Entropy - a state function

In order to account for spontaneity or directionality of processes, the concept of entropy is defined and incorporated into what is known as the second law of thermodynamics. Roughly speaking, entropy (symbolized S) is a quantitative measure of the number of ways that energy can be distributed within a system. Entropy can be defined and measured as a thermodynamic quantity. Furthermore, it can be shown that entropy is a state function. These features of entropy are essential to the formulation of the second law. Spontaneous processes such as the attainment of thermal equilibrium or the mixing of gases or liquids can be shown to result in an increase in total entropy, meaning the entropy of the system plus the entropy of the surroundings (S_univ). Indeed, the statement that in any spontaneous process, ΔS_univ > 0 can be taken as a statement of the second law of thermodynamics.

Understanding entropy

What is entropy? Is entropy "a measure of disorder in a system"? Is the seeming tendency of your room to get messy an example of the inexorable increase in entropy? What is a microstate? Find out the answers to these important questions and many more at this highly recommended Entropy website by Professor Frank Lambert of Occidental College.

Qualitative assessment of entropy changes

We can develop some general rules of thumb for qualitative assessment of entropy changes for various types of processes. First of all, any system that undergoes a process that increases its temperature also increases in entropy; i.e. ΔS_sys > 0. This is evident from the classical definition of entropy, since q is positive for any real system undergoing a process in which ΔT_sys > 0. (Note: In this case, where T is not constant, the classical definition becomes
dS = dq_rev/ T, which must be integrated in order to obtain ΔS_sys for the process. The integral is always non-negative for dq positive, i.e. heat passing into the system from the surroundings. The quantities dS and dq_rev are called differentials, and represent infinitesimal changes in the quantities S and q_rev. The differential dq_rev, an infinitesimal amount of heat transfer, requires further description, which is given below.) For a phase change, such as the melting of ice to liquid water, or evaporation of water to water vapor, the change from the more to less condensed phase is always associated with ΔS > 0. For a given substance, the relative change in entropy for the evaporation of a liquid is much greater than that for the melting of the solid. In chemical reactions, the change from more numerous simpler, smaller molecules to fewer, larger and more complex molecules generally represents a decrease in entropy. (Note that this is in spite of the fact that a larger molecule has greater entropy than a small one. The kinetic energy in a system of molecules can be more effectively dispersed by an increase in the number of particles with translational freedom than it can by the increased number of vibrational modes within a coherent collection of the same particles linked by chemical bonds.) In reactions where all reactants and products are gases, the entropy change will be positive in the direction of the reaction that produces more total moles of gas. In reactions occurring in solution, or in reactions of heterogeneous systems, production of a gas will make a significant positive contribution to the overall entropy change for the reaction. If a precipitation reaction occurs, this generally makes a a negative contribution to the entropy of reaction, although in this case we need to be careful to consider the entropic contribution of water molecules released when precipitating ions are desolvated.

Engines and entropy

The history of the development of the steam engine is not only part of the story about the dawn of the industrial age, but is also intimately associated with the development of classical thermodynamics. The theoretical study of engines led to the definition and measurement of entropy in terms of heat and temperature. The early steam engines were quite inefficient, and there were many attempts to improve them. By the latter half of the 1700s, the Scottish inventor James Watt had made much progress, but even Watt's best engines had an efficiency of only about 5%. Efficiency is defined for an engine by the ratio of amount of input energy - here heat from a fire - to the work output. For a perfect engine, that ratio would be one, or an efficiency of 100%. Now, the first law of thermodynamics says you can't get something for nothing - no engine could ever exceed 100% efficiency - but might someday an engine approaching 100% efficiency be built? In 1824, a Frenchman, Sadi Carnot, published his opus, Reflections on the Motive Power of Fire, that answered this question with a resounding no. Carnot brilliantly showed that the efficiency of even an idealized engine could be no greater than the ratio (T_h – T_c) / T_h, where T_h is the temperature of the hot reservoir, and T_c the temperature of the cold reservoir.

Under these conditions, the efficiency can be computed according to the first law:

ΔU  =  q_tot +  w_tot =  q_h + q_c + w_tot =  0       or:   – w_tot = q_h + q_c

This says that the total work that can be done by the engine is equal to the heat absorbed from the hot reservoir minus that exhausted to the cold reservoir. This means that efficiency, e, is given by

e = – w_tot/ q_h =  (q_h + q_c) / q_h

Since the two isotherms are joined by two adiabats, it can be shown that

(V_D / V_C) = (V_A/ V_B), so that

– w_tot = q_h + q_c = RT_hln(V_B/ V_A)  –  RT_cln(V_B/ V_A) = Rln(V_B / V_A){T_h – T_c}

Thus, efficiency can be expressed as

e = – w_tot/ q_h =  {Rln(V_B/ V_A)}{T_h – T_c}/ RT_hln(V_B/ V_A) = (T_h – T_c) / T_h.

Exercise: Show that equating the two above expressions for efficiency e of a heat engine leads to the relation

{ q_h / T_h } + { q_c / T_c} = 0

for the Carnot cycle, suggesting that the quantity q_rev/T provides the basis for the definition of a state function (where q_rev is the heat transferred to the system along a reversible path.

Clausius was the first to define entropy in the classical sense, which he did after realizing what we showed in the exercise above, and he defined entropy as ΔS = q_rev/T (or the integral of dq_rev/T for a process in which temperature is changing). This definition is a really a way to compute entropy for an isothermal process, since q_rev/T - the heat transferred to the system in a reversible process divided by the temperature at which it occurs - is a state function, whereas q is not. But no matter how we carry out a change in state, ΔS is the same for the same change in state, or (equivalently) ΔS is zero for a cyclic process. In all irreversible (= not reversible = spontaneous) processes, q is not a state function, and is path-dependent, and we can show that ΔS_univ > 0. Mathematically, this means that the integral of dq is not Δq = q(final) - q(initial). Actually, we never write Δq for just this reason. The integral of dq is also path dependent , and the mathematicians have a more succinct way of describing this by saying dq is not an exact differential (and the notation δq is often used to signify this, whereas dU, dH, dS, etc. are all exact differentials).