Reversibility infinitesimal changes

It is not difficult to show that the entropy of a closed system in an equilibrium state is an extensive property. Suppose a system of uniform temperature T is divided into two closed subsystems A and B. When a reversible infinitesimal change oeeurs, the entropy changes of the subsystems are d5A = dqp,/T and d e = dq /T and of the system d5 = dqj T. But dq is the sum of d A and dq, which gives d = d A + d B- Thus, the entropy changes are additive, so that entropy must be extensive 5 =Sa+Sb. ... 

A particular path from a given initial state to a given final state is the reversible process, one in which after each infinitesimal step the system is in equilibrium with its surroundings, and one in which an infinitesimal change in the conditions (constraints) would reverse the direction of the change. 

Equation (A2.1.15) involves only state fiinctions, so it applies to any infinitesimal change in state whether the actual process is reversible or not (although, as equation (A2.1.14) suggests, dS is not experimentally accessible unless some reversible path exists). 

Essentially this requirement means that, during die irreversible process, innnediately inside die boundary, i.e. on the system side, the pressure and/or the temperature are only infinitesimally different from that outside, although substantial pressure or temperature gradients may be found outside the vicinity of the boundary. Thus an infinitesimal change in p or T would instantly reverse the direction of the energy flow, i.e. the... 

The coordinates of thermodynamics do not include time, ie, thermodynamics does not predict rates at which processes take place. It is concerned with equihbrium states and with the effects of temperature, pressure, and composition changes on such states. For example, the equiUbrium yield of a chemical reaction can be calculated for given T and P, but not the time required to approach the equihbrium state. It is however tme that the rate at which a system approaches equihbrium depends directly on its displacement from equihbrium. One can therefore imagine a limiting kind of process that occurs at an infinitesimal rate by virtue of never being displaced more than differentially from its equihbrium state. Such a process may be reversed in direction at any time by an infinitesimal change in external conditions, and is therefore said to be reversible. A system undergoing a reversible process traverses equihbrium states characterized by the thermodynamic coordinates. 

Changing the pressure will have a similar effect. If we increase p by dp, the solid melts. This process can be reversed at any time by decreasing the pressure by dp. Note that at p = 1 atm (101.325 kPa), only at T = 273.15 K can the phase change be made to occur reversibly because this is the temperature where solid and liquid are in equilibrium at this pressure. If we tried to freeze liquid water aip— atm and a lower temperature such as 263.15 K, the process, once started, would proceed spontaneously and could not be reversed by an infinitesimal change in p or T. 

To calculate the work of reversible, isothermal expansion of a gas, we have to use calculus, starting at Eq. 3 written for an infinitesimal change in volume, dV ... 

The work done by any system on its surroundings during expansion against a constant pressure is calculated from Eq. 3 for a reversible, isothermal expansion of an ideal gas, the work is calculated from Eq. 4. A reversible process is a process that can be reversed by an infinitesimal change in a variable. 

A thermodynamic process is reversible if an infinitesimal change in an external variable (e.g. pressure) can change the direction in which the process occurs. 

A special case of gaseous expansion, and one that can only be approached but not exactly realized, is that in which the external pressure is adjusted continuously so that it differs only infinitesimally from the pressure of the gas. By an infinitesimal change in the external pressure, the direction can be reversed, hence, the designation reversible. 

For any infinitesimal change in a reversible system, the change in energy may be written... 

It is used internally and rarely enters directly into ealens, but rather in the form of its increments or changes. Entropy is arrived at in thermodynamics in the form of the conception of a change in the entropy of the system, which is equal to the heat taken up during each infinitesimal change of a reversible or isothermal process, divided by the temperature at which it is absorbed. For the entire change in the system, the change in entropy is equal to the summation of the infinitesimal terms as denoted by the equation ... 

A review of the concept of infinitesimal change, and its relevance in chemistry, in view of the links to the concept of reversability in thermodynamics. 

Before using these equations to calculate the work of a reversible process, let s examine the meaning of reversible in this context. In everyday language, a reversible process is one that can take place in either direction. This common usage is refined in science in thermodynamics, a reversible process is one that can be reversed by an infinitesimal change in a variable. For example, if the external pressure exactly matches the pressure of the gas in the system, the piston moves in neither direction. If the external pressure is increased by an infinitesimal amount, the piston moves in. If, instead, the external pressure is reduced by an infinitesimal amount, the piston moves out. 

Expansion against an external pressure that differs by a finite (measurable) amount is an irreversible process in the sense that an infinitesimal change in the external pressure does not reverse the direction of travel of the piston. For instance, if the pressure of the system is 2.0 atm... 

EH We met the concept of a reversible change in Section 6.7 a reversible change is a process that can be reversed by an infinitesimal change in the conditions. 

A thermodynamic process is said to have taken place if a change is observed to have taken place in any macroscopic property of the system. An infinitesimal process is a process in which there is only an infinitesimal change in any macroscopic property of the system. A natural process is an infinitesimal process that occurs spontaneously in real systems an unnatural process is one that cannot occur spontaneously in real systems. Reversible processes are either natural or unnatural processes which can occur in either direction between two states of equilibrium... 

Consider any number of systems that may do work on each other and also transfer heat from one to another by reversible processes. The changes of state may be of any nature, and any type of work may be involved. This collection of systems is isolated from the surroundings by a rigid, adiabatic envelope. We assume first that the temperatures of all the systems between which heat is transferred are the same, because of the requirements for the reversible transfer of heat. For any infinitesimal change that takes place within the isolated system, the change in the value of the entropy function for the ith system is dQJT, where Qt is the heat absorbed by the ith system. The total entropy change is the sum of such quantities over all of the subsystems in the isolated system, so... 

The oscillations of the piston assembly are damped out because the viscous nature of the gas gradually converts gross directed motion of the molecules into chaotic molecular motion. This dissipative process transforms some of the work initially done by the gas in accelerating the piston back into internal energy of the gas. Once the process is initiated, no infinitesimal change in external conditions can reverse its direction the process is irreversible. 

The development of thermodynamics is facilitated by the introduction of a special kind of nonflow process characterized as reversible. A process is reversible when its direction can be reversed at any point by an infinitesimal change in external conditions. 

No infinitesimal change in external conditions can reverse process direction. 

In thermodynamics, a reversible process is one that can be reversed by infinitesimal changes in some property of the system without loss of energy. In chemistry, this normally means a transition (e.g. chemical reaction or phase transformation) from some initial state to some final state. If, after transitioning to the final state, the process is... 

The engines being reversible, their direction of working can be reversed by only infinitesimal change in the constraints. Thus, engine II will work taking up Q2 wao mX of heat from reservoir at T2 and discharging amount of heat to the reservoir at Tj provided W2 work... 

This is an example of a real process and it is not a reversible process since infinitesimal change in constraints cannot bring back the previous state of the system. How to calculate entropy change in such a process ... 

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