Spontaneous Processes, Entropy and the Second Law of Thermodynamics

• Although a reaction may be energetically favorable (i.e. products have lower energy than reactants), reaction rates are largely determined by the magnitude of the activation energy barrier
• For example, molecules require a minimum kinetic energy to collide and react
• The higher the activation energy barrier, the fewer the molecules in the population with enough energy to overcome the barrier
• At equilibrium, opposing reaction rates (e.g. forward and reverse reaction rates) are equal in magnitude
• Reaction rates are related to the product of an intrinsic rate constant and the concentration of reactant(s)
• If equilibrium is related to reaction rates and reaction rates are related to energy, then equilibrium must be related to energy in some way
• We have used the magnitude and sign of reaction enthalpies to predict whether a reaction will proceed or not
• Reactions with negative enthalpies indicate heat release; the products are lower in energy than the reactants and the reaction will proceed
• Reactions with positive enthalpies indicate that heat must be input into the system for the reaction to proceed. The products are higher in energy than the reactants and the reaction will not usually proceed spontaneously
• However, we have seen that a consideration of reaction enthalpies alone is not enough to predict whether a reaction (or physical process) will proceed
• NH₄SO₃(s) dissolves in water spontaneously even though the enthalpy of solvation is positive (it absorbs heat and is the basis of instant ice packs)

The first law of thermodynamics:

• The Law of Conservation of Energy
• Any energy lost by a system must be gained by the surroundings, and vice versa. Energy is neither created nor destroyed by a process (it is only transformed from one form to another)
• Energy, E, can be associated with work, w (i.e. force * distance, in the displacement of an object) or the transfer of heat, q

DE = q + w

DE = the change in the internal energy of a system

q = heat absorbed by the system from the surroundings

w = the work done on the system by the surroundings

We can keep track of the work done on a system and the heat absorbed or emitted by a system, however, since the overall change in energy for (the system + surroundings) = 0, we cannot use the change in energy of a system to predict whether a reaction will occur

• If the internal energy of a system is increased, then the energy of the surroundings must decrease, and vice versa
• When the effects of the energy change of the surroundings are taken into account, the net energy change = 0

Some processes are known to always occur:

• Highly ordered systems become more disordered
• Eggs have a tendency to break, yet I have never seen an intact egg spontaneously reassemble from a pile of egg bits
• A pile of leaves in my garden will eventually distribute themselves over my entire lawn. Unfortunately, I have never seen the leaves on my lawn spontaneously assemble into a nice neat pile by themselves
• I have seen ocean waves destroy an elaborate sand castle, but have never seen a sand castle spontaneously rise up and form out of the sandy surf.

• By all accounts, in the absence of some outside influence, the above spontaneous processes are irreversible

Although a process may be spontaneous, that does not mean that it will necessarily be fast

• Thermodynamics provides information about the direction a reaction or process will proceed (i.e. which direction of a reaction is spontaneous, but not about how fast it will go
• Spontaneity of a process depends not only upon the enthalpy change (e.g. exothermic) but also upon how the disorder of the system changes

Consider the following two experiments:

1. Behavior of gas in a container

• I have two glass containers connect by a valve. In the left-hand side container I have a sample of nitrogen gas. In the right-hand side container there is nothing (i.e. a vacuum):

• I then open the valve connecting the two containers. What happens?

• There is a spontaneous process that involves the flow of gas molecules from the left chamber to fill in the empty right-hand side chamber
• There was no work performed either on the system or on the surroundings (w = 0)
• There was no heat input or output from the system to the surroundings (q = 0)
• The spontaneous reaction results in pressure uniformity
• Although the gases spontaneously moved from the left chamber to the right, it is highly unlikely (impossible) that the gases might spontaneously move back into the left-hand side chamber (to produce the original condition). Thus, the process appears to be spontaneous and irreversible (in the absence of some outside influence)

2. The behavior of objects of different temperature

• I have two cubes of metal. One cube I stick in the oven until it gets red hot. The other I put in the freezer until it gets really cold. Then I put the two blocks next to each other:

• Over time what will happen to the temperature of the two blocks?

• There is a spontaneous process that involves the flow of heat energy from the hot block to the cold block (the net heat energy is unchanged, however)
• There was no work performed either on the system or on the surroundings (w = 0)
• There was no heat input or output from the system to the surroundings (q = 0)
• The spontaneous reaction results in temperature uniformity
• Although the heat energy spontaneously moved from the hot block to the cold block, it is highly unlikely (impossible) that heat energy might spontaneously move back into the left-hand side block (to produce the original condition). Thus, the process appears to be spontaneous and irreversible (in the absence of some outside influence)

In both of the above cases, we had a situation where the starting condition was highly ordered (gas on left, vacuum on right; hot block on left, cold block on right)

• The final condition was one of uniformity (of pressure or temperature)

The disorder is expressed by a thermodynamic quantity called entropy (S)

• The more disordered a state, the larger its entropy (a large magnitude for S means a lot of disorder)
• Entropy is a state function: DS = S_final - S_initial
• In other words, the pathway you choose to get to S_final is not important, what is important is discussing the change in entropy of a system is a comparison of the initial and final values of the entropy
• A positive value for DS indicates an increase in disorder. A negative value for DS indicates a decrease in disorder

How is the change in the enthalpy of a system (DS) defined?

(where T is some constant temperature)

• Obviously an enthalpy change is in some way proportional to some kind of energy term. If a system does not do any mechanical work, then heat energy must be involved. (Note: heat, like work, is not an entity but a method of energy transfer)
• If the system is at constant T, then the non-mechanical energy flow is not being used to change the temperature (it is associated with the changing order of the system)
• Why is DS inversely proportional to the absolute (K) temperature?
• At high temperatures what will the system be like? It will be a highly energetic gas and it will be difficult to get much more disorder out of it (thus at high temperatures, the change in disorder (DS) will be small for a given amount of non-mechanical energy transfer to the system.
• At low temperatures the system will be more ordered (highest order is at absolute zero) and the change in disorder (DS) will be larger for a given amount of non-mechanical energy transfer to the system

One of the classic examples of entropy changes in response to non-mechanical energy transfer at a constant temperature is the melting of ice at 0°C (i.e. 273K)

• At 0K liquid water is in equilibrium with solid (i.e. ice)
• Non-mechanical energy can be transferred into the system without raising the temperature (this is true as long as there is some ice and water present; if things get to the point where all the ice is melted, then the temperature will raise as you heat the water; if all the liquid water is frozen, then you lower the temperature of the ice as heat is removed; otherwise, the effect of energy flow into and out of the system results in either more or less ice being present, but no temperature change)
• The amount of heat transferred to the system during the fusion of ice is the heat of fusion, DH_fus (6.01kJ/mole). Thus,

DS = 6.01kJ/mole / 273K = 22 J/mol K

• DS is positive, indicating that the liquid form of water has greater disorder compared to the solid form (ice) (Note: this is an equilbrium situation, not an irreversible spontaneous reaction. Thus, input energy must increase the disorder of the system)

We can also define the entropy change in relationship to expansion of a gas at constant T (i.e. isothermal expansion

DS = nR ln (V_final/V_initial)
(for an ideal gas at some constant T)

The law that expresses the idea that there is an inherent direction in which processes occur is called the second law of thermodynamics

We must consider the change in entropy of the system and the surroundings

• Together, the system and the surroundings constitute the universe

DS universe = DS system + DS surroundings

• For an irreversible (spontaneous) process, we have an increase in entropy. Since it is irreversible, the Universe has gained entropy:

DS universe = (DS system + DS surroundings) > 0

• For a reversible process (not spontaneous, but manipulable by heat flow) there is no net DS as far as the Universe is concerned (i.e. if we input heat energy into the system, then the heat energy of the surroundings decreases)

DS universe = (DS system + DS surroundings) = 0

• Thus, there is no case where DS of the universe decreases

Chemical reactions follow this same law

• Consider an exothermic reaction that is less disordered when complete (DS decreases; the disorder decreases)

O₂(g) + 2H₂(g) ® 2H₂O(g)

• Let's perform this reaction in the following way: We will keep the temperature constant. Since the reaction is exothermic, we do this by having the surroundings absorb the released heat.
• The product is two molecules of water. The reactants comprise three molecules (one of O₂ and two of H₂)
• Since temperature is constant, the overall change in entropy of the system (reactants and products) is reduced (products are fewer gas molecules due to atoms being bonded together - this reduces the entropy of the atoms because since they are bonded together they are restricted in their movements)
• The released heat is taken up by the surroundings
• The surroundings are kept at same temperature.
• Since temperature is constant, the energy absorbed by surroundings is manifest as increased disorder. The surroundings become more disordered

-q_sys/T = +q_surr/T (where T is constant)

in other words

• If the entire process were reversible, DS universe = 0
• If any small part is irreversible, DS > 0