## Thermodynamics

The third law of thermodynamics states that the entropy of a perfect crystal at absolute zero (0K) is zero. This means that all molecular motion ceases in a perfect crystal at absolute zero. Pay attention to the definition of the third law of thermodynamics.

### Law of Thermodynamics

Thermodynamics is a very important branch of both physics and chemistry. It deals with the study of energy, the conversion of energy between different forms and the ability of energy to do work. As you go through this article, I am pretty sure that you will begin to appreciate the importance of thermodynamics and you will start noticing how laws of thermodynamics operate in your daily lives! There are essentially four laws of thermodynamics.

### Zeroth Law of Thermodynamics

“If two systems are in thermal equilibrium with a third system, then they are in thermal equilibrium with one another.”

Let’s first define what thermal equilibrium is. When two systems are in contact with each other and no energy flow takes place between them, then the two systems are said to be in thermal equilibrium with each other. In simple words, thermal equilibrium means that the two systems are at the same temperature. Thermal equilibrium is a concept that is so integral to our daily lives.

For example, let's say you have a bowl of hot soup and you put it in the freezer. What will happen to the soup? The soup will, of course, start cooling down with time. You all know that. And you also probably know that the soup will continue to cool down until it reaches the same temperature as the freezer. Even if you are familiar with this concept, what you may not realize is that this is an excellent example of thermal equilibrium. Here, heat flows from the system at a higher temperature (bowl of soup) to the system at a lower temperature (freezer). Heat flow stops when the two systems reach the same temperature.

In other words, now the two systems are in thermal equilibrium with each other and there is no more heat flow taking place between the two systems. Let’s assume we have three systems - system 1, system 2 and system 3. Their temperatures are T1, T2, and T3 respectively.

Diagram showing the zeroth law

The Zeroth law states that if the temperature of system 1 is equal to temperature of system 3, and the temperature of system 2 is equal to temperature of system 3; then the temperature of system 1 should be equal to the temperature of system 2. The three systems are said to be in thermal equilibrium with each other.

That is; if $T_1=T_3$ and $T_2=T_3$, then $T_1=T_2$

The zeroth law is analogous to the basic rule in algebra, if A=C and B=C, then A=B.

This law points to an important fact: temperature affects the direction of heat flow between systems. Heat always flows from high temperature to low temperature. Heat flow is mathematically denoted as Q.

### First Law of Thermodynamics

This law is essentially the law of conservation of energy. Energy can neither be created nor destroyed; it can just be converted from one form to another.

In simple words, the first law of thermodynamics states that whenever heat energy is added to a system from outside, some of that energy stays in the system and the rest gets consumed in the form of work. Energy that stays in the system increases the internal energy of the system. This internal energy of the system can be manifested in various different forms – kinetic energy of molecules, potential energy of molecules or heat energy (that simply raises the temperature of the system).

#### What is Internal Energy?

It is defined as the sum total of kinetic energy, which comes from motion of the molecules, and potential energy which comes from the chemical bonds that exist between the atoms and any other intermolecular forces that may be present.

We can state the first law of thermodynamics conventionally as: “The change in internal energy of a closed system is equal to the energy added to it in the form of heat (Q) plus the work (W) done on the system by the surroundings.”

We can write this mathematically as:

$\Delta E_{internal} = Q + W$

Conventional definition of the first law is based on the system gaining heat and the surrounding doing work. The opposite scenario can occur too, in which case the signs in the equation will have to be changed appropriately. We will discuss this shortly.

Lots of times you will notice that $\Delta E_{internal}$ is also denoted as $\Delta\cup$.

Now you must be wondering what is meant by a closed system. Let’s try to understand this through a simple example. Imagine we have two saucepans containing water: 1. with a lid and 2. without a lid. Both are kept on a heated stove.

Two saucepans on a heated stove: One has a lid and is a closed system the other does not have a lid and is an opened system.

Both the saucepans shown above will absorb heat from the stove and get heated up. So there is exchange of energy taking place in both the cases from the stove (surroundings) to the system (water). But do you notice a difference? Saucepan, with the lid, prevents any addition of mass to it or removal of mass from it; whereas in the case of the saucepan, without the lid, we can easily add coffee and sugar from outside and thus change the mass of the contents.

Basically, the lid is preventing matter from entering the saucepan and leaving the saucepan. Here, the saucepan with a lid is an example of a closed system; while the saucepan without a lid is an example of an open system.

Thus, we can define an open system as a system that freely exchanges energy and matter with its surroundings. A closed system is a system that exchanges only energy with its surroundings, not matter.

PS: Coming back to the sign conventions for the first law of thermodynamics ($\Delta E_{internal} = Q + W$); let’s be clear about the following norms:

• if heat flows out of the system, then Q will be negative.
• if heat flows into the system, then Q will be positive.
• if work is done by the system, then W will be negative.
• if work is done on the system, then W will be positive.

Let’s look at the following example:

We have a gas in a sealed container (closed system; no matter exchange can take place with the surroundings). A piston is attached, on top of which is placed a block of wood. We provide heat (Q) from outside to this system. This heat energy leads to expansion of the gas, which in turn pushes the piston up. So, the gas does work (W) in expanding itself, which results in pushing of the piston.

If you recall from the ideal gas laws; for an ideal gas, work (W) = PV = nRT

Diagram showing work put into a piston and the piston going up.

Let’s now mathematically define the change in the internal energy of the above system.

Change in internal energy of a system (∆E) = Q + W

Let’s try to apply the rules we talked about earlier,

In this example, *heat flows into the system, so Q will be positive and the work is done by the system (gas in this case), so W will be negative *

Thus, (∆E) = Q – W = Q - P∆V

[Q is the external energy provided, P is the pressure of the gas, and ∆V is the change in volume of the gas]

Now let’s attempt a couple of problems dealing with the first law of thermodynamics. Always be sure to use the correct units while solving any numerical!!!

#### Problem 1

In an exothermic process, the volume of a gas expanded from 186 mL to 1997 mL against a constant pressure of 745 torr. During the process, 18.6 calories of heat energy were given off. What was the internal energy change for the system in joules? Also, (1 L atm = 101.3 J)

Q = heat given off by system = 18.6 cal = 18.6 x 4.184 J (remember: 1 cal = 4.184 J)

= 77.82 J

Since heat flows out of the system, Q will be negative. So Q = -77.82 J

Work (W) = P∆V, where P = constant pressure of gas, ∆V = change in volume of gas

Let’s first convert pressure and volume into the correct units:

760 torr = 1 atm, so 745 torr of pressure = 745/ 760 = 0.98 atm

Volume should be in litres (L), thus ∆V = (1997 – 186) ml = 1811 mL = 1811 X 10-3 L

Thus, W = P∆V = 0.98 atm X 1.811 L = 1.77 atm L

Also, the problem statement gives conversion between atm L and Joules (1 L atm = 101.3 J). So, 1.77 atm L = 1.77 X 101.3 = 179.30 J

In this example, work is being done by the system in expansion of the gas, so W is negative (remember: if work is done by the system, then W will be negative). Therefore, W = -179.30 J So, net change in internal energy of the system (∆U) = Q + W = -77.82 + (-179.30) = -257.12 J

This result indicates that the energy of the gas is decreases by 257.12 J. This means, at the end of the process the gas has less energy than it had in the beginning.

#### Problem 2

The work done when a gas is compressed in a cylinder is 820 J. At the same time, the system lost 320 J of heat to the surrounding. What is the energy change of this system?

Here the gas is the system. First you must decide the signs of W and Q using the convention discussed earlier. Work is done on the system, so W = + 820 J and heat is lost by the system, so Q = -320 J.

Therefore ∆E = Q + W = -320 J + 820 J = 500 J

This result indicates that the energy of the gas is increased by 500 J. This implies, at the end of the process the gas has more energy than it had in the beginning.

### Second Law of Thermodynamics

This second law of thermodynamics states, “The total change in entropy of a system plus its surroundings will always increase for a spontaneous process.”

Entropy is defined as the measure of disorder or randomness of a system. Every system wants to achieve a state of maximum disorder or randomness. A commonly observed daily life example of something constantly moving toward a state of randomness is a kid’s room. Every time their parent cleans up the room, within minutes the room looks like this:

The natural state in which a kid’s room wants to exist – a mess. ☺

Another commonly encountered entropy driven process is the melting of ice into water. This happens spontaneously as soon as ice is left at room temperature. Ice is a solid with an ordered crystalline structure as compared to water, which is a liquid in which molecules are more disordered and randomly distributed. All natural processes tend to proceed in a direction which leads to a state that has more random distribution of matter and energy.

All of these processes take place spontaneously, meaning that once they start, they will proceed to the end if there is no external intervention. You will never witness the reverse of this process, in which water converts back to ice at room temperature. In other words, it would be inconceivable that this process could be reversed without tampering with the external conditions (you will have to put water in the freezer to force it to form ice). So what determines the direction in which a process will go under a given set of conditions? Now you know the answer - all these processes are driven by entropy.

We can state the second law of thermodynamics mathematically as:

$\Delta S_{universe}=\Delta S_{system}+\Delta S_{surroundings}>0$

Where, $\Delta S_{universe}$ = net change in entropy of the universe.

$\Delta S_{universe}$ = net change in entropy of the system.

$\Delta S_{universe}$ = net change in entropy of the surroundings.

Take home message: Entropy of the universe is constantly increasing. Entropy is mathematically calculated as Q/ T (heat absorbed or released by the system or surroundings divided by the temperature of the system or surroundings). Entropy is expressed in Joules per Kelvin (J/ K).

#### Why is the Second Law of Thermodynamics so Important?

Second law of thermodynamics is very important because it talks about entropy and as we have discussed, entropy dictates whether or not a process or a reaction is going to be spontaneous.

I want you to realize that any natural process happening around you is driven by entropy!!

Let’s take a daily life example:

We drink coffee every day. What happens to our cup of hot coffee in say 10 minutes? The coffee starts getting cold, or in thermodynamic terms you will tell me that the hot coffee gives out heat to the surroundings and in turn the coffee cools down. You are 100% correct, but the thing you might not have realized is that this very obvious daily life phenomenon is governed by entropy. Let’s try to prove this mathematically.

As shown below, the following two scenarios are possible:

Two cups of coffee, one giving off heat to the environment, and the other is absorbing heat from the environment.

The temperature of the surroundings and the coffee are 25o C and 45o C respectively.

#### Scenario 1

10 J of heat is absorbed (from the surroundings) by coffee, so surroundings lose 10 J and the system (coffee) gains 10 J. Thus, $Q_{system} = + 10 J$ and $Q_{surroundings} = -10 J$

$\Delta S_{universe} = \Delta S_{system} + \Delta S_{surroundings}$

$= Q_{system}/ T_{system} + Q_{surroundings}/T_{surroundings}$

= 10 /(45 + 273) + (-10)/ (25 + 273) [temperatures are to be converted into Kelvin]

= -0.0021 Joules/ Kelvin

This violates the second law of thermodynamics ($\Delta S_{universe}$ should be greater than zero), so the above process cannot occur spontaneously.

#### Scenario 2

10 J of heat is released (to the surroundings) by hot coffee, so surroundings gain 10 J and system (coffee) loses 10 J. Thus, $Q_{system} = - 10 J$ and $Q_{surroundings} = +10 J$

$\Delta S_{universe} = \Delta S_{system} + \Delta S_{surroundings}$

$= Q_{system}/ T_{system} + Q_{surroundings}/ T_{surroundings}$

= -10/ (45 + 273) + 10/ (25 + 273) [temperatures have to be converted into Kelvin]

= +0.0021 Joules/ Kelvin

This obeys the second law of thermodynamics ($\Delta S_{universe} > 0$), so the above process occurs spontaneously.

In the case of a cup of coffee this was pretty intuitive. But this is not true for chemical reactions. Just by looking at a chemical reaction, we cannot predict if it will take place spontaneously or not. So, calculating the entropy change for that particular reaction becomes important.

Table summarizing the relationship between ∆S and if a spontaneous, non-spontaneous reaction occurs.

#### Gibb’s Free Energy (G): Predictor of Spontaneity of a Chemical Reaction

Gibb’s free energy is defined as the energy associated with a chemical reaction that can be used to do work. The free energy (G) of a system is the sum of its enthalpy (H) minus the product of the temperature (T) and the entropy (S) of the system:

G = H - TS

Gibbs free energy combines the effect of both enthalpy and entropy. The change in free energy (ΔG) is equal to the sum of the change of enthalpy (∆H) minus the product of the temperature and the change of entropy (∆S) of the system.

∆G = ∆H - T∆S

*ΔG predicts the direction in which a chemical reaction will go under two conditions: (1) constant temperature and (2) constant pressure. *

Rule of thumb: If ΔG is positive, then the reaction is not spontaneous (it requires the input of external energy to occur) and if it is negative, then it is spontaneous (occurs without the input of any external energy).

Let’s attempt a problem involving ΔG:

Calculate ΔG for the following reaction at 25o C

NH3(g) + HCl(g) → NH4Cl(s)

but first we need to convert the units for ΔS into kJ/ K (or convert ΔH into J) and temperature into Kelvin. So,

ΔS = −284.8 J/ K = −0.2848 kJ/ K

T = 273.15 K + 25 = 298 K

*(1 kJ = 1000 J)

Now, ΔG = ΔH − TΔS

ΔG = −176.0 kJ − (298 K)(−0. 284.8 kJ/ K)

ΔG = −176.0 kJ − (−84.9 kJ)

ΔG = −91.1 kJ

Answer: Yes, this reaction is spontaneous at room temperature since ΔG is negative.

The beauty of the Gibb’s free energy equation is its ability to determine the relative importance of the enthalpy (ΔH) and the entropy (ΔS) of the system. The change in free energy of the system measures the balance between the two driving forces (ΔH and ΔS), which together determine whether a reaction is spontaneous or not.

Let's tabulate this information to make it easier to comprehend ΔG = ΔH − TΔS:

Favorable Reaction Conditions Unfavorable Reaction Conditions
∆H < 0 ∆H > 0
∆S > 0 ∆S < 0
∆G < 0 ∆G > 0

• If ∆H < 0 and ∆S > 0, without even doing any calculations you can say that the reaction will be spontaneous because ΔG = ΔH – TΔS will be negative.

• If ∆H >0 and ∆S < 0, without even doing any calculations you can say that the reaction will not be spontaneous because ΔG = ΔH – TΔS will be positive.

• Actual calculations become necessary when out of the two parameters, ∆H and ∆S, one is favorable and the other is not. In such a case, ΔG has to be calculated to predict the spontaneity of the reaction.

### Third Law of Thermodynamics

“The entropy of a perfect crystal of any pure substance approaches zero as the temperature approaches absolute zero.”

Whenever I think of this law, I am reminded of the beautiful documentary made on the survival tactics of penguins March of the Penguins. Those who have watched this documentary will recall: in order to survive the extremely cold weathers of Antarctica (where temperatures approach −80 °F or 210.92 K), penguins form colonies consisting of a group of huddles; and in each colony the penguins are tightly packed and do not move at all, and all the penguins face in the same direction as shown in the image on the right.

Image showing three penguins scattered and a group of penguins huddled together.

Now assume these penguins to be atoms. Analogous to the penguins, at a temperature of zero Kelvin the atoms in a pure crystalline substance get aligned perfectly and do not move around. Matter is in a state of maximum order (least entropy) when the temperature approaches absolute zero (0 Kelvin). In other words, the entropy of a perfect crystal approaches zero when the temperature of the crystal approaches 0 Kelvin. This is, in fact, the third law of thermodynamics. The consequence of this law is that any thermal motion ceases in a perfect crystal at 0 K. Conversely, if there will be any thermal motion within the crystal at 0 K, it will lead to disorder in the crystal because atoms will start moving around, violating the third law of thermodynamics.