9.2: The Second and Third Laws of Thermodynamics

Learning Objectives

By the end of this section, you will be able to:

• State and explain the second and third laws of thermodynamics
• Calculate entropy changes for phase transitions and chemical reactions under standard conditions

The Second Law of Thermodynamics

In the quest to identify a property that may reliably predict the spontaneity of a process, a promising candidate has been identified: entropy. Processes that involve an increase in entropy of the system (ΔS > 0) are very often spontaneous; however, examples to the contrary are plentiful. By expanding consideration of entropy changes to include the surroundings, we may reach a significant conclusion regarding the relation between this property and spontaneity. In thermodynamic models, the system and surroundings comprise everything, that is, the universe, and so the following is true

\[ \Delta S_\mathrm{univ} = \Delta S_\mathrm{sys} + \Delta S_\mathrm{surr} \]

To illustrate this relation, consider again the process of heat flow between two objects, one identified as the system and the other as the surroundings. There are three possibilities for such a process:

1. The objects are at different temperatures, and heat flows from the hotter to the cooler object. This is always observed to occur spontaneously. Designating the hotter object as the system and invoking the definition of entropy yields the following:$${}_{}$$

\[ \Delta S_{\mathrm{sys}}=\frac{-q_{\mathrm{rev}}}{T_{\mathrm{sys}}} \space \space \mathrm{and} \space \space \Delta S_{\mathrm{surr}}=\frac{q_{\mathrm{rev}}}{T_{\mathrm{surr}}} \]

The magnitudes of −q_rev and q_rev are equal, their opposite arithmetic signs denoting loss of heat by the system and gain of heat by the surroundings. Since T_sys > T_surr in this scenario, the entropy decrease of the system will be less than the entropy increase of the surroundings, and so the entropy of the universe will increase:

\[ \begin{aligned} & \left|\Delta S_{\text {sys }}\right|<\left|\Delta S_{\text {surr }}\right| \\ & \Delta S_{\text {univ }}=\Delta S_{\text {sys }}+\Delta S_{\text {surr }}>0\end{aligned} \]

2. The objects are at different temperatures, and heat flows from the cooler to the hotter object. This is never observed to occur spontaneously. Again designating the hotter object as the system and invoking the definition of entropy yields the following

\[ \Delta S_{\mathrm{sys}}=\frac{q_{\mathrm{rev}}}{T_{\mathrm{sys}}} \space \space \mathrm{and} \space \space \Delta S_{\mathrm{surr}}=\frac{-q_{\mathrm{rev}}}{T_{\mathrm{surr}}} \]

The arithmetic signs of q_rev denote the gain of heat by the system and the loss of heat by the surroundings. The magnitude of the entropy change for the surroundings will again be greater than that for the system, but in this case, the signs of the heat changes (that is, the direction of the heat flow) will yield a negative value for ΔS_univ. This process involves a decrease in the entropy of the universe.

3. The objects are at essentially the same temperature, T_sys ≈ T_surr, and so the magnitudes of the entropy changes are essentially the same for both the system and the surroundings. In this case, the entropy change of the universe is zero, and the system is at equilibrium.

\[ \begin{aligned} & \left|\Delta S_{\text {sys }}\right| \approx\left|\Delta S_{\text {surr }}\right| \\ & \Delta S_{\text {univ }}=\Delta S_{\text {sys }}+\Delta S_{\text {surr }}=0\end{aligned} \]

These results lead to a profound statement regarding the relation between entropy and spontaneity known as the second law of thermodynamics: all spontaneous changes cause an increase in the entropy of the universe. A summary of these three relations is provided in Table 16.1.

The Second Law of Thermodynamics

ΔSuniv > 0	spontaneous
ΔSuniv < 0	nonspontaneous (spontaneous in opposite direction)
ΔSuniv = 0	at equilibrium

Table 16.1

For many realistic applications, the surroundings are vast in comparison to the system. In such cases, the heat gained or lost by the surroundings as a result of some process represents a very small, nearly infinitesimal, fraction of its total thermal energy. For example, combustion of a fuel in air involves transfer of heat from a system (the fuel and oxygen molecules undergoing reaction) to surroundings that are infinitely more massive (the earth’s atmosphere). As a result, q_surr is a good approximation of q_rev, and the second law may be stated as the following:

\[ \Delta S_{\text {univ }}=\Delta S_{\text {sys }}+\Delta S_{\text {surr }}= \Delta S_{\text {sys}} + \frac{q_{\text {surr }}}{T} \]

We may use this equation to predict the spontaneity of a process as illustrated in Example 16.4.

Example 16.4

Will Ice Spontaneously Melt?

The entropy change for the process

$${\text{H}}_2\text{O}(s)⟶{\text{H}}_2\text{O}(l)$$

is 22.1 J/K and requires that the surroundings transfer 6.00 kJ of heat to the system. Is the process spontaneous at −10.00 °C? Is it spontaneous at +10.00 °C?

Solution

We can assess the spontaneity of the process by calculating the entropy change of the universe. If ΔS_univ is positive, then the process is spontaneous. At both temperatures, ΔS_sys = 22.1 J/K and q_surr = −6.00 kJ.

At −10.00 °C (263.15 K), the following is true:

$$\begin{array}{cc}\hfill \text{Δ}{S}_{\text{univ}}& =\text{Δ}{S}_{\text{sys}}+\text{Δ}{S}_{\text{surr}}=\text{Δ}{S}_{\text{sys}}+\frac{q_{\text{surr}}}{T}\hfill \\ & =\text{22.1 J/K}+\frac{\mathrm{-6.00}\times {10}^3\text{J}}{\text{263.15 K}}=\mathrm{-0.7}\text{J/}\hfill \end{array}$$

S_univ < 0, so melting is nonspontaneous (not spontaneous) at −10.0 °C.

At 10.00 °C (283.15 K), the following is true:

$$\begin{array}{}\\ \text{Δ}{S}_{\text{univ}}=\text{Δ}{S}_{\text{sys}}+\frac{q_{\text{surr}}}{T}\\ \\ =22.1\text{J/K}+\frac{\mathrm{-6.00}\times {10}^3\text{J}}{\text{283.15 K}}=\text{+0.9 J/}\end{array}$$

S_univ > 0, so melting is spontaneous at 10.00 °C.

Check Your Learning

Using this information, determine if liquid water will spontaneously freeze at the same temperatures. What can you say about the values of S_univ?

Answer:

Entropy is a state function, so ΔS_freezing = −ΔS_melting = −22.1 J/K and q_surr = +6.00 kJ. At −10.00 °C spontaneous, +0.7 J/K; at +10.00 °C nonspontaneous, −0.9 J/K.