FIRST PRINCIPLE OF THERMODYNAMICS

Thermodynamics is a branch of physics which studies the thermal properties of macroscopic systems without explicitly considering the microscopic structure of matter. The behavior of the systems is described by macroscopic concepts such as those of temperature, pressure and heat. Thermodynamics is based on several principles, which are a generalization of numerous observations and experiments. Although thermodynamics was developed before the microscopic nature of matter was well understood, the principles of thermodynamics are ultimately explained by a statistical treatment of the random motion of atoms and molecules.

INTERNAL ENERGY AND WORK OF MANY-PARTICLE SYSTEM

Any thermal system is composed of large number of particles. The energy transfer between the system and its surroundings is given by the work-energy relation

.

If we choose C-frame of reference, the orbital kinetic energy becomes zero and we have U = W ext, where U = Eint is the internal energy and Wext is the work of external forces on the system. That is, the increment of the internal energy is equal to the work done by the external forces on the system. There are various mechanisms of energy exchange between the system and its surroundings. In this connection, it was found convenient to decompose Wext into two parts, called mechanical work or simply work and heat.

WORK

The energy exchange between a system of many particles with its surroundings is called work when it is associated with a collective displacement under the action of a macroscopic force. By convention, the work done by the system on its surroundings is usually taken as positive and denoted by W. For example, when a gas in a cylinder expands against the piston (Fig. 1), it does work. The force on the piston of area A is F = pA. The infinitesimal work dW done by the gas to move the piston a distance dx is where dV = Adx is the infinitesimal change in volume of the gas. For a finite change from V1 to V2 the work done by the gas is

This equation is valid for the work in any change of the volume of gas. To compute the integral we need the relation between p and V (shown in the p - V diagram on Fig. 1). The work done by the gas is given by the area under the curve AB in the p - V diagram. This result shows that the work depends also on the type of process. We shall calculate W for some special transformations.

Fig. 1

The process in which the system returns to its initial state is called cycle (Fig. 2). The work in a cycle is determined by . In the p - V diagram the work is given by the area enclosed within the curve representing the cycle. The sign of W depends on the direction of the cycle. If the cycle is realized in the clockwise direction, the work done by the gas is positive and vice versa.

Fig. 2

In the case of an isobaric process, p = const,

From the ideal gas law we have so that . The work in isobaric process is proportional to the change in temperature of the ideal gas.
For an isothermal transformation of an ideal gas we have:

and substituting this result for p, we find

HEAT

There exist some energy exchanges between the system and its surroundings which are not associated with a collective displacement and are, therefore, not associated with the mechanical work. This is the case of energy transfer between the system and another body which is hotter or colder than the system. In this case, the change of the internal energy is a result of interactions between the particles of systems put into contact. The molecules, which are in random motion, collide each other. At each collision, a small amount of energy is exchanged. Energy is transferred also at distance by emission and absorption of electromagnetic radiation. The concept of heat is referred to the energy transfer due to the microscopic exchanges of energy in the random collisions between the molecules of the system and its surroundings, without any collective displacement. The amount of heat Q is defined as the net energy transferred in this way to the system. By convention, it is positive when the heat is absorbed by the system and it is negative when the heat is given off by the system. When heat is neither absorbed nor given off, the transformation is called adiabatic.
When the temperature of two systems is the same (that is, they are in thermal equilibrium), there is no heat transfer between them. That means that, on the average, the same amount of energy is transferred in one direction as in the other. When the temperature is different, heat is transferred from the system at higher temperature to the system at lower temperature.

FIRST LAW OF THERMODYNAMICS

Using the notions of work and heat, we may rewrite the work-energy relation in the form:

U = Q - W     (1)

That is, the increment of internal energy of a system is equal to the heat absorbed minus the work done by the system on its surroundings.
This statement, called the first law of thermodynamics, is the law of conservation of energy applied to many-particle systems.
In order to account the energy exchange by a radiation transfer we may write

U = Q - W +R     (2)

where R is the radiation energy absorbed by the system. This is a more general form of the first law of thermodynamics.
The internal energy depends only on the state of the system and it does not depend on the type of the process. We have seen that the work depends on the process. Now, writing the first law of thermodynamics in the form

Q = U + W     (3)

we see that the heat also depends on the type of the process because it is related to W. We shall apply the first law of thermodynamics to various transformations.

CYCLE

When the transformation is cyclic, the initial and final states are the same and the internal energy does not vary, U = 0. Then (Eq. 1) becomes:

Q = W     (4)

Thus, the work done by the system in a cyclic process is equal to the heat absorbed by the system.

For an adiabatic transformation, by definition, Q = 0, and the first law of thermodynamics takes the form

U = - W     (5)

That is, the internal energy decreases by an amount equal to the work done by the system. Since the internal energy of an ideal gas depends only on the temperature, when the gas expands (the work is positive), its temperature decreases and vice versa. It can be shown that in a quaei-static adiabatic process the pressure and volume of ideal gas are related by the equation:

pV = const (m fixed)     (6)

where = Cp/Cv - The molar heat capacities Cp and Cv are considered in the next section.

ISOCHORIC PROCESS

In the case of isochoric transformation V= 0, and, therefore W= 0. Then the first law becomes

U = Q     (7)

When no work is done, the change in internal energy is equal to the heat added to the system.

For a radiative process, when energy is transferred only by emission and absorption of radiation,

U = R     (8)

HEAT CAPACITY

It was found experimentally that the heat required to produce a given rise of temperature of a given body is proportional to the mass of a body and to the change of its temperature. If the body contains nm moles of substance, we may write:

dQ = CnmdT     (9)

where C is the molar heat capacity of the substance. It is defined as the amount of heat required to rise the temperature of one mole of the substance by one degree. That is,

(10)

Since the heat depends on the process, the heat capacity also depends on the process of heating. The most important heat capacities are the heat capacity at constant pressure, Cp, and the heat capacity at constant volume, Cv, defined by

(11, 12)

In the case of isochoric process Q = U so that:

(13)

In the case of isobaric process:

(14)

For ideal monatomic gas we use the expression for the internal energy U = 3/2 nm.R.T and the expression for the work in isobaric process W = nm.R.T. Thus we obtain:
(15, 16)

 SECOND PRINCIPLE OF THERMODYNAMICS ENTROPY. IRREVERSIBLE AND REVERSIBLE PROCESSES.     Consider a cycle composed of compression and expansion of a gas in a cylinder with piston (Fig. 1). In the initial state (a) the gas is at equilibrium. If we move the weight to the top of the piston, the gas is rapidly compressed as shown in (b). We may expand the gas to restore the initial state. If the weight is removed the external pressure drops and the gas expands up to the initial position on the piston. The gas approaching equilibrium returns to its initial state, as shown in (c). The cyclic process is completed. However, the weight is now at the bottom. That is, in a cycle consisted of non-equilibrium processes, the system returns to its initial state, but a permanent change is produced in the surroundings. The system cannot be restored to its original state without also making some change in the surroundings. We say that such process is irreversible.     We consider now a cycle composed of equilibrium compression and expansion of the same system (Fig. 2). Firstly we move one small weight to the top of the piston shown in (a). The external pressure increases slightly and the equilibrium of the gas is slightly disturbed. The piston compresses the gas until equilibrium is restored. Then we may repeat the process a number of times by moving the next weight until the piston approaches the lowest position, shown in (b). To restore the gas to the initial state, we have to place back, in the reverse order, the same weights until the cycle is completed leaving negligibly small change in the surroundings (c). That is, it is possible to restore the system to its original state making some infinitesimally small change in the surroundings. Such process is called reversible. Fig. 1                     Fig. 2 DEFINITION OF ENTROPY     If a system, at temperature T, absorbs heat dQ during an infinitesimal reversible transformation, the entropy of the system is defined as a quantity whose change is determined by: For a finite reversible transformation     When the transformation is irreversible, it may be replaced by a reversible transformation, which connects the initial and the final state. It is important, that the change in entropy is independent on the type of the reversible process. In the case of a cycle, we may decompose a cyclic transformation into two transformations between points 1 and 2 and write That is, in a cycle, the change in entropy of a system is zero dQ = TdS and therefore: For a cycle U = Q - W = 0 and where the integral depends on the particular reversible transformation. THE ENTROPY OF AN IDEAL GAS     To calculate the entropy of ideal gas we start from the definition and apply the first law of thermodynamics Using the formulae for the internal energy U = 3/2nmRT and the ideal gas law pV= nmRT, we obtain: SECOND LAW OF THERMODYNAMICS     There exist some processes that are unidirectional. The irreversible processes are always unidirectional: heat is transferred from a hot body to a colder one, gases expand into vacuum etc. The reverse of any of these processes is not observed although that it would not violate any of the known conservation laws. In the case of a heat transfer the change in entropy of the system consisted of two bodies is: If T1 < T2, then Q1 = - Q2 > 0 and     In the general case of processes in isolated system, it was found that the total entropy of the system remains constant if the processes are reversible or increases if the processes are irreversible, or     This is the second law of thermodynamics. Processes which occur in an isolated system are those in which energy is conserved and entropy increases or remain constant.