8.10: The Second Law of Thermodynamics

Entropy measures available microstates; more microstates = more entropy

Chemical and physical changes in a system may be accompanied by either an increase or a decrease in the disorder of the system, corresponding to an increase in entropy (ΔS > 0) or a decrease in entropy (ΔS < 0), respectively. As with any other state function, the change in entropy is defined as the difference between the entropies of the final and initial states: ΔS = S_f − S_i.

We can illustrate the concepts of microstates and entropy using a deck of playing cards, as shown in Figure \(\PageIndex{2}\). In any new deck, the 52 cards are arranged by four suits, with each suit arranged in descending order. If the cards are shuffled, however, there are approximately 10⁶⁸ different ways they might be arranged, which corresponds to 10⁶⁸ different microscopic states. The entropy of an ordered new deck of cards is therefore low, whereas the entropy of a randomly shuffled deck is high. Card games assign a higher value to a hand that has a low degree of disorder. In games such as five-card poker, only 4 of the 2,598,960 different possible hands, or microstates, contain the highly ordered and valued arrangement of cards called a royal flush, almost 1.1 million hands contain one pair, and more than 1.3 million hands are completely disordered and therefore have no value. Because the last two arrangements are far more probable than the first, the value of a poker hand is inversely proportional to its entropy.

Instead of a deck of cards, let’s now consider 1 L of an ideal gas at standard temperature and pressure (STP), which contains 2.69 × 10²² molecules (6.022 × 10²³ molecules/22.4 L). If we allow the sample of gas to expand into a second 1 L container, the probability of finding all 2.69 × 10²² molecules in one container and none in the other at any given time is extremely small, approximately \(\frac{2}{2.69 \times 10^{22}}\). The probability of such an occurrence is effectively zero. Although nothing prevents the molecules in the gas sample from occupying only one of the two bulbs, that particular arrangement is so improbable that it is never actually observed. The probability of arrangements with essentially equal numbers of molecules in each bulb is quite high, however, because there are many equivalent microstates in which the molecules are distributed equally. Hence a macroscopic sample of a gas occupies all of the space available to it, simply because this is the most probable arrangement.

The Second Law of Thermodynamics

For reasons we will not elaborate greatly on here, for a special type of process known as a reversible process, the amount of entropy gained or lost by the surroundings is related to the amount of heat lost or gained by the system. This is because the quantity of heat transferred from the system (q_rev) is directly proportional to the absolute temperature of the surroundings (T) according to the equation below.

\(\Delta S_{\textrm{surr}}=\dfrac{q_{\textrm{rev}}}{T}=\dfrac{\Delta H_{\textrm{sys}}}{T} \

Because the numerator (q_rev) is expressed in units of energy (joules), the units of ΔS are joules/kelvin (J/K).

As an example, let's consider the reversible melting of a sample of ice right at its melting point of 0°C and 1 atm. The enthalpy of fusion of ice is 6.01 kJ/mol, which means that 6.01 kJ of heat are absorbed reversibly from the surroundings when 1 mol of ice melts at 0°C, as illustrated in Figure \(\PageIndex{6}\). The system is the ice, and the surroundings are a sample of low-density carbon foam that is thermally conductive and we can use to measure temperature. Because the carbon foam is initially warmer than the ice, heat will flow towards the ice as depicted by the arrow in the thermogram below going from red (warm carbon foam) to blue (cold ice). From this process, we can write the entropy of fusion of ice as follows:

The amount of heat lost by the surroundings is the same as the amount gained by the ice, so the entropy of the universe does not change. However, the entropy change of the system is ΔS_fus = (6.01 kJ/mol)/(273 K) = 22.0 J/(mol•K) = ΔS_sys. On the other hand, the amount of heat lost by the surroundings is the same as the amount gained by the ice, so ΔS_surr = q_rev/T = −(6.01 kJ/mol)/(273 K) = −22.0 J/(mol•K). Once again, we see that the entropy of the universe does not change:

ΔS_univ = ΔS_sys + ΔS_surr = 22.0 J/(mol•K) − 22.0 J/(mol•K) = 0

This reversible melting of ice is an example of a principle that is true of all reversible processes and constitutes part of the second law of thermodynamics: the entropy of the universe remains constant in a reversible process, whereas the entropy of the universe increases in an irreversible (spontaneous) process.

Processes can also be irreversible, but the second law still holds. As an example of an irreversible process, consider the entropy changes that accompany the spontaneous and irreversible transfer of heat from a hot object to a cold one, as occurs when lava spewed from a volcano flows into cold ocean water. The cold substance, the water, gains heat (q > 0), so the change in the entropy of the water can be written as ΔS_cold = q/T_cold. Similarly, the hot substance, the lava, loses heat (q < 0), so its entropy change can be written as ΔS_hot = −q/T_hot, where T_cold and T_hot are the temperatures of the cold and hot substances, respectively. The total entropy change of the universe accompanying this process is therefore

\[\Delta S_{\textrm{univ}}=\Delta S_{\textrm{cold}}+\Delta S_{\textrm{hot}}=\dfrac{q}{T_{\textrm{cold}}}+\left(-\dfrac{q}{T_{\textrm{hot}}}\right) \label{Eq6}\]

The numerators on the right side of Equation \(\ref{Eq6}\) are the same in magnitude but opposite in sign. Whether ΔS_univ is positive or negative depends on the relative magnitudes of the denominators. By definition, T_hot > T_cold, so −q/T_hot must be less than q/T_cold, and ΔS_univ must be positive. As predicted by the second law of thermodynamics, the entropy of the universe increases during this irreversible process. Any process for which ΔS_univ is positive is, by definition, a spontaneous one that will occur as written. Conversely, any process for which ΔS_univ is negative will not occur as written but will occur spontaneously in the reverse direction. We see, therefore, that heat is spontaneously transferred from a hot substance, the lava, to a cold substance, the ocean water. In fact, if the lava is hot enough (e.g., if it is molten), so much heat can be transferred that the water is converted to steam (Figure \(\PageIndex{7}\)).

Summary

For a given system, the greater the number of microstates, the higher the entropy. During a spontaneous process, the entropy of the universe increases. \[\Delta S=\frac{q_{\textrm{rev}}}{T}\]

A measure of the disorder of a system is its entropy (S), a state function whose value increases with an increase in the number of available microstates. A reversible process is one for which all intermediate states between extremes are equilibrium states; it can change direction at any time. In contrast, an irreversible process occurs in one direction only. The change in entropy of the system or the surroundings is the quantity of heat transferred divided by the temperature. The second law of thermodynamics states that in a reversible process, the entropy of the universe is constant, whereas in an irreversible process, such as the transfer of heat from a hot object to a cold object, the entropy of the universe increases.