`The Second Law of Thermodynamics states that the state of entropy of the entire universe, as an isolated system, will always increase over time. The second law also states that the changes in the entropy in the universe can never be negative.`

## Introduction

Why is it that when you leave an ice cube at room temperature, it begins to melt? Why do we get older and never younger? And, why is it whenever rooms are cleaned, they become messy again in the future? Certain things happen in one direction and not the other, this is called the “arrow of time” and it encompasses every area of science. The thermodynamic arrow of time (entropy) is the measurement of disorder within a system. Denoted as ΔS, the change of entropy suggests that time itself is asymmetric with respect to order of an isolated system, meaning: a system will become more disordered, as time increases.

### Major players in developing the Second Law

Nicolas Léonard Sadi Carnot was a French physicist, who is considered to be the “father of thermodynamics,” for he is responsible for the origins of the Second Law of Thermodynamics, as well as various other concepts. The current form of the second law uses entropy rather than caloric, which is what Sadi Carnot used to describe the law. Caloric relates to heat and Sadi Carnot came to realize that some caloric is always lost in the motion cycle. Thus, the thermodynamic reversibility concept was proven wrong, proving that irreversibility is the result of every system involving work.

Rudolf Clausius was a German physicist, and he developed the Clausius statement, which says “Heat generally cannot flow spontaneously from a material at a lower temperature to a material at a higher temperature.”

William Thompson, also known as Lord Kelvin, formulated the Kelvin statement, which states “It is impossible to convert heat completely in a cyclic process.” This means that there is no way for one to convert all the energy of a system to work, without losing energy.

Constantin Carathéodory, a Greek mathematician, created his own statement of the second low arguing that “In the neighbourhood of an initial state, there are states which cannot be approached arbitrarily close through adiabatic changes of state.”

### Probabilities

If a given state can be accomplished in more ways, then it is more probable than the state that can only be accomplished in a fewer/one way.

Assume a box filled with jigsaw pieces were jumbled in its box, the probability that a jigsaw piece will land randomly, away from where it fits perfectly, is very high. Almost every jigsaw piece will land somewhere away from its ideal position. The probability of a jigsaw piece landing correctly in its position, is very low, as it can only happen one way. Thus, the misplaced jigsaw pieces have a much higher multiplicity than the correctly placed jigsaw piece, and we can correctly assume the misplaced jigsaw pieces represent a higher entropy.

### Derivation and Explanation

To understand why entropy increases and decreases, it is important to recognize that two changes in entropy have to consider at all times. The entropy change of the surroundings and the entropy change of the system itself. Given the entropy change of the universe is equivalent to the sums of the changes in entropy of the system and surroundings:

ΔSuniv=ΔSsys+ΔSsurr=qsysT+qsurrT(1.1)

In an isothermal reversible expansion, the heat q absorbed by the system from the surroundings is

qrev=nRTlnV2V1(1.2)

Since the heat absorbed by the system is the amount lost by the surroundings, qsys=−qsurr

.Therefore, for a truly reversible process, the entropy change is

ΔSuniv=nRTlnV2V1T+−nRTlnV2V1T=0(1.3)

If the process is irreversible, however, the entropy change is

ΔSuniv=nRTlnV2V1T>0(1.4)

If we put the two equations for ΔSuniv

together for both types of processes, we are left with the second law of thermodynamics,

ΔSuniv=ΔSsys+ΔSsurr≥0(1.5)

where ΔSuniv

equals zero for a truly reversible process and is greater than zero for an irreversible process. In reality, however, truly reversible processes never happen (or will take an infinitely long time to happen), so it is safe to say all thermodynamic processes we encounter every day are irreversible in the direction they occur.

The second law of thermodynamics can also be stated that “all spontaneous processes produce an increase in the entropy of the universe”.

#### Gibbs Free Energy

Given another equation:

ΔStotal=ΔSuniv=ΔSsurr+ΔSsys(1.6)

The formula for the entropy change in the surroundings is ΔSsurr=ΔHsys/T

. If this equation is replaced in the previous formula, and the equation is then multiplied by T and by -1 it results in the following formula.

−TΔSuniv=ΔHsys−TΔSsys(1.7)

If the left side of the equation is replaced by G

, which is known as Gibbs energy or free energy, the equation becomes

ΔG=ΔH−TΔS(1.8)

Now it is much simpler to conclude whether a system is spontaneous, non-spontaneous, or at equilibrium.

ΔH

refers to the heat change for a reaction. A positive ΔH means that heat is taken from the environment (endothermic). A negative ΔH

means that heat is emitted or given the environment (exothermic).

ΔG

is a measure for the change of a system’s free energy in which a reaction takes place at constant pressure (P) and temperature (T

).

According to the equation, when the entropy decreases and enthalpy increases the free energy change, ΔG

, is positive and not spontaneous, and it does not matter what the temperature of the system is. Temperature comes into play when the entropy and enthalpy both increase or both decrease. The reaction is not spontaneous when both entropy and enthalpy are positive and at low temperatures, and the reaction is spontaneous when both entropy and enthalpy are positive and at high temperatures. The reactions are spontaneous when the entropy and enthalpy are negative at low temperatures, and the reaction is not spontaneous when the entropy and enthalpy are negative at high temperatures. Because all spontaneous reactions increase entropy, one can determine if the entropy changes according to the spontaneous nature of the reaction.