§ 6 Entropy
Usually, any process in which the system moves from one state to
another, occurs in a way that can not be done this process in the
opposite direction so that the system passed through the same
intermediate states, while in others the bodies are not any changes.
This is due to the fact that in the process of the energy is
dissipated, for example, due to friction, radiation, etc. Thus. almost
all processes in nature are irreversible. In any process of the energy
is lost. To characterize the energy dissipation is introduced the
concept of entropy. (The entropy characterizes the thermal state of the
system and determines the probability of a given state of the body.
The more likely this condition, the greater the entropy.) All natural
processes are accompanied by an increase in entropy. Entropy is
constant only in the case of an idealized reversible process that occurs
in a closed system, ie a system in which there is an exchange of
energy with the external to this system bodies.
Thermodynamic entropy and its meaning:
- is a function of the system, an infinitesimal change in a reversible
process which is the ratio of the infinitesimal amount of heat
introduced into the process, to the temperature at which it was
All in a reversible process the entropy change can be calculated by the formula:
where the integration is one of the initial state 1 of the system to the final state 2.
Since entropy is a state function, the integral is
the property of its independence from the shape of the contour (path),
on which it is calculated, so the integral is determined only by the
initial and final states of the system.
In any reversible process the change in entropy is equal to 0
2) In thermodynamics, it is proved that the system undergoes irreversible S cycle increases
ΔS> 0 (2)
Equations (1) and (2) apply only to closed systems, but if the system is exchanged heat with the environment, its S can behave in any way.
Of (1) and (2) can be written as the Clausius inequality
ΔS ≥ 0
entropy of a closed system can either increase (in the case of
irreversible processes) or remain constant (in the case of reversible
If the system performs the equilibrium transition from state 1 to state 2, the entropy change
where dU and δA is recorded for a particular process. According to this formula ΔS
is determined up to an additive constant. The physical meaning is not
the entropy, entropy difference. We find the entropy change in the
process of an ideal gas.
ie changes in the entropy S ΔS1 → 2 ideal gas at its transition from state 1 to state 2 is independent of the process.
Because for an adiabatic process δQ = 0, ΔS = 0 => S = const, ie adiabatic reversible process takes place at constant entropy. So it is called isentropic.
An isothermal process (T = const; T1 = T2: )
When isochoric process (V = const; V1 =V2; )
Entropy is additive: the entropy of the system is the sum of the entropies of bodies in the system. S = S1 + S2 + S3
+ ... Qualitative difference of the thermal motion of molecules from
other forms of movement is its state of chaos, randomness. Therefore,
to characterize the thermal motion to introduce a quantitative measure
of the degree of molecular disorder. If we consider any given
macroscopic state of the body with some average values, then it is
nothing but the continuous change of close microstates differing
distribution of molecules in different parts of the volume and the
energy is distributed between the molecules. These continuous
successive microstates characterizes the degree of disorder of the
macroscopic state of the entire system, ϖ is called thermodynamic
probability of the microstate. Thermodynamic probability ϖ of the
system - is the number of ways that this can be done state macroscopic
system, or the number of microstates carrying this microstate (ϖ ≥ 1,
and the mathematical probability of ≤ 1).
As a measure of unexpected events have agreed to take the logarithm
of its probability, the negative of: unexpected state is = - ln ? ω
According to Boltzmann, the entropy S of the system and the thermodynamic probability linked as follows:
where k - Boltzmann constant, . Thus, the entropy
is defined logarithm of the state in which this can be achieved
microstate. Entropy can be considered as a measure of the probability
of the state thermodynamic system. Boltzmann formula allows us to give
the following statistical interpretation of entropy. Entropy is a
measure of disorder in a system. In fact, the greater the number of
microstates realizing this microstate, the greater the entropy. In the
equilibrium state of the system - the most probable state of the system -
the maximum number of microstates, with the maximum and entropy.
real processes are irreversible, it can be argued that all the
processes in a closed system leads to an increase in its entropy - the
principle of entropy increase. In the statistical interpretation
of entropy, this means that the process in a closed system are to
increase the number of microstates, in other words, from less probable
to more probable, as long as the probability of the state will not be
§7 The second law of thermodynamics
The first law of thermodynamics, expressing energy
conservation and transformation of energy, does not establish the
direction of the flow thermodynamic processes. You can also submit a
set of processes that do not contradict the beginning I law
thermodynamic, in which energy is conserved, but in nature they are not
implemented. Possible formulation of the second law of thermodynamics:
1) the law of increasing entropy of a closed system in irreversible
processes: any irreversible process in a closed system is such that the
entropy of the system is on the increase ΔS ≥ 0 (irreversible)
2) ΔS ≥ 0 (S = 0 for a reversible and ΔS ≥ 0 for an irreversible process)
The processes that take place in a closed system, entropy does not decrease.
2) From the Boltzmann S = k ln ω > 0,
and consequently, an increase in the entropy of the system means the
transition from a less probable to a more probable state.
According to Kelvin: circular process is not possible, the only result
of which is the conversion of heat received from the heater into an
In the Clausius: circular process is not possible, the only result is
to transfer heat from the less heated body to a warmer.
To describe the
thermodynamic systems at 0 K using Theorem Nernst-Planck (third
thermodynamics): the entropy of all bodies in equilibrium tends to zero
as the temperature approaches 0 K
From Theorem Nernst-Planck equation, it follows that Cp = Cv = 0 at 0 K.
§ 8 Thermal and refrigerators.
Carnot cycle and its efficiency
From the wording of the second law of thermodynamics on Kelvin
that a perpetual motion machine of the second kind is impossible.
(Perpetual motion - this batch engine does work by cooling a heat
Thermostat - is thermodynamic system that can exchange heat with the bodies without a change in temperature.
The principle of operation of the heat engine: the thermostat with temperature T1 - heater for a cycle is subtracted the amount of heat Q1, a thermostat with temperature T2 (T2 < T1), refrigerator, for a series of heat transferred to Q2, while work is done A = Q1 - Q2
process or cycle is a process by which the system is going through a
number of states, is reset. Cycle in the state diagram depicted a
closed curve. Cycle, performed by an ideal gas can be divided into
processes of expansion (1-2) and compression (2-1), the work of expansion is positive A1-2 > 0, because V2> V1, the work of compression is negative A1-2 < 0, because V2 < V1.
Hence, the work done by the gas per cycle, determined by the area
covered by a closed curve 1-2-1. If the cycle is done positive work loop clockwise), then the cycle is called direct if - reverse cycle (cycle occurs in a counter-clockwise).
Direct cycle is used in heat engines - periodically a motor does the
work by producing heat from the outside. Reverse cycle is used in
refrigerators - periodically existing installations, in which through
the work of external forces, heat is transferred to the body with a
As a result of the circular process, the system returns to its initial
state, and therefore, the total change in internal energy is zero. Then
start the I law of thermodynamics for a circular process
that is, the work done per cycle is the amount of heat received from the outside, but
Q = Q1 - Q2
Q1 - the amount of heat received by the system
Q2 - the amount of heat given system.
Thermal efficiency for cyclic process is the ratio of the work done by the system, to the amount of heat supplied to the system:
To η = 1, the condition Q2 = 0, ie, heat engine should have one heat source Q1, but this contradicts the second law of thermodynamics.
The reverse process is happening in the heat engine is used in the refrigeration machine.
The thermostat with temperature Т2 deducted the amount of heat Q2 transferred to the thermostat and the temperature T1, the amount of heat Q1.
Q = Q2 - Q1 < 0 so A < 0.
Without doing the work can not take away the heat from a hot body, and at least give it a warmer.
Based on the second law of thermodynamics, Carnot led theorem.
Carnot's theorem: From time to time all heat engines operating with the same heater temperature (T1) and refrigerators (T2), the highest efficiency have a reversible machine. Efficiency reversible machine with equal T1 and T2 are equal and do not depend on the nature of the working fluid.
Working body - the body to make circular process to exchange energy to other bodies.
Carnot cycle - the most economical reversible cycle consisting of 2 isotherms and 2 adiabats.
1-2-isothermal expansion at T1 heater, gas is supplied to the heat Q1 and work is done
2 - 3 - adiabats. expansion, the gas does work A2-3 > 0 on external bodies.
3 - 4-isothermal compression at T2 refrigerator, heat is taken Q2 and work is done
4-1-adiabatic compression work is done on the gas A4-1 < 0 external bodies.
An isothermal process U = const, so Q1 = A12
Adiabatic expansion Q2-3 = 0, and the work done A23 gas by the internal energy A23 = -U
The amount of heat Q2, uploaded from gas to refrigerator at isothermal compression equal to the work of compression А3-4
The work of the adiabatic compression
The work performed by a circular process
and equal to the area of ??the curve 1-2-3-4-1.
Thermal efficiency Carnot cycle
From the equation for adiabatic processes 2-3 and 3-4 we obtain
efficiency Carnot cycle is determined only by the temperature heater
and refrigerator. To increase the efficiency necessary to increase the
difference Т1 - Т2.