# Entropy

## Entropy: A Study Guide

Discussion Questions

• What is entropy? What is its symbol?
• How is entropy related to energy?
• What does it mean by "state function"?
• What are phase transitions? How are entropies of phase transitions evaluated?
• What is the state of a substance at zero degree Kelvin?
• How is standard molar entropy defined and measured? How is it related to the structure and molar mass of molecules?
• How is entropy related to disorder?
• How are standard entropies of reaction measured?

## Entropy

Entropy is a chemical concept that is very difficult to explain, because a one-sentence definition will not lead to a comprehensive statement. Thus, few people understand what entropy really is. You are not alone if you have some difficulty with this concept. The word entropy is used in many other places and for many other aspects. We confine our discussion to thermodynamics (science dealing with heat and changes) and to chemical and physical processes.

We have define energy as the driving force for changes, entropy is also a driving force for physical and chemical changes (reactions). Entropy, symbol S, is related to energy, but it a different aspect of energy. This concept was developed over a long period of time. Human experienced chemical and physical changes that cannot be explained by energy alone. A different concept is required to explain spontaneous changes such as the expansion of a gas into an abailable empty space (vacumm) and heat transfer from a hot body into a cold body. These changes cause an increase in entropy for the system under consideration, but energy is not transferred into or out of the system.

Traditionally, the entropy concept is associated with the second and third laws of thermodynamics. Entropy is related to the energy distribution of energy states of a collection of molecules, and this aspect is usually discussed in statistical mechanics.

#### Second Law of Thermodynamics

When a system receives an amount of energy q at a constant temperature, T, the entropy increase DS is defined by the following equation.

DS = q / T.

Entropy is the amount of energy transferred divided by the temperature at which the process takes place. Thus, entropy has the units of energy unit per Kelvin, J K-1. If the process takes place over a range of temperature, the quantity can be evaluated by adding bits of entropies at various temperatures. This sum can take the form of integration if the temperature various contineously. You have learned the concept of integration in a calculus course.

Entropy is a state function in that it depends only on the initial and final state of the system, regardless of the path by which the changes take place. However, the changes are supposedly take place slowly over a long period of time, or in an almost equilibrium or reversible condition. If the change takes place quickly in an irreversible manner, the entropy is greater than what is evaluated, because the temperature increase is not uniform.

Nature has a tendency for entropy S to increase, and the system changes in response to this tendency. Such a change is called a spontaneous process. Thus, the driving force for a spontaneous process in an isolated system is an increase in entropy. This statement is one of the acceptable statement of second law of thermodynamics. Sorry for being so formal, but just so that you know you know something classical.

Example 1

Calculate entropy change when 36.0 g of ice melts at 273 K and 1 atm. If you look up the enthalpy of fusion for ice in a table, you would get a molar enthalpy of 6.01 kJ/mol.

SOLUTION

dS = (6.01 kJ mol-1)/272 K * (36 g)/(18 g mol-1)
= 1.22 kJ / K

DISCUSSION
Gibbs Paradox illustrates an interesting aspect of entropy.

#### Third Law of Thermodynamics

By definition, the change in entropy can be evaluated by measuring the amount of energy transferred. Entropy contained in a system, say in a mole of a pure substance, is a theoretical quantity that takes account of all heat transferred to it since the lowest atainable temperature, 0 K. At absolute zero Kelvin, the substance contains no removable energy. A substance is in a completely ordered crystalline state, at which the moleules contains no removalbe vibrational, rotational, translational, or even thermal disorder energy.

As energy q is absorbed by a substance, its temperature increases by dT. If the heat capacity is C, then

q = C dT

The area below the curve of a plot of q / T versus T from 0 K to T is then the absolute entropy S of the system. Of course, the heat capacity C may also vary with temperature.

q / T
|                .
|          |    .
|   Phase  V   .|
|transition   . |
|           .   |
|          .    |
|          .    |
|          .    |
|       .       |
|    .          |
|  .            |
| .    S        |
|.              |
---------------------
0          T


In a phase transition, heat is absorbed, but the temperature remain constant. The entropy increase is DH / T.

Recall that thermodynamic values at standard condition are called standard values. Thus, the entropies S so evaluated T = 298 K are called standard entropies. They have been carefully measured for many substances. For example, the standard molar entropy of some solids are given below:

                            (white)     (rhombic)
diamond  graphite  sodium  phosphorus  sulfur    silver
2.38      5.74      51.3    41.1        31.8      42.6 J (K mol)-1


The standard molar entropies (standard entropy per mole) for gases are usually higher because heat of melting and heat of vaporization must be included. The standard molar entropies for noble gases are:

    He     Ne     Ar     Kr     Xe (all in gaseous state)
126.0  146.2  154.7  164.0  169.6 J (K mol)-1


Note that the entropy increase as the atomic mass increase. The same trends is also found for the halogens, but the entropies for these diatomic gases are much greater than those of monoatomic noble gases.

    H2     N2      O2  (all in gaseous state)
130.6  191.5  205.0  J K-1

F2     Cl2    Br2     I2 (all in gaseous state)
203.7  222.9  245.4  260.6  J K-1


Standard entropy of some compounds have also been measured. For example,

   H2O(l)   H2O(g)   NH3(g)   H2O2(l)  CH3OH(l)  CH3Cl(l)  CHCl3(l)
69.9    188.7    192.5    110.0    126.9    145.3     294.9 J (K mol)-1

CO       CO2      NO      NO2    N2O4    SO2  (all in gases state)
197.8    213.6   210.6   240.4  304.3   248.4 J (K mol)-1

CH4   C2H2  C2H4  C2H6O4 (all in gases state)
186   201  221  230 J (K mol)-1


From the variations of standard entropies of these substances, one can conclude that the more complicate the molecule and the heavier the molar mass, the higher the standard entropy.

### How is Entropy Related to Disorder?

The relationship between entropy and disorder is studied in a discipline called statistical mechanics. Simple illustrations are used here to explain the relationship of entropy and disorder.

Entropy is also related to probability, as a measure of randomness or disorder, and entropy is proportional to the logarithm of the probability. The formula given by (and inscribed on the tomstone of) Ludwig Boltzmann,

S = k ln W

sugests that the entropy, S, is proportional to the natural logarithm (ln) of the number of possible state, W. The proportional constant k (=1.38*10-23 J/K) is called the Boltzmann's constant.

When you toss three coins, the possibility of having 2 heads and 1 tail or 2 tails and 1 head are much higher than having 3 heads or 3 tails, becuase the possibility of having the former is three times higher than the latter groupings.

Similarly, when two gases are placed in a container, the chances of having them separated in the two halves are much less than having them mixed. The mixed state have a higher entropy than the unmixed state.

Thus, the general rule says: A natural process (reaction) causes an increase of entropy or entropy increase is the driving force for natural reactions.

### How to Evaluate Standard Entropy of Reaction?

Standard entropies of reaction, DSoreaction, equals the entropy of products minus the entropy of reactants. Since the entropies of most substances have been measured and tabulated in handbooks and data banks, standard entropies of reactions can be evaluated in a similar manner as enthalpies of reactions.

Similarly, when you put a drop of ink into a cup of water, the ink disperse into the water, or water disperse into the ink. In this process, the entropy is higher, because an increase of disorder results in an increase of entropy.

Example 2

Evaluate the entropy change for the reaction:

CO + 3 H2 -> CH4 + H2O

in which all reactants and products are gaseous.

SOLUTION
Standard entropies of reaction, DSoreaction, equals the entropy of products minus the entropy of reactants.

The standard entropies of the reactants and products have been given above, and for clarity, the entropies are given below the formula of the reactants and products: (data are given in 3 significant digits)

CO + 3 H2 -> CH4 + H2O
198  131     186   189


DS = ((186 + 189) - (198 + 3*131)) J (K mol)-1
= -216 J (K mol)-1

DISCUSSION
There are four (4) moles of reactants and two (2) moles of products (all in gaseous state), hense the large negative change in entropy for the reaction.

Is the entropy change for the electrolytic decomposition of water,

H2O(l) -> 2 H2 + O2,

positive or negative?

Gibbs Paradox illustrates an interesting aspect of entropy.

The description above only covered some useful aspect of this important concept of entropy DHoreaction.

## Questions

1. Calculate the entropy change if one mole of water evaporate at 373 K and 1 atm. If you look up the enthalpy of vaporization, for water in a table, you would get a molar enthalpy of 40.7 kJ/mol.

2. What is the entropy change for the electrolytic decomposition of water,

H2O(l) -> 2 H2 + O2,

3. Arrange the following molecules, all in gas states, in the order of increasing standard molar entropy:

O2, N2, H2, C2H2

4. At the same state, which has a higher standard molar entropy, H2O or NH3?

5. Which has a higher value of standard molar entropy, H2O(l) or H2O(g)?

6. The molar bond energy for O=O (double bonded oxygen molecule) is 502 kJ/mol. Calculate the standard molar entropy change for the reaction:

O2 -> 2 O

## Solutions

1. 1091 J/K

Skill -
Evaluate entropy changes for phase transitions from the enthalpies of phase transition.

2. +326 J/K

Skill -
Evaluate entropy changes of reactions.

3. H2, N2, O2, C2H2

Skill -
Give trends of variation of standard molar entropies based on molecular structures and molar mass.

4. NH3 (amonia)

Discussion
Actually, the difference is not very large for these two similar compounds.

5. The gas state.

Skill -
Calculate the entropy of phase transition.

At standard condition, the stable phase for water is liquid. However, for standard molar entropy of water vapour is DHvaporization/T higher than that of water.

6. 1.685 kJ/K

Discussion
This estimate is very crude, because the process takes place only at elivated temperatures.