heterogeneous, if physical state of all its constituents is not the same.

Example: Mixture of oil and water

Surrounding:

Everything in the universe that is not the part of the system is called surroundings.

Boundary:

Anything which separates the system from its surrounding is called boundary.

Types of Systems:

There are three types of thermodynamic systems depending on the nature of the boundary.

Isolated System:

A system which can exchange neither matter nor energy with its surroundings is called an isolated system. Here boundary is sealed and insulated. Hot water contained in a thermos flask, is an example for an isolated system. In this isolated system both energy (heat) and matter (water vapour) neither enter nor leave the system.

Closed System:

A system which can exchange only energy but not matter with its surroundings is called a closed system. Here the boundary is sealed but not insulated. Hot water contained in a closed beaker is an example for a closed system. In this system energy (heat) is transferred to the surroundings but no matter (water vapour) can escape from this system. A gas contained in a cylinder fitted with a piston constitutes a closed
system.

Open System:

A System which can exchange both matter and energy with its surrounding is called an open system. Hot water contained in an open beaker is an example for open system. In this system both matter (water vapour) and energy (heat) is transferred to the surrounding. All living things and chemical reactions are open systems because they exchange matter and energy with the surroundings.

Properties of the System:

Intensive and Extensive Properties

Some of the properties of a system depend on its mass or size whereas other properties do not depend on its mass or size. Based on this, the properties of a system are grouped as extensive property and intensive property.

Extensive Properties:

The property that depends on the mass or the size of the system is called an extensive property.

Examples:
Volume, Number of moles, Mass, Internal energy, etc.,

Intensive Properties:

The property that is independent of the mass or the size of the system is called an intensive property.

Examples:
Refractive index, Surface tension, density, temperature, Boiling point, Freezing point, molar volume, etc.,

Table: 7.1 Typical List of Extensive and Intensive properties

Extensive properties	Intensive properties
volume, mass, amount of substance (mole), energy, enthalpy, entropy, free energy, heat capacity	molar volume, density, molar mass, molarity, mole fraction, molality, specific heat capacity

Thermodynamic Processes

The method of operation which can bring about a change in the system is called thermodynamic process. Heating, cooling, expansion, compression, fusion, vaporization etc., are some examples of a thermodynamic process.

Types of Processes:

A thermodynamic process can be carried out in different ways and under different conditions. The processes can be classified as follows:

Reversible process:

The process in which the system and surrounding can be restored to the initial state from the final state without producing any changes in the thermodynamic properties of the universe is called a reversible process. There are two important conditions for the reversible process to occur. Firstly, the process should occur infinitesimally slowly and secondly throughout the process, the system and surroundings must be in equilibrium with each other.

Irreversible Process:

The process in which the system and surrounding cannot be restored to the initial state from the final state is called an irreversible process. All the processes occurring in nature are irreversible processes. During the irreversible process the system and surroundings are not in equilibrium with each other.

Adiabatic Process:

An adiabatic process is defined as one in which there is no exchange of heat (q) between the system and surrounding during the process. Those processes in which no heat can flow into or out of the system are called adiabatic processes.

This condition is attained by thermally insulating the system. In an adiabatic process if work is done by the system its temperature decreases, if work is done on the system its temperature increases, because, the system cannot exchange heat with its surroundings.

For an adiabatic process q = 0

Isothermal Process:

An isothermal process is defined as one in which the temperature of the system remains constant, during the change from its initial to final state. The system exchanges heat with its surrounding and the temperature of the system remains constant. For this purpose the experiment is often performed in a thermostat. For an isothermal process dT = 0

Isobaric Process

An isobaric process is defined as one in which the pressure of the system remains constant during its change from the initial to final state. For an isobaric process dP = 0 . Isochoric process. An isochoric process is defined as the one in which the volume of system remains constant during its change from initial to final state. Combustion of a fuel in a bomb calorimeter is an example of an isochoric process.

For an isochoric process, dV= 0.

Cyclic Process:

When a system returns to its original state after completing a series of changes, then it is said that a cycle is completed. This process is known as a cyclic process.

For a cyclic process dU = 0, dH = 0, dP = 0, dV = 0, dT = 0

Overview of the Process and its Condition

Process	Condition
Adiabatic	q = 0
Isothermal	dT = 0
Isobaric	dP = 0
Isochoric	dV = 0
Cyclic	dE = 0, dH = 0, dP = 0, dV = 0 , dT = 0

State Functions, Path Functions:

State Function

A thermodynamic system can be defined by using the variables P, V, T and ‘n’. A state function is a thermodynamic property of a system, which has a specific value for a given state and does not depend on the path (or manner) by which the particular state is reached.

Example: Pressure (P), Volume (V), Temperature (T), Internal energy (U), Enthalpy (H), free energy (G) etc.

Path Functions:

A path function is a thermodynamic property of the system whose value depends on the path by which the system changes from its initial to final states.

Example: Work (w), Heat (q).

Work (w) will have different values if the process is carried out reversibly or irreversibly.

Internal Energy (U)

The internal energy is a characteristic property of a system which is denoted by the symbol U. The internal energy of a system is equal to the energy possessed by all its constituents namely atoms, ions and molecules. The total energy of all molecules in a system is equal to the sum of their translational energy (U_t), vibrational energy (U_v), rotational energy (U_r), bond energy (U_b), electronic energy (U_e) and energy due to molecular interactions (U_i).

Thus:

U = U_t + U_v + U_r + U_b + U_e + U_i

The total energy of all the molecules of the system is called internal energy. In thermodynamics one is concerned only with the change in internal energy (ΔU) rather than the absolute value of energy.

Importance of Internal Energy

The internal energy possessed by a substance differentiates its physical structure. For example, the allotropes of carbon, namely, graphite (C_graphite) and diamond (C_diamond), differ from each other because they possess different internal energies and have different structures.

Characteristics of Internal Energy (U):

The internal energy of a system is an extensive property. It depends on the amount of the substances present in the system. If the amount is doubled, the internal energy is also doubled.

The internal energy of a system is a state function. It depends only upon the state variables (T, P, V, n) of the system. The change in internal energy does not depend on the path by which the final state is reached.

The change in internal energy of a system is expressed as ΔU = U_f – U_i

In a cyclic process, there is no internal energy change. ΔU_(cyclic) = 0

If the internal energy of the system in the final state (U_f) is less than the internal energy of the system in its initial state (U_i), then ΔU would be negative.

ΔU = U_f – U_i = – ve (U_f < U_i)

If the internal energy of the system in the final state (U_f) is greater than the internal energy of the system in its initial state (U_i), then ΔU would be positive.

The heat (q) is regarded as an energy in transit across the boundary separating a system from its surrounding. Heat changes lead to temperature differences between system and surrounding. Heat is a path function.

Units of Heat:

The SI unit of heat is joule (J). Heat quantities are generally measured in calories (cal). A calorie is defined as the quantity of heat required to raise the temperature of 1 gram of water by 1° C in the vicinity of 15° C.

Sign Convention of Heat:

The symbol of heat is q. If heat flows into the system from the surrounding, energy of a system increases. Hence it is taken to be positive (+q).

If heat flows out of the system into the surrounding, energy of the system decreases. Hence, it is taken to be negative (-q).

Work (w)

Work is defined as the force (F) multiplied by the displacement (x).

– w = F. x ……………….. (7.1)

The negative sign (-) is introduced to indicate that the work has been done by the system by spending a part of its internal energy.

The work,

1. Is a path function.
2. Appears only at the boundary of the system.
3. Appears during the change in the state of the system.
4. In thermodynamics, surroundings is so large that macroscopic changes to surroundings do not happen.

Units of Work:

The SI unit of work is joule (J), which is defined as the work done by a force of one Newton through a displacement of one meter (J = Nm). We often use kilojoule (kJ) for large quantities of work. 1 kJ = 1000 J.

Sign Convention of Work:

The symbol of work is ‘w’. If work is done by the system, the energy of the system decreases, hence by convention, work is taken to be negative (- w). If work is done on the system, the energy of the system increases, hence by convention, the work is taken to be positive (+ w).

Pressure – Volume Work

In elementary thermodynamics the only type of work generally considered is the work done in expansion (or compression) of a gas. This is known as pressure-volume work, PV work or expansion work.

Work Involved in Expansion an Compression Processes:

In most thermodynamic calculations we are dealing with the evaluation of work involved in the expansion or compression of gases. The essential condition for expansion or compression of a system is that there should be difference between external pressure (P_ext) and internal pressure (P_int).

For understanding pressurevolume work, let us consider a cylinder which contains ‘n’ moles of an ideal gas fitted with a frictionless piston of cross sectional area A. The total volume of the gas inside is V_i and pressure of the gas inside is P_int.

If the external pressure P_ext is greater than P_int, the piston moves inward till the pressure inside becomes
equal to P_ext. Let this change be achieved in a single step and the final volume be V_f.

In this case, the work is done on the system (+w). It can be calculated as follows

w = -F.Δx ………….. (7.2)

where dx is the distance moved by the piston during the compression and F is the force acting on the gas.

Since work is done on the system, it is a positive quantity. If the pressure is not constant, but changes during the process such that it is always infinitesimally greater than the pressure of the gas, then, at each stage of compression, the volume decreases by an infinitesimal amount, dV. In such a case we can calculate the work done on the gas by the relation

In a compression process, P_ext the external pressure is always greater than the pressure of the system.
i.e P_ext = (P_int + dP).

In an expansion process, the external pressure is always less than the pressure of the system
i.e; P_ext = (P_int – dP).

When pressure is not constant and changes in infinitesimally small steps (reversible conditions) during compression from V_i to V_f, the P-V plot looks like in fig 7.4. Work done on the gas is represented by the shaded area.

In general case we can write,

P_ext = (P_int ± dP). Such processes are called reversible processes. For a compression process work can be related to internal pressure of the system under reversible conditions by writing equation

For a given system with an ideal gas

If V_f > V_i (expansion), the sign of work done by the process is negative.

If V_f < V_i (compression) the sign of work done on the process is positive.

Zeroth Law of Thermodynamics:

The zeroth law of thermodynamics, also known as the law of thermal equilibrium, was put forward much after the establishment of the first and second laws of thermodynamics. It is placed before the first and second laws as it provides a logical basis for the concept of temperature of the system.

The law states that ‘If two systems are separately in thermal equilibrium with a third one, then they tend to be in thermal equilibrium with themselves’. According to this law, if systems B and C separately are in thermal equilibrium with another system A, then systems B and C will also be in thermal equilibrium with each other. This is also the principle by which thermometers are used.

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Introduction of Thermodynamics

The term ‘Thermodynamics’ means flow of heat and is derived from the Greek ‘Thermos’ (heat) and ‘dynamics’ (flow), In our daily life, we come across many useful reactions such as burning of fuel to produce heat energy, flow of electrons through circuit to produce electrical energy, metabolic reactions to produce the necessary energy for biological functions and so on.

Thermodynamics, the study of the transformation of energy, explains all such processes quantitatively and allows us to make useful predictions. In the 19th century, scientists tried to understand the underlying principles of steam engine which were already in operation, in order to improve their efficiency.

The basic problem of the investigation was the transformation of heat into mechanical work. However, over time, the laws of thermodynamics were developed and helped to understand the process of steam engine. These laws have been used to deduce powerful mathematical relationships applicable to a broad range of processes.

Thermodynamics evaluates the macroscopic properties (heat, work) and their inter relationships. It deals with properties of systems in equilibrium and is independent of any theories or properties of the individual molecules which constitute the system. The principles of thermodynamics are based on three laws of thermodynamics.

The first two laws (First and second law) summarise the actual experience of inter conversion of different forms of energy. The third law deals with the calculation of entropy and the unattainability of absolute zero Kelvin. Thermodynamics carries high practical values but bears certain limitations. It is independent of atomic and molecular structure and reaction mechanism.

The laws can be used to predict whether a particular reaction is feasible or not under a given set of conditions, but they cannot give the rate at which the reaction takes place. In other words, thermodynamics deals with equilibrium conditions quantitatively, but does not take into account the kinetic approach to the equilibrium state.

Thermodynamics is the study of the energy, principally heat energy, that accompanies chemical or physical changes. Others absorb heat energy and are called endothermic reactions, and they have a positive enthalpy change. But thermodynamics is concerned with more than just heat energy.

Thermodynamics is the study of the relations between heat, work, temperature, and energy. The laws of thermodynamics describe how the energy in a system changes and whether the system can perform useful work on its surroundings.

Basic Principles of Thermodynamics. A thermodynamical system is an arbitrarily but suitable chosen region of the space where certain phenomena are investigated. The system is enclosed by its surroundings. The system and its surroundings may be in equilibrium or may be interactions between them.

Thermodynamics is the science of the relationship between heat, work and the properties of substances. While the Zeroth Law provides the basis of measurement of Temperature, the First and Second Laws serve to define the two properties, Energy and Entropy, and deal with the conservation and degradation of energy.

Thermodynamics is a very important branch of both physics and chemistry. It deals with the study of energy, the conversion of energy between different forms and the ability of energy to do work.

Here are some more applications of thermodynamics: Sweating in a crowded room: In a crowded room, everybody (every person) starts sweating. The body starts cooling down by transferring the body heat to the sweat. Sweat evaporates adding heat to the room.

Thermodynamics is that part of science which is concerned with the conditions that material systems may assume and the changes in conditions that may occur either spontaneously or as a result of interactions between systems. The word “thermodynamics” was derived from the Greek words thermé (heat) and dynamics (force).

One of the most important things we can do with heat is to use it to do work for us. A heat engine does exactly this it makes use of the properties of thermodynamics to transform heat into work. Gasoline and diesel engines, jet engines, and steam turbines that generate electricity are all examples of heat engines.

The most common practical application of the First Law is the heat engine. Heat engines convert thermal energy into mechanical energy and vice versa. Most heat engines fall into the category of open systems.

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Liquefaction of Gases

For important commercial operations such as LPG and rocket fuels, we require gases in their liquid state. The liquefication methods are based on the Joule-Thomson effect. He observed appreciable cooling when the compressed gas is forced through an orifice plug into a low-pressure region. This phenomenon of lowering of temperature when a gas is made to expand adiabatically from a region of high pressure into a region of low pressure is known as JouleThomson effect.

This effect is observed only below a certain temperature, which is a characteristic one for each gas. This temperature below which a gas obeys Joule-Thomson effect is called inversion temperature (Ti). This value is given us van der waals constants a and b.

T₁ = \(\frac{2a}{Rb}\) …………… (6.34)

Gases like O₂, He, N₂ and H₂ have very low T_c, hence Joule-Thomson effect can be applied for cooling effectively. At the inversion temperature, no rise or fall in temperature of a gas occurs while expanding. But above the inversion temperature, the gas gets heated up when allowed to expand through a hole.

There are different methods used for liquefaction of gases:

1. In Linde’s method, Joule-Thomson effect is used to get liquid air or any other gas.

2. In Claude’s process, the gas is allowed to perform mechanical work in addition to Joule-Thomson effect so that more cooling is produced.

3. In Adiabatic process, cooling is produced by removing the magnetic property of magnetic material such as gadolinium sulphate. By this method, a temperature of 10^-4 K i.e. as low as 0 K can be achieved.

Liquefaction of gases is physical conversion of a gas into a liquid state (condensation). The liquefaction of gases is a complicated process that uses various compressions and expansions to achieve high pressures and very low temperatures, using, for example, turboexpanders.

In gases, the intermolecular distance is very large. But when we apply high pressure and lower the temperature the gas converts to liquid. Thus, the most favourable conditions to liquefy a gas are high pressure and low temperature. Thus, the correct option is (B) low temperature and high pressure.

The critical temperature signifies the force of attraction between the molecules. The higher the critical temperature, higher is the intermolecular force of attraction and easier is the liquefaction of the gas. Gases require cooling and compression both for liquefaction.

The liquefaction of a gas takes place when the intermolecular forces of attraction become so high that they bind the gas molecules together to form the liquid state. For each gas, there is a particular temperature above which it cannot be liquefied, howsoever, high pressure may be applied on the gas.

Liquefaction takes place when loosely packed, water-logged sediments at or near the ground surface lose their strength in response to strong ground shaking. Liquefaction occurring beneath buildings and other structures can cause major damage during earthquakes.

In general, gases can be liquefied by one of three methods:

(1) By compressing the gas at temperatures less than its critical temperature

(2) By making the gas do some kind of work against an external force, which causes the gas to lose energy and change to the liquid state; and

(3) By making gas do work against its

The permanent gases have weak intermolecular forces of interaction which makes the process of liquefaction impossible to carry out. Since the options have hydrogen, oxygen and nitrogen, it is clear that they are permanent gases. Only chlorine can be liquified easily by applying the suitable pressure on it.

But, the only difference between Linde Claude’s process of liquefaction of air, or other gases is that in Claude’s process there is an isentropic expansion. That’s why Claude’s process is more efficient than Linde’s process.

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Pressure-Volume Isotherms of Carbon Dioxide

Andrew’s Isotherm

Thomas Andrew gave the first complete data on pressure-volumetemperature of a substance in the gaseous and liquid states. He plotted isotherms of carbon dioxide at different temperatures which is shown in Figure. 6.12. From the plots we can infer the following.

At low temperature isotherms, for example, at 13° C as the pressure increases, the volume decreases along AB and is a gas until the point B is reached. At B, a liquid separates along the line BC, both the liquid and gas co-exist and the pressure remains constant.

At C, the gas is completely converted into liquid. If the pressure is higher than at C, only the liquid is compressed so, there is no significant change in the volume. The successive isotherms shows similar trend with the shorter flat region. i.e. The volume range in which the liquid and gas coexist becomes shorter.

At the temperature of 31.1° C the length of the shorter portion is reduced to zero at point P. In other words, the CO₂ gas is liquefied completely at this point. This temperature is known as the liquefaction temperature or critical temperature of CO₂. At this point the pressure is 73 atm. Above this temperature CO₂ remains as a gas at all pressure values. It is then proved that many real gases behave in a similar manner to carbon dioxide.

Though the nature of isotherm remains similar, the critical temperature, the corresponding pressure and volume are characteristics of a particular gas.

Now we can define the critical constants as follows. Critical temperature (T_c) of a gas is defined as the temperature above which it cannot be liquefied even at high pressure. Critical pressure (Pc) of a gas is defined as the minimum pressure required to liquefy 1 mole of a gas at its critical temperature. Critical volume (V_c) is defined as the volume occupied by 1 mole of a gas at its critical temperature and critical pressure. The critical constants of some common gases are given in Table 6.2.

Derivation of Critical Constants from van der Waals Constant:

The van der Waals equation for n moles is

(P + \(\frac{\mathrm{a} \mathrm{n}^{2}}{\mathrm{~V}^{2}}\)) (V – nb) = nRT ………….. (6.22)

For 1 mole

(P + \(\frac{\mathrm{a}}{\mathrm{V}^{2}}\)) (V – b) = RT ………….. (6.23)

From the equation we can derive the values of critical constants P_c, V_c and T_c in terms of a and b, the van der Waals constants, On expanding the above equation

PV + \(\frac{a}{V}\) – Pb – \(\frac{\mathrm{ab}}{\mathrm{V}^{2}}\) – RT = 0 ………… (6.24)

Multiply equation (6.24) by V² / P

When the above equation is rearranged in powers of V

V³ – [\(\frac{RT}{P}\) + b]V² + \(\frac{a}{P}\)V – \(\frac{ab}{P}\) = 0 ………. (6.26)

The equation (6.26) is a cubic equation in V. On solving this equation, we will get three solutions. At the critical point all these three solutions of V are equal to the critical volume V_C. The pressure and temperature becomes P_c and T_c respectively.

i.e., V = V_C
V – V_C = 0
(V – V_C)³ = 0
V³ – 3 V_CV² + 3V_C²V – V_C³ = 0 ………… (6.27)

As equation (6.26) is identical with equation (6.27), we can equate the coefficients of V², V and constant terms in (6.26) and (6.27).

Divide equation (6.30) by equation (6.29)

when equation (6.31) is substituted in (6.29)

substituting the values of V_c and P_c in equation (6.28),

The critical constants can be calculated using the values of van der waals constant of a gas and vice versa.

a = 3 V²_CP_C and b = \(\frac{\mathrm{V}_{\mathrm{C}}}{3}\)

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Deviation form Ideal Gas Behaviour

The kinetic theory of gases (postulates of kinetic theory of gases are described in physics text book) which is the basis for the gas equation (PV=nRT), assumes that the individual gas molecules occupy negligible volume when compared to the total volume of the gas and there is no attractive force between the gas molecules.

Gases whose behaviour is consistent with these assumptions under all conditions are called ideal gases. But in practice both these assumptions are not valid under all conditions. For example, the fact that gases can be liquefied shows that the attractive force exists among molecules. Hence, there is no gas which behaves ideally under all conditions. The non-ideal gases are called real gases. The real gases tend to approach the ideal behaviour under certain conditions.

Compressibility Factor Z

The deviation of real gases from ideal behaviour is measured in terms of a ratio of PV to nRT. This is termed as compressibility factor. Mathematically,

Z = \(\frac{PV}{nRT}\)

For ideal gases PV = nRT, hence the compressibility factor, Z = 1 at all temperatures and pressures. For these gases the plot of Z vs P should be a straight line parallel to the pressure axis. When a gas deviates from ideal behaviour, its Z value deviates from unity.

For all gases, at very low pressures and very high temperature the compressibility factor approaches unity and they tend to behave ideally. The plot of the compressibility factor vs pressure for some common gases are shown in Figure 6.8.

When the pressure is low, the volume of the container is very large compared to the volume of the gas molecules so that individual volume of the gas molecules can be neglected. In addition, the molecule in a gas are far apart and attractive forces are negligible.

As the pressure increases, the density of gas also increases and the molecules are much closer to one another. Hence, the intermolecular force becomes significant enough to affect the motion of the molecules and the gas will not behave ideally.

At high temperatures the average kinetic energy of the molecules is very high and hence inter moleclular attractions will become insignificant. As the temperature decreases, the average kinetic energy of molecules also decreases, hence the molecular attraction is enhanced.

The temperature at which a real gas obeys ideal gas law over an appreciable range of pressure is called Boyle temperature or Boyle point. The Boyle point varies with the nature of the gas. Above the Boyle point, for real gases, Z > 1, ie., the real gases show positive deviation.

Below the Boyle point, the real gases first show a decrease for Z, reaches a minimum and then increases with the increase in pressure. So, it is clear that at low pressure and at high temperature, the real gases behave as ideal gases.

Compressibility Factor for Real Gases

The compressibility factor Z for real gases can be rewritten

Where V_real is the molar volume of the real gas and V_ideal is the molar volume of it when it behaves ideally.

Van der Waals Equation

J. D. Van der Waals made the first mathematical analysis of real gases. His treatment provides us an interpretation of real gas behaviour at the molecular level. He modified the ideal gas equation PV = nRT by introducing two correction factors, namely, pressure correction and volume correction.

Pressure Correction:

The pressure of a gas is directly proportional to the force created by the bombardment of molecules on the walls of the container. The speed of a molecule moving towards the wall of the container is reduced by the attractive forces exerted by its neighbours. Hence, the measured gas pressure is lower than the ideal pressure of the gas. Hence, van der Waals introduced a correction term to this effect.

Van der Waals found out the forces of attraction experienced by a molecule near the wall are directly proportional to the square of the density of the gas.

P α ρ²
ρ = \(\frac{n}{V}\)

where n is the number of moles of gas and V is the volume of the container

where a is proportionality constant and depends on the nature of gas

Therefore,

Volume Correction

As every individual molecule of a gas occupies a certain volume, the actual volume is less than the volume of the container, V. Van der Waals introduced a correction factor V’ to this effect. Let us calculate the correction term by considering gas molecules as spheres.

V = excluded volume
Excluded volume for two molecules
= \(\frac{4}{3}\)π(2r)³
= 8(\(\frac{4}{3}\)πr³) = 8 V_m

where v_m is a volume of a single molecule

Excluded volume for single molecule
\(\frac{8 \mathrm{~V}_{\mathrm{m}}}{2}\) = 4 V_m

Excluded volume for n molecule
= n(4 V_m) = nb

Where b is van der waals constant which is equal to 4 V_m
⇒ V’ = nb
V_total = V – nb ……………… (2)

Replacing the corrected pressure and volume in the ideal gas equation PV=nRT we get the van der Waals equation of state for real gases as below,

(P + \(\frac{\mathrm{an}^{2}}{\mathrm{~V}^{2}}\)) (V – nb) = nRT ………….. (6.22)

The constants a and b are van der Waals constants and their values vary with the nature of the gas. It is an approximate formula for the non-ideal gas.