ENTROPY AND THE SECOND LAW

The results of Example 18-10 about the flow of heat from a higher to a lower temperature, or the mixing of  substances at different temperatures, are characteristic of all natural [that is, irreversible] processes. When we include the entropy changes of all the systems taking part in the process, the increases in entropy are always greater than the decreases. In the special case of reversible process, the increases and decreases are equal. Hence we can state the general principle: When all systems taking part in a process are included, the entropy either remains constant or increases. In other words, no process is possible in which the total entropy decreases, when all systems taking part in the process are included.  This is an alternative statement of the second law of thermodynamics in terms of entropy. Thus it is equivalent to the “engine” and “refrigerator” statements discussed earlier.

The increase of entropy in every natural, irreversible process measures the increase of disorder or randomness in the universe associated with that process. Consider again the example of mixing hot and cold water. We might have used the hot and cold water as the high- and low-temperature reservoirs of a heat engine. While removing heat from the hot water and giving heat to the cold water, we could have obtained some mechanical work. But once the hot and cold water have been mixed and have come to a uniform temperature, this opportunity to convert heat to mechanical work is lost irretrievably. The lukewarm water will never unmix itself and separate into hotter and colder portions. No decrease in energy occurs when the hot and cold water are mixed. What has been lost is not an energy, but opportunity, the opportunity to convert part of the heat from the hot water into mechanical work. Hence when entropy increases, energy becomes less available, and the universe becomes more random or ”run down”. 

The statement of the second law is in terms of entropy:

“The entropy of a closed system never decreases or equivalently: The change in entropy of the universe is always greater than or equal to zero”:

Equivalent Statements Of The Second Law Of Thermodynamics

The Kelvin statement:

For example, it is easy to convert mechanical work completely into thermal energy, but it is impossible to remove thermal energy from a system and convert it completely into mechanical work with no other changes. This experimental fact is one statement of the second law of thermodynamics.

“It is impossible to remove thermal energy from a system at a single temperature and convert it to mechanical work without changing the system or surroundings in some other way”.

Second Law of Thermodynamics: Kelvin Statement

The Clausius statement:

If we place an ice cube on a hot day, the ice cube will melt. From the point of view of energy, what has happened is that some of the heat from the surrounding air enters the ice cube, raising its temperature, and eventually melting it. The surrounding air subsequently cools somewhat. However, nothing from energy conversion, or Newton’s law in general, would prevent heat from leaving the ice cube, making the ice cube colder and the surrounding air warmer. Why then doesn’t this later phenomena occur?

A common example of the conversion of mechanical energy into thermal energy is movement with friction. For example, when a block slides along a rough table, the initial mechanical [kinetic] energy of the block is converted into thermal energy as the block and the table are heated. The reverse process never occurs- a block and table that are warm will never spontaneously cool by converting their thermal energy into kinetic energy that sends the block sliding across the table. Thus there is a lack of symmetry in the roles played by heat and work. This lack of symmetry is related to the fact that some processes are irreversible. It may be mentioned that thermodynamic processes that occur in nature are all irreversible.

Let us take the case of heat conduction which is an irreversible process. If we place a hot body in contact with a cold body, heat will flow from the hot body to the cold body until they are at the same temperature. The reverse does not occur; heat does not flow from one to the other making one colder and the other warmer.

The answer lies in the Clausius statement of the 2nd law of thermodynamics, which can be written as:

“A process whose only final result is to transfer thermal energy from a cooler object to a hotter one is impossible”.

Second Law of Thermodynamics: Clausius Statement

In other words, “Heat never flows spontaneously from low temperature to high temperature”.

The Refrigerator statement:

Refrigerator:

It is impossible to make heat flow from a body at a lower temperature to a body at a higher temperature without doing external work on the working substance. Energy will not flow spontaneously from a low temperature object to a higher temperature object. This precludes a perfect refrigerator. 

“No cyclic process can transfer heat from a colder place to a hotter place with no input of mechanical work”.

Second Law of Thermodynamics: Refrigerator Statement

 

Units of Heat:

Before scientists realized that heat is transferred energy, heat was measured in terms of its ability to raise the temperature of water. Thus, the calorie (cal) was defined as the amount of heat that would raise the temperature of 1 g of water from 14.5⁰C to 15.5⁰C.

In 1948, the scientific community decided that since heat (like work) is transferred energy, the SI unit for heat should be the one we use for energy, namely, the joule. The calorie is now defined to be 4.1860 J (exactly). The “calorie” used in nutrition is really is kilocalorie.

1.                   Calorie (cal): It is the amount of heat required to increase the temperature of 1 g of water from 14.5^oC to 15.5^oC. (1 cal=4.186 J)

2.                   Kilocalorie (kcal): It is the amount of heat required to raise the temperature of 1 kg of pure water through 1^oC.

3.                   British Thermal Units (BTU): It is defined as the quantity of heat required to raise the temperature of 1 pound (lb) of pure water through 1^oF. It is also referred to as pound-degree Fahrenheit unit.

Conversion:            1 B.T.U = 251.996 Cal ( or 252 Cal)
1 calorie = 4.186 Joule ( or 4.2 joule)

Temperature and Heat

If you take a can of cola from the refrigerator and leave it on the kitchen table, its temperature will rise- rapidly at first but then more slowly – until the temperature of the cola equals that of the room(the two are then in thermal equilibrium).

In generalizing this situation, we describe the cola or coffee as a system (with temperature T_S) and the relevant part of the kitchen as the environment (with temperature T_E) of that system. Our observation is that if T_S is not equal to T_E, then T_S will change until the two temperatures are equal and thus thermal equilibrium is reached.

Such a change in temperature is due to the transfer of energy between the thermal energy of the system and the system’s environment. It may be mentioned that thermal energy is an internal energy that consists of the kinetic and potential energies associated with the random motions of the atoms, molecules and other microscopic bodies within an object. The transferred energy is called heat and is symbolized Q. Heat is positive when energy is transferred to a system’s thermal energy from its environment (we say that heat is absorbed). Heat is negative when energy is transferred from a system’s thermal energy to to its environment ( we say that heat is released or lost).

We are then led to this definition of heat:

“Heat is the energy that is transferred between a system and its environment because of a temperature difference that exists between them.”

Recall that energy can also transferred between a system and its environment as work W via a force acting on a system. Heat and work, unlike temperature, pressure, and volume, are not intrinsic properties of a system. They have meaning only as they describe the transfer of energy into or out of a system.

Let us now look into the Molecular Theory of Matter for an explanation of heat and temperature. Molecular Theory of Matter states that matter is made up of tiny particles called molecules. These particles are in constant motion within the bounds of the material. Since the relationship between kinetic energy of an object and its velocity is: KE = ½ mv², which means that the more energy an object has, the faster it is traveling (or vice versa).

Thus, when you provide extra energy to an object, you cause its molecules to speed up. Those molecules, in turn, can cause other molecules to speed up. The sum effect of the speed or energy of these molecules is the phenomenon we call heat. Molecules can go into high-energy motion, causing heat, from various energy sources such as Light, Chemical reactions, Electrical resistance, Friction and nuclear reactions.

Heat is defined as "The total kinetic energy of all the molecules of a body" and temperature is a measure of “the average internal molecular kinetic energy of an object”.

The Zeroth Law of Thermodynamics

The "zeroth law" states that if two systems are in thermal equilibrium with a third system, then they must be in thermal equilibrium with each other.

This law states that if object A is in thermal equilibrium with object B, and object B is in thermal equilibrium with object C, then object C is also in thermal equilibrium with object A.

The message of the zeroth law is: “Every body has a property called temperature”. Two objects are defined to have the same temperature if they are in thermal equilibrium with each other. This law allows us to build thermometers. For example the length of a mercury column (object B) may be used as a measure to compare the temperatures of the two other objects.

The zeroth law, which has been called a logical afterthought, came to light only in 1930s, long after the first and second laws of thermodynamics had been discovered and numbered. Because the concept of temperature is fundamental to those two laws, the law that establishes temperature as a valid concept should have the lowest number-hence the zero.

The zeroth law has a fairly straightforward statistical interpretation and this will allow us to begin to establish the equivalence between the statistical definitions and the conventional thermodynamic ones.

Thermal equilibrium and Temperature:

The central concept of thermodynamics is temperature. Temperature is familiar to us all as the measure of the hotness or coldness of objects. We shall learn afterward that temperature is a measure of the average internal molecular kinetic energy of an object.

It is observed that a higher temperature object which is in contact with a lower temperature object will transfer heat to the lower temperature object. The objects will approach the same temperature, and in the absence of loss to other objects, they will then maintain a constant temperature. They are then said to be in thermal equilibrium. Thermal equilibrium is the subject of the Zeroth Law of Thermodynamics.

Thermal expansion of solids and liquids

A mercury thermometer utilizes thermal expansion: the phenomenon that most substances increase in volume as their temperature increases. A rod that is heated will change in length (Δ L) according to Δ L = α L₀ Δ T, where L₀ is the original length and Δ T (delta T) change in temperature. The constant α (Greek letter alpha) is the average coefficient of linear expansion. This value is found in tables of coefficients for different materials and is measured in units of (degrees C)⁻¹.
Not only does length change with a change in temperature, but area and volume change also. Thus, Δ A = γ A₀Δ T, where Δ A is the change in the original area A₀. The Greek letter gamma (γ) is the average coefficient of area expansion, which equals 2α. For change in volume, Δ V = β V₀, Δ T, where Δ V is the change in the original volume V₀. The Greek letter beta (β) is the average coefficient of volume expansion, which is equal to 3α.
Example 1: As an example of the application of these equations, consider heating a steel washer. What will be the area of the washer hole with original cross-sectional area of 10 mm² if the steel has α = 1.1 × 10⁻⁵ per °C and is heated from 20 degrees C to 70 degrees C?
Solution: The hole will expand the same as a piece of the material having the same dimensions. The equation for increase in area leads to the following:

Therefore, the new area of the hole will be 10.011 mm².

Water is an exception to the usual increase in volume with increasing temperature. Note in Figure 2 that the maximum density of water occurs at 4 degrees Celsius.

Figure2:  The density of water changes as the temperature changes. Figure2:  The density of water changes as the temperature changes.

This characteristic of water explains why a lake freezes at the surface. To see this, imagine that the air cools from 10 degrees Celsius to 5 degrees Celsius. The surface water in equilibrium with the air at these temperatures is denser than the slightly warmer water below it; therefore, the colder water sinks and warmer water from below comes to the surface. This occurs until the air temperature decreases to below 4 degrees when the surface water is less dense than the deeper water of about 4 degrees; then, the mixing ceases. As the temperature of the air continues to fall, the surface water freezes. The less dense ice remains on top of the water. Under these conditions, life near the bottom of the lake can continue to survive because only the water at or near the surface is frozen. Life on earth might have evolved quite differently if a pool of water froze from the bottom up.

Thermometers and temperature scales

The sense of touch provides some indication of the temperature of an object but is unreliable. For example, the metal shelf in the refrigerator feels colder than the food sitting on the shelf, even though they are in thermal equilibrium. The metal feels colder because the metal conducts the heat from your hand more efficiently.
Thermometers are instruments that define and measure the temperature of a system. The common thermometer consists of a volume of mercury that expands into a capillary tube when heated. When the thermometer is in thermal equilibrium with an object, the temperature can be read from the thermometer scale.

Three temperature scales are commonly used: Celsius, Fahrenheit, and Kelvin (also called absolute). Comparisons of the Celsius and Fahrenheit thermometers are shown in Figure 1 .

Figure:  Comparison of Celsius and Fahrenheit thermometers. Figure:  Comparison of Celsius and Fahrenheit thermometers.

On the Celsius scale, the ice point is 0, and the steam point is 100. The interval between these temperatures is divided into 100 equal parts called degrees. As shown in Figure 1 , on the Fahrenheit scale, the ice point is 32 degrees, and the steam point is 212 degrees. The interval between these temperatures is divided into 180 equal parts. The following equations relate temperature in Celsius (C) and Fahrenheit (F):

The Kelvin scale (K) has degrees of the same size as the Celsius scale, but the zero is shifted to the triple point of water. The triple point of water exists when water within a closed vessel is in equilibrium in all three states: ice, water, and vapor. This point is defined as 273.16 Kelvin and equals .01 degrees Celsius; therefore, to convert Celsius to Kelvin, simply add 273.15. Note that because the degrees are the same in the two scales, temperature differences are the same in either Celsius or Kelvin.

Temperature

Thermodynamics is the field of physics that successfully explains the properties of matter visible in our everyday, macroscopic world and the correlation between those properties and the mechanics of atoms and molecules.

Temperature measures how hot or cold a body is with respect to a standard object. To discuss temperature changes, two basic concepts are important: thermal contact and thermal equilibrium. Two objects are in thermal contact if they can affect each other's temperature. Thermal equilibrium exists when two objects in thermal contact no longer affect each other's temperature. For example, if a carton of milk from the refrigerator is set on the kitchen countertop, the two objects are in thermal contact. After several hours, their temperatures are the same, and they are then in thermal equilibrium.