Entropy change reversible temperature changes

The electric polarization P and the entropy S of the crystal depend on the same quantities AT. A reversible temperature change AT implies an entropy... 

The entropy increase A5 i.2 by a reversible temperature change from T to T2 is now calculated from (4.14)... 

The second law of thermodynamics also consists of two parts. The first part is used to define a new thermodynamic variable called entropy, denoted by S. Entropy is the measure of a system s energy that is unavailable for work.The first part of the second law says that if a reversible process i f takes place in a system, then the entropy change of the system can be found by adding up the heat added to the system divided by the absolute temperature of the system when each small amount of heat is added ... 

It must be emphasised that the heat q which appears in the definition of entropy (equation 20.137) is always that absorbed (or evolved) when the process is conducted reversibly. If the process is conducted irreversibly and the heat absorbed is q, then q will be less than q, and q/T will be less than AS the entropy change (equation 20.137). It follows that if an irreversible process takes place between the temperatures Tj and 7 , and has the same heat intake q at the higher temperature 7 2 as the corresponding reversible process, the efficiency of the former must be less than that of the latter, i.e. 

The equation (16) shows that the increase of bound energy in a reversible isothermal change is equal to the increase of entropy multiplied by the absolute temperature, so that the entropy may be regarded as the capacity for bound energy in such changes. B will evidently contain the arbitrary term / IT. 

To derive an expression for the change in entropy when a system is heated, we first note that Eq. 1 applies only when the temperature remains constant as heat is supplied to a system. Except in special cases, that can be true only for infinitesimal transfers of heat so we have to break down the calculation into an infinite number of infinitesimal steps, with each step taking place at a constant but slightly different temperature, and then add together the infinitesimal entropy changes for all the steps. To do this is we use calculus. For an infinitesimal reversible transfer dgrev at the temperature T, the increase in entropy is also infinitesimal and, instead of Eq. 1, we write... 

A change in entropy is equal to the heat supplied reversibly to a system divided by the temperature at which the transfer takes place, entropy of vaporization (AS ) The entropy change per mole accompanying vaporization (the conversion of a substance from the liquid state into the vapor state). 

In most processes, a reversible absorption of heat is accompanied by a change in temperature, and a calculation of the corresponding entropy change requires an evaluation of the integral of q/T. The term q is related to the heat capacity of the system which is usually expressed as a function of temperature. In a constant volume process, for example, the entropy change is... 

At a constant pressure, the entropy of any pure substance can be calculated for any temperature through the use of the procedure that is herein being described. The entropy change taking place during an isothermal reversible process, it may be recalled, is equal to the heat change involved divided by the absolute temperature ... 

Because the entropy change for the H2/O2 reaction is negative, the reversible potential of the H2/O2 fuel cell decreases with an increase in temperature by 0.84 mV/°C (assuming reaction product is liquid water). For the same reaction, the volume change is negative therefore, the reversible potential increases with an increase in pressure. 

If an isobaric temperature change is carried out reversibly, the heat exchanged in the process can be substituted into the expression for the entropy change, and the equations at constant pressure when no work is performed other than PV work are... 

The first consists of two steps (1) an isothermal reversible expansion at the temperature Ta until the volume V is reached, and (2) an adiabatic reversible expansion from V to Vj,. The entropy change for the gas is given by the sum of the entropy changes for the two steps ... 

A reversible adiabatic expansion of an ideal gas has a zero entropy change, and an irreversible adiabatic expansion of the same gas from the same initial state to the same final volume has a positive entropy change. This statement may seem to be inconsistent with the statement that 5 is a thermodynamic property. The resolution of the discrepancy is that the two changes do not constitute the same change of state the final temperature of the reversible adiabatic expansion is lower than the final temperature of the irreversible adiabatic expansion (as in path 2 in Fig. 6.7). 

To calculate the change in entropy in this irreversible flow, it is necessary to consider a corresponding reversible process. One process would be to allow an ideal gas to absorb reversibly the quantity of heat Q at the temperature T2. The gas then can be expanded adiabatically and reversibly (therefore with no change in entropy) until it reaches the temperature Ti. At Ti the gas is compressed reversibly and evolves the quantity of heat Q. During this reversible process, the reservoir at T2 loses heat and undergoes the entropy change... 

The formation of water from gaseous hydrogen and oxygen is a spontaneous reaction at room temperature, although its rate may be unobservably small in the absence of a catalyst. At 298.15 K, the heat of the irreversible reaction at constant pressure is — 285,830 J mol . To calculate the entropy change, we must carry out the same transformation reversibly, which can be performed electrochemicaUy with a suitable set of electrodes. Under reversible conditions, the heat of reaction for Equation (6.99) is —48,647 J mol. Hence, for the irreversible or reversible change... 

The heat absorbed by the surrounding reservoir during the irreversible reaction is 285,830 J, and this heat produces the same change in state of the reservoir as the absorption of an equal amount of heat supplied reversibly. If the surrounding reservoir is large enough to keep the temperature essentially constant, its entropy change is... 

In the fourth step, which is adiabatic and reversible, no entropy change occurs, but the temperature increases to the initial value T2- As the process is cyclic... 

The thermodynamic definition of entropy says that the change in entropy dS in a process carried out reversibly is the heat absorbed in the process d Qrev divided by the temperature... 

Here the reversible formal potential, E° of the redox couple of interest has been measured at a series of temperatures versus a reference electrode maintained at constant temperature. Values of AShr have been interpreted in terms of differences in water-ligand hydrogen bonding [1] and have been compared to the entropy change for spin crossover in iron complexes [2]. 

Assuming the heat capacity of an ideal gas is constant with temperature, calculate the entropy change associated with lowering the temperature of 1.47 mol of monatomic ideal gas reversibly from 99.32°C to — 78.54°C at (a) constant pressure and (b) constant volume. 

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