2.4 The Second Law

2.4 The Second Law#

The first law of thermodynamics addresses energy exchange between a system and it’s surroundings. But, it does not address if a reaction or process will occur under a given set of conditions (\(P\), \(T\), concentration, etc) That is, it does not address if the chemical process is spontaneous. The second law introduces a state function called entropy (\(S\)) to address spontaneity.

Spontaneous processes#

Spontaneous processes are defined as processes that does occur under a specific set of conditions. Unlike in common parlance, spontaneous processes do not indicate “fast” or suddenly occurring reactions. Reaction rates are not addressed in thermodynamics, where all state functions are compared is systems assumed to already be at equilibrium under given set of conditions.

Processes that result in a decrease in the energy of a system often are spontaneous – i.e., these processes reach a most stable lower energy level. In such reactions, heat is given off to the surroundings (exothermic processes) However, endothermic processes can also be spontaneous.

Example: Spontaneous chemical processes

All reactions shown below occur spontaneously at \(\pu{25 ^\circ C}\). It is also evident that first two reactions are exothermic, while the last two reactions are endothermic.

Reaction	\(\Delta_{rxn}H^\circ\), \(\pu{kJ mol-1}\)
\(\ce{CH4(g) + 2 O2(g) -> CO2(g) + 2 H2O(l)}\)	\(-890.4\)
\(\ce{H+ (aq) + OH- (aq) <=> H2O(l) }\)	\(-56.2\)
\(\ce{H2O(s) -> H2O(l)}\)	\(6.01\)
\(\ce{NH4NO3(s) ->[H2O] NH4+ (aq) + NO3- (aq)}\)	\(25.0\)

Because \(\Delta_{rxn}H^\circ\) is not adequate to predict spontaneity of the process, we have to define new terms such as entropy and Gibbs free energy.

Entropy#

Imagine two well-insulated bottles somewhere in the universe (Fig. 31) and that within one side of bottle was filled with a gas and the other side of the bottle contains vacuum, separated by a valve. If this valve were to be removed, you would intuitively know that gas will enter the side under vacuum spontaneously. We have neither added energy to nor taken energy from the system, hence the first law says nothing about this process. Nor did opening the valve cause the reaction. This is apparent from the observation that if we reclose the valve, the gas will not return to the original side. This phenomenon suggests there is something very fundamental and universal about this. This process is addressed in the second law of thermodynamics.

https://openstax.org/apps/archive/20221109.213337/resources/9eafa2a59a63c98a307979ec929c1e6ae7d47898 — Fig. 31 An isolated system consists of an ideal gas in one flask that is connected by a valve to a second flask containing a vacuum. Once the valve is opened, the gas spontaneously becomes evenly distributed between the flasks. Image source: 12.1 Spontaneity - Chemistry: Atoms First | OpenStax#

Entropy (\(S\)) of a system is a measure of how spread out or how dispersed the system’s energy is. The simplest interpretation of this is how spread out a system’s energy is in space. Because the gas molecules that were originally confined to one side of the container are moving, they possess motional energy. When the barrier is removed, the motional energy of molecules will cause the as molecules to spread out and occupy a larger volume. This dispersal of system’s motional energy causes an increase in the system’s entropy. Just as spontaneity is favored by a process being exothermic, spontaneity is also favored by an increase in the system’s entropy. Whether it is the enthalpy change, the entropy change, or both, for a process to be spontaneous, something must favor spontaneity.

What is entropy? - Jeff Phillips | TED-Ed

Entropy is an extensive property (see Table 5) and has the units \(\pu{J K−1}\). Molar entropy is entropy normalized to amount (moles) of a substance (\(\pu{J K-1 mol−1}\)) and is, therefore, an intensive property. Mathematically, \(S\) is described as the heat absorbed (\(q\)) in a reversible reaction that occurred in a closed system divided by absolute temperature (\(T\), an intensive property).

(11)#\[\delta S = \frac{\delta q}{T}\]

Note that \(\delta\) indicates a small change. As seen in the above example, most reactions that occur in nature are irreversible the above equation is rewritten as

(12)#\[\delta S > \frac{\delta q}{T}\]

Standard entropy, \(S^\circ\)#

It is the absolute entropy of a substance at \(\pu{1 atm}\) (they are function of temperature, but standard values are expressed at \(\pu{25 ^\circ C}\). The units of entropy are \(\pu{J K-1 mol−1}\). The \(S^\circ\) of substances (elements and compounds) are always positive (i.e., \(S^\circ > 0\)), even for elements in their standard states.

General trends in entropy (\(S^\circ\))

Entropy (\(S^\circ\)) of selected substances are shown below. Observe the trends in the table.

Substance	\(S,\ \pu{J K-1 mol−1}\)
\(\ce{H2O(l)}\)	\(69.9\)
\(\ce{H2O(g)}\)	\(188.7\)
\(\ce{Na(s)}\)	\(51.05\)
\(\ce{Na(l)}\)	\(57.56\)
\(\ce{Na(g)}\)	\(153.7\)
\(\ce{He(g)}\)	\(126.1\)
\(\ce{Ne(g)}\)	\(146.2\)
\(\ce{C}\) (diamond)	\(2.4\)
\(\ce{C}\) (graphite)	\(5.69\)
\(\ce{O2(g)}\)	\(205.0\)
\(\ce{O3(g)}\)	\(237.6\)
\(\ce{F2(g)}\)	\(203.34\)
\(\ce{Au(s)}\)	\(47.7\)
\(\ce{Hg(l)}\)	\(77.4\)

The table of data shown above highlight the following trends:

For a given substance, is increases in the following order: \(S_\text{gas} \gg S_\text{liquid}> S_\text{solid}\). This can be explained by greater molecular motion in gases compared with liquids, and in liquids compared with solids (see Fig. 32).
For two monatomic species, the one with the larger molar mass has the greater \(S^\circ\). E.g., compare \(\ce{He}\) vs. \(\ce{Ne}\).
For two substances in the same phase, and with similar molar masses, the substance with the more complex molecular structure has the greater \(S^\circ\). The more complex a molecular structure, the more different types of motion the molecule can exhibit. E.g., \(\ce{O2(g)}\) vs. \(\ce{O3(g)}\).
When an element exists in two or more allotropic forms (e.g., graphite vs. diamond), the form in which the atoms are more mobile has the greater entropy. In diamond, the carbon atoms occupy fixed positions in a three-dimensional array. In graphite, although the carbon atoms occupy fixed positions within the two-dimensional sheets, the sheets are free to move with respect to one another, which increases the mobility and, therefore, total number of possible arrangements of atoms within the solid.

https://openstax.org/apps/archive/20221109.213337/resources/f3f96b7f897d7cd062c326b3e451634f6c3d5a20 — Fig. 32 The entropy of a substance increases (\(S^\circ > 0\)) as it transforms from a relatively ordered solid, to a less-ordered liquid, and then to a still less-ordered gas. The entropy decreases (\(S^\circ < 0\)) as the substance transforms from a gas to a liquid and then to a solid. Image source: 12.2 Entropy - Chemistry: Atoms First | OpenStax#

Entropy of a system (\(\Delta _{sys} S^\circ\)) can be calculated similar to mathematical procedure for calculating \(\Delta _{sys} H^\circ\).

(13)#\[\Delta _{sys} S^\circ = \sum \Delta_{prod}S - \sum \Delta_{reac}S\]

Qualitative prediction of \(\Delta _{sys}S^\circ\)#

Equation (13) helps us calculate \(\Delta _{sys}S^\circ\) for any process where \(S^\circ\) of all reactants and products are known. But, we can qualitatively predict the sign of \(\Delta _{sys}S^\circ\) by evaluating the chemical process.

Processes that lead to increase in \(S^\circ\) of a system

Melting - when solids melt, molecules have greater energy and are more mobile. They go from being in fixed positions in the solid, to being free to move about in the liquid.
Vaporization or sublimation - a dramatic increase in energy/mobility, and in the number of possible arrangements of a system’s molecules when the molecules go from a condensed phase to the gas phase.
Temperature increase - the energy of the system’s molecules increases due to increase in their average kinetic energy.
Reaction resulting in a greater number of gas molecules – since entropy of a substance in the gas phase is always significantly greater than its entropy in either the liquid or solid phase, a reaction that results in an increase in the number of gas molecules causes an increase in the system’s entropy.
Dissolving a substance - dissolving causes dispersal of the molecules (and consequently, of the system’s energy) into a larger volume and greater entropy. This rule breaks down for multiple-charged ions in water as these ions attract water molecules (hydration) around their surfaces resulting in lower entropy.

\(S^\circ\) changes in the universe#

Every system we study is part of the universe. We call the environment that exists outside the system as the surroundings. Together, the system and its surroundings make up the universe. The transfer of energy from the system to its surroundings was addressed in the first law. Based on our understanding of entropy, we can also describe entropy changes in the surroundings. To correctly predict if a system is spontaneous, we need to address entropy change in the surroundings as well.

Consider melting of an ice cube at room temperature (endothermic process). Because we can expect ice cube to melt spontaneously, we would expect \(\Delta _{sys} S^\circ >0\) (melting of a solid). Consequent to this melting of ice, the air in the room becomes cooler causing a decrease in entropy of the surrounding (\(\Delta _{surr}S^\circ <0\)). Now consider the opposite process of a cup of boiling water placed at room temperature (exothermic process). Because the water in the cup is cooling, \(\Delta _{sys} S^\circ >0\) and since the surroundings are warmer than before, \(\Delta _{surr}S^\circ >0\).

Now, let’s examine combustion of graphite (\(\ce{C}\)) in the same room as above. This process would result in formation of \(\ce{CO2(g)}\) and heat (exothermic process). Though this reaction is producing a gas (\(\Delta _{sys} S^\circ >0\)), it is an exothermic reaction resulting in \(\Delta _{surr}S^\circ >0\).

Therefore, in exothermic processes, heat transferred from the system to the surroundings increases the temperature of the molecules in the surroundings causing entropy of the surroundings to increase. Conversely, in endothermic processes, heat is transferred from the surroundings to the system, decreasing the entropy of the surroundings. There is a clear correlation between \(\Delta _{sys} H^\circ\), \(\Delta _{surr}S^\circ\), and \(T\) in the system (see Eq. (11)):

(14)#\[\begin{split}\begin{aligned} \Delta _{surr}S^\circ &\propto \Delta _{sys} H^\circ \\ \Delta _{surr}S^\circ &\propto \frac{1}{T} \end{aligned}\end{split}\]

The negative sign refers to an exothermic process. In endothermic processes, the signs would be reversed. Combining, above two expressions, we have

(15)#\[\Delta _{surr}S^\circ = - \frac{\Delta _{sys} H^\circ}{T}\]

Second Law - Definition#

It was clear from the above discussion that both systems and surroundings can undergo changes in entropy during a chemical or physical process. These changes can be summarized as:

(16)#\[\Delta _{universe} S^\circ = \Delta _{sys} S^\circ + \Delta _{surr}S^\circ\]

Second law of thermodynamics defines that increases in a spontaneous process and remains unchanged at equilibrium. This law is summarized in Table 6

Table 6 \(\Delta _{univ} S^\circ\) and spontaneity of processes in systems.#
\(\Delta _{univ} S^\circ\) condition	Reaction condition
\(\Delta _{univ} S^\circ >0\)	Reaction is spontaneous
\(\Delta _{univ} S^\circ =0\)	Reaction is at equilibrium
\(\Delta _{univ} S^\circ <0\)	Reaction is nonspontaneous

Equations (15) and (16) can be combined to focus on the system properties as follows:

(17)#\[\Delta _{universe} S^\circ = \Delta _{sys} S^\circ - \frac{\Delta _{sys} H^\circ}{T}\]

We can now use Eq. (17) to determine if a chemical process occurs spontaneously.