BOLTZMANN ENTROPY, THERMODYNAMICS, AND COFFEE

       Many of us consider mornings to be incomplete without a cup of coffee. But what does a cup of coffee has to do with thermodynamics?
       Thermodynamics is a branch of physics that deals with heat, energy, and work. Thermodynamics is concerned with the thermodynamic system, its interaction with the surrounding and the universe. A cup of coffee is our system here, the room is our surroundings and all these combine to form our universe. Apart from this, we must know about entropy. Entropy is the measure of the randomness/chaos/disorder of a system. 

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        Now let's come back to our cup of black coffee. Our coffee has a pair of stories to tell-why it cools down and why milk/cream spontaneously mixes with it. The coffee is hot and the cup is itself cool. Now heat transfer starts, and the coffee(system) exchanges matter(water vapor) and energy(by means of heat transfer) with its surrounding(cup and air of the room). The coffee in a cup is thus called an open system. The cooling of coffee is an irreversible process and it is known that entropy in case of an irreversible process always increases. So what is entropy? Entropy has two statements. The one given by Clausius is ΔS=Q/T where ∆S=change in entropy, Q= flow of heat from a system to a large reservoir whose temperature is T kelvin above absolute zero for a closed, reversible process. But since an ideal reversible process is not possible so what happens is that ΔS>Q/T for an irreversible process. The cooling of coffee is spontaneous and irreversible no matter what. There is no way through which a cup of coffee left aside will keep itself hot forever. The cup of coffee exchanges heat with its surroundings, spontaneously. As the coffee cools, heat energy agitates the surrounding air molecules and makes the arrangement a bit disordered thereby increasing the entropy of the surrounding. This in turn, increases the entropy of the universe. So the local entropy of the cup of coffee decreases(during cooling the agitation of the hot coffee+water molecules calm down as they release energy into the surrounding). But the entropy of the universe increases, because energy is lost. So this is why a cup of coffee always cools down even if we want it to stay hot forever.

       The second story is about why a quart of cream/milk will mix spontaneously with coffee. Here comes Ludwig Boltzmann and statistical mechanics. He stated that S= k.logW where S=measure of entropy, k=Boltzmann constant, and logW=natural logarithm of the available microstates of a system that corresponds to the macrostate of the system. By macrostate, we mean pressure, temperature, volume of a system. After characterizing the macrostate of a system, with the help of statistical mechanics we can tell about a system's microstate i.e the behavior of each and every molecule of the system. In other words, Boltzmann entropy gives us the relationship between entropy and the number of ways the atoms and molecules of a thermodynamic system that can be arranged. When we add cream to our coffee it's more probable for the cream molecules to be arranged with the coffee molecules. Since the mixing of cream with coffee is again a spontaneous reaction so there is no way for the cream molecules to get unmixed once it is dispersed among the coffee. And because a cup of coffee is an open system so at the end of the day entropy of the universe ultimately increases. Boltzmann entropy also plays an important role apart from this. The above equation explains the arrow of time-the order of progress of a reaction which only moves forward. Since the entropy of the universe always increases so time also moves in the forward direction and never backward. For example, there is absolutely no way to unbreak an egg i.e to convert an omelet to a healthy full egg. This happens because the broken pieces have a larger probability of arranging themselves in disordered state(higher entropy) than to arrange themselves in an ordered state(lower entropy, full egg). The same is with the burning of matches which are nothing but spontaneous processes. If the process was non-spontaneous then an interesting thing would have happened. The egg would be broken and then the broken pieces would arrange themselves again into a full egg. Entropy would first increase and then decrease, thereby reverting to its original state. But that thing never happens.

           An analogy to understand entropy in simple terms: Suppose there is a word called "FOREVER". You can arrange the letters F,O,R,E,V in only one meaningful way and a thousand gibberish combinations. So the arranged word "FOREVER" has low entropy(ordered). The other gibberish seven letter word combinations have high entropy(disordered). 

          Thus, thermodynamics is an interesting branch of physics. With the knowledge of thermodynamics, we can explain a lot of natural phenomena starting from the behavior of car engines, cooling of a cup of coffee to stars, galaxies, and the evolution of our universe.