## A Simple Example of a Thermodynamic Process

Now let’s take a simple example that everyone understands from ordinary life experience. Let’s put some cold water in an insulated container, so no heat will flow into or out of the water from or to the outside. What happens when we drop a hot stone into cold water? The stone cools off and the water warms up. Heat flows spontaneously from the hot stone into the cold water. Eventually the stone and water reach the same temperature. We can wait as long as we like, but heat will never flow the other way. Before we dropped the stone into the water, the heat was concentrated in one place, the stone. There was little heat in the cold water. Most of the heat was in one place and not in another. To that extent the heat was organized or ordered. When most of the heat flowed out of the stone into the water, the heat became disorganized or less ordered, because it was spread out through both the stone and the water. This is what we mean when we say there was a net increase in entropy.

Increases in entropy are closely associated with irreversibility. By themselves the stone will never get hot again while the water at the same time gets colder. The experiment of dropping the hot stone into the cold water is irreversible because there was an increase in entropy or disorder.

Of course we can take the stone out, dry it off, heat it up again, and refrigerate the water. We can reverse the effect of the experiment. To do so, we have to do work or expend energy on the stone and the water to return the two parts of the experiment to their original state of lower net entropy or greater organization. Our work must be intelligent. We have to separate the stone from the water before heating the stone and refrigerating the water. We can’t restore the stone and water to their original states by heating the stone and water together and then putting both into the refrigerator. Separating the stone from the water requires an intelligent, perceptive analysis of their properties. For instance, noting that the stone is solid and that water is liquid, we may pour off the water into another container while holding the stone in place. This doesn’t require great intelligence, but it is something that mindless random action can never do. The stone will never have a lucky fluctuation so great that it will hop out of the water all by itself.

From the foregoing we see that entropy and the second law of thermodynamics are closely associated with intelligence. Without intelligent work no house will clean itself, no mess will straighten itself out, and no heat will concentrate itself. The tendency toward disorder is universal and irreversible by itself. Reversing disorder, or organizing, requires work and intelligence. With sufficient intelligence one can perhaps design a machine or structure to do the work. Any machine that cleans or organizes, and any structure that reduces entropy, must have come from intelligence.

In the simple situation of the stone and the water we can easily estimate the increase in entropy. We need only know the amount of heat that flowed and the initial temperatures of the stone and water. Refining the estimate requires knowing the heat capacity of both the stone and the water. But a reasonable estimate will serve the needs of this example. The change in the internal energy of any object is the amount of energy that flows in, minus the amount that flows out. When the hot stone drops into the cold water, no heat flows into the stone, but some heat flows out of it into the water. The change in the amount of heat in the stone is therefore negative. That is the same as saying that the amount of heat in the stone decreases. At the same time, heat from the stone flows into the water, but no heat flows out of the water. The change in the amount of heat in the water is therefore positive. In other words, the amount of heat in the water increases. Notice that the size or magnitude of the change in the water’s amount of heat is exactly equal to the magnitude of the stone’s change in amount of heat. The two changes are equal in magnitude but opposite in sign. The amount of heat that flowed is equal to the magnitude of the change in amount of heat in either the water or the stone.

The final temperature is the temperature reached after the flow of heat ceases. The final temperature of the water is the same as the final temperature of the stone. It is higher than the initial temperature of the water and lower than the initial temperature of the stone.

The change in entropy is the change in the amount of heat divided by a certain temperature that we will call the appropriate temperature. The mathematical operation of division introduces an inverse relation between change in entropy and temperature. In an inverse relationship, when one quantity is big, the inverse quantity is small. Dividing by a big number gives a smaller answer than dividing by a small number. One hundred divided by 25 gives 4, which is smaller than 100 divided by 20 (= 5), which in turn is smaller than 100 divided by 5 (= 20). The appropriate temperature lies between the initial temperature and the final temperature. The stone’s temperature was high at the start of the experiment, and always remained higher than the water’s temperature, until at last both the stone and water reached the same final temperature and the heat stopped flowing. The appropriate temperature for the stone is therefore a high temperature, somewhere above the final temperature but below the initial temperature of the stone. Therefore the stone’s change of entropy is low in magnitude. This does not affect the sign of the stone’s change in entropy. The change in the amount of heat in the stone is negative, so the change in the stone’s entropy is negative. The hot stone’s entropy decreases when the stone drops into the cold water. The decrease in the stone’s entropy is small in magnitude because the stone’s appropriate temperature is high.

Now consider the water. The same heat that flowed out of the stone flowed into the water. The heat in the water increased. The water was at a low temperature at the start of the experiment. The water’s temperature was always lower than the stone’s temperature until the heat stopped flowing. Therefore the appropriate temperature for the water is low, somewhere below the final temperature but higher than the initial low temperature of the water. The water’s entropy increases by a large amount when the heat of the stone flows into the water.

To get the change in entropy of the combination of the stone and the water, we must add the water’s change of entropy to the stone’s change of entropy. The water’s change of entropy was positive and large. The stone’s change of entropy was negative and small in magnitude. Adding a large positive number to a small-magnitude negative number is the same as subtracting the small number’s magnitude from the large positive number. The result is a smaller number, but the number is positive. The change in entropy of the stone-and-water combination is positive. When we drop a hot stone into cold water, we raise the entropy of the stone-and-water combination. The stone cools down, and its entropy decreases by a small amount. The water warms up, and its entropy increases by a large amount. The overall entropy increases.

Increases in entropy are closely associated with irreversibility. By themselves the stone will never get hot again while the water at the same time gets colder. The experiment of dropping the hot stone into the cold water is irreversible because there was an increase in entropy or disorder.

Of course we can take the stone out, dry it off, heat it up again, and refrigerate the water. We can reverse the effect of the experiment. To do so, we have to do work or expend energy on the stone and the water to return the two parts of the experiment to their original state of lower net entropy or greater organization. Our work must be intelligent. We have to separate the stone from the water before heating the stone and refrigerating the water. We can’t restore the stone and water to their original states by heating the stone and water together and then putting both into the refrigerator. Separating the stone from the water requires an intelligent, perceptive analysis of their properties. For instance, noting that the stone is solid and that water is liquid, we may pour off the water into another container while holding the stone in place. This doesn’t require great intelligence, but it is something that mindless random action can never do. The stone will never have a lucky fluctuation so great that it will hop out of the water all by itself.

From the foregoing we see that entropy and the second law of thermodynamics are closely associated with intelligence. Without intelligent work no house will clean itself, no mess will straighten itself out, and no heat will concentrate itself. The tendency toward disorder is universal and irreversible by itself. Reversing disorder, or organizing, requires work and intelligence. With sufficient intelligence one can perhaps design a machine or structure to do the work. Any machine that cleans or organizes, and any structure that reduces entropy, must have come from intelligence.

In the simple situation of the stone and the water we can easily estimate the increase in entropy. We need only know the amount of heat that flowed and the initial temperatures of the stone and water. Refining the estimate requires knowing the heat capacity of both the stone and the water. But a reasonable estimate will serve the needs of this example. The change in the internal energy of any object is the amount of energy that flows in, minus the amount that flows out. When the hot stone drops into the cold water, no heat flows into the stone, but some heat flows out of it into the water. The change in the amount of heat in the stone is therefore negative. That is the same as saying that the amount of heat in the stone decreases. At the same time, heat from the stone flows into the water, but no heat flows out of the water. The change in the amount of heat in the water is therefore positive. In other words, the amount of heat in the water increases. Notice that the size or magnitude of the change in the water’s amount of heat is exactly equal to the magnitude of the stone’s change in amount of heat. The two changes are equal in magnitude but opposite in sign. The amount of heat that flowed is equal to the magnitude of the change in amount of heat in either the water or the stone.

The final temperature is the temperature reached after the flow of heat ceases. The final temperature of the water is the same as the final temperature of the stone. It is higher than the initial temperature of the water and lower than the initial temperature of the stone.

The change in entropy is the change in the amount of heat divided by a certain temperature that we will call the appropriate temperature. The mathematical operation of division introduces an inverse relation between change in entropy and temperature. In an inverse relationship, when one quantity is big, the inverse quantity is small. Dividing by a big number gives a smaller answer than dividing by a small number. One hundred divided by 25 gives 4, which is smaller than 100 divided by 20 (= 5), which in turn is smaller than 100 divided by 5 (= 20). The appropriate temperature lies between the initial temperature and the final temperature. The stone’s temperature was high at the start of the experiment, and always remained higher than the water’s temperature, until at last both the stone and water reached the same final temperature and the heat stopped flowing. The appropriate temperature for the stone is therefore a high temperature, somewhere above the final temperature but below the initial temperature of the stone. Therefore the stone’s change of entropy is low in magnitude. This does not affect the sign of the stone’s change in entropy. The change in the amount of heat in the stone is negative, so the change in the stone’s entropy is negative. The hot stone’s entropy decreases when the stone drops into the cold water. The decrease in the stone’s entropy is small in magnitude because the stone’s appropriate temperature is high.

Now consider the water. The same heat that flowed out of the stone flowed into the water. The heat in the water increased. The water was at a low temperature at the start of the experiment. The water’s temperature was always lower than the stone’s temperature until the heat stopped flowing. Therefore the appropriate temperature for the water is low, somewhere below the final temperature but higher than the initial low temperature of the water. The water’s entropy increases by a large amount when the heat of the stone flows into the water.

To get the change in entropy of the combination of the stone and the water, we must add the water’s change of entropy to the stone’s change of entropy. The water’s change of entropy was positive and large. The stone’s change of entropy was negative and small in magnitude. Adding a large positive number to a small-magnitude negative number is the same as subtracting the small number’s magnitude from the large positive number. The result is a smaller number, but the number is positive. The change in entropy of the stone-and-water combination is positive. When we drop a hot stone into cold water, we raise the entropy of the stone-and-water combination. The stone cools down, and its entropy decreases by a small amount. The water warms up, and its entropy increases by a large amount. The overall entropy increases.