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The thermodynamic entropy S, often simply called the entropy in the context of chemistry and thermodynamics, is a measure of the amount of energy in a physical system which cannot be used to do work. It is also a measure of the disorder present in a system. The SI unit of entropy is joule per kelvin, or J/K or JK-1 - which is the same unit as heat capacity.
The concept of entropy was introduced in 1865 by Rudolf Clausius. He defined the change in entropy of a thermodynamic system, during a reversible process in which an amount of heat ΔQ is applied at constant absolute temperature T, as
Clausius' identification of S as a significant quantity was motivated by the study of reversible and irreversible thermodynamic transformations. In the next few sections, we will examine the steps leading to this identification, and its consequences for thermodynamics.
A thermodynamic transformation is a change in a system's thermodynamic properties, such as its temperature and volume. A transformation is said to be reversible if, at each successive step of the transformation, the system is infinitesimally close to equilibrium; otherwise, the transformation is said to be irreversible. As an example, consider a gas enclosed in a piston chamber, whose volume may be changed by moving the piston. A reversible volume change is one that takes place so slowly that the density of the gas always remains homogeneous. An irreversible volume change is one that takes place so quickly that pressure waves are created within the gas, which is a state of disequilibrium. Reversible processes are sometimes referred to as quasi-static processes.
A heat engine is a thermodynamic system that can undergo a sequence of transformations which ultimately return it to its original state. This sequence is called a cycle. During some transformations, the engine may exchange heat with large systems known as heat reservoirs, which have a fixed temperature and can absorb or provide an arbitrary amount of heat. The net result of a cycle is (i) work done by the system (which may be negative, which is the same as positive work done on the system), and (ii) heat transferred between the heat reservoirs. By the conservation of energy, the heat lost by the heat reservoirs is exactly equal to the work done by the engine plus the heat gained by the heat reservoirs. (See cyclic processA cyclic process is a thermodynamic process which begins from and finishes at the same thermostatic state. It is a closed loop on a P-V diagram. The area enclosed by the loop is the work done by the process: :. This work is equal to the balance of heat tr.)
If every transformation in the cycle is reversible, the cycle is reversible. This means that it can be run in reverse, i.e. the heat transfers occur in the opposite direction and the amount of work done switches sign. The simplest reversible cycle is a Carnot cycleA heat engine is an engine that uses heat to produce mechanical work by carrying a working substance through a cyclic process. The Carnot heat engine uses a particular thermodynamic cycle studied by Nicolas Leonard Sadi Carnot in the 1820s. The Carnot cyc, which exchanges heat with two heat reservoirs.
In thermodynamics, absolute temperature is defined in the following way. Suppose we have two heat reservoirs. If a Carnot cycle absorbs an amount of heat Q from the first reservoir and delivers an amount of heat Q′ to the second, then the respective temperatures T and T′ are given by
Now consider a cycle of an arbitrary heat engine, during which the system exchanges heats Q1, Q2, ..., QN with a sequence of N heat reservoirs that have temperatures T1, ..., TN. We take each Q to be positive if it represents heat received by the system, and negative if it represents heat emitted by the system. We will show that
where the equality sign holds if the cycle is reversible.
To prove this, we introduce an additional heat reservoir at some arbitrary temperature T0, as well as N Carnot cycles that have the following property: the j-th such cycle operates between the T0 reservoir and the Tj reservoir, transferring heat Qj to the latter. From the above definition of temperature, this means that the heat extracted from the T0 reservoir by the j-th cycle is
We now consider one cycle of our arbitrary heat engine, accompanied by one cycle of each of the N Carnot cycles. At the end of this process, each of the reservoirs T1, ..., TN have no net heat loss, since the heat extracted by the heat engine is replaced by one of the Carnot cycles. The net result is (i) an unspecified amount of work done by the heat engine, and (ii) a total amount of heat extracted from the T0 reservoir, equal to
If this quantity is positive, this process would function as a perpetual motion machine of the second kind. The second law of thermodynamicsIn physics, the second law of thermodynamics in its many forms, is a statement about the quality and direction of energy flow, and it is closely related to the concept of entropy. This law and its derivatives, such as the law of friction, define the arrow states that this is extremely improbable, so
as claimed. It is easy to show that the equality holds if the engine is reversible, by repeating the above argument for the reverse cycle.
It is important to note that we have used Tj to refer to the temperature of each heat reservoir with which the system comes into contact, not the temperature of the system itself. If the cycle is not reversible, then heat always flows from higher temperatures to lower temperatures, so that
where T is the temperature of the system while it is in thermal contact with the heat reservoir.
However, if the cycle is reversible, the system is always infinitesimally close to equilibrium, so its temperature must be equal to any reservoir with which it is contact. In that case, we may replace each Tj with T. In the limiting case of a reversible cycle consisting of a continuous sequence of transformations,
where the integral is taken over the entire cycle, and T is the temperature of the system at each step.
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