 This is Thermodynamics 4, entropy, and the second law. Welcome. Can we make a machine that pulls energy out of thin air? After all, the air is composed of a vast number of molecules bouncing around, each with some kinetic energy. As we saw in a previous video for an ideal monatomic gas of n-atoms at temperature T, this internal energy is three-halves n-k-t, where k is Boltzmann's constant. At 20 degrees Celsius and standard atmospheric pressure, this works out to around 100 joules per liter of gas. It seems like, maybe, we should be able to devise a machine that can trap this energy and use it to, say, power our lights. The reason why we can't do this is stated in the celebrated Second Law of Thermodynamics. Let's jump right in and give some statements of this. Later we'll work out what these actually mean. No heat engine is more efficient than the Carnot cycle. This is sometimes known as Carnot's principle. Clausius stated the second law as, heat cannot of itself pass from a colder to a hotter body. Planck wrote, it is impossible to construct an engine which will work in a complete cycle and produce no effect except the raising of a weight and the cooling of a heat reservoir. Arguably, the most precise statement of the Second Law is, the entropy of a closed system can never decrease. Of course, this presupposes an understanding of the term entropy, possibly one of the most confusing and often misunderstood concepts in physics. Hopefully, by the end of this video, all of these statements will make sense and be seen to be equivalent statements of a single principle. Sadi Carnot's 1824 publication, Reflections on the Motive Power of Fire, is commonly considered the beginning of the modern science of thermodynamics. In addition to describing the Carnot cycle, he presented Carnot's theorem, no heat engine is more efficient than the Carnot cycle. Influenced by Carnot's work, Rudolf Clausius advanced thermal science in a series of papers starting around 1850 and summarized in his 1876 book, The Mechanical Theory of Heat. He developed the Second Law of Thermodynamics, the topic of this video. Clausius coined the word entropy based on the Greek word tropi, signifying a transformation or turning, with an N prefix to create a term similar to the word energy. As we'll see, entropy and energy are intimately connected. Let's review the Carnot cycle. We begin with the gas, the yellow region, maximally compressed and in contact with a reservoir at temperature T-hot. It undergoes isothermal expansion, which extracts heat from the reservoir and converts it to work. Then we remove the heat reservoir and insulate the gas. This is followed by adiabatic expansion until the gas temperature is reduced to D-cold. The gas is put in contact with a reservoir at temperature T-cold and isothermally compressed, which converts work into heat transferred to the reservoir. The reservoir is removed and the gas is insulated. Then the gas is adiabatically compressed until its temperature is T-hot. Now the cycle can be repeated. On a PV diagram, the cycle begins in the upper left at state one. Isothermal expansion takes the system to state two. Heat QS is transferred to the system and converted to work WE done on the environment, equal to NK T-hot natural log X2 over X1. Then adiabatic expansion takes the system from state two to state three. Internal energy is converted into work WE done on the environment, equal to three-halves NK T-hot minus T-cold, and the gas cools the temperature T-cold. Isothermal compression then takes the system from state three to state four. Work WS is done on the system and converted to heat QE transferred to the environment, equal to NK T-cold natural logarithm X2 over X1. Finally, adiabatic compression takes the system from state four to state one. Work WS equal to three-halves NK T-hot minus T-cold is done on the system and converted to internal energy as the gas warms to temperature T-hot. In the upper right corner we track the heat extracted from the hot reservoir Q-hot and the heat delivered to the cold reservoir Q-cold. Now consider a reversed Carnot cycle. We start at state one, move to state four through adiabatic expansion, move to state three by isothermal expansion, move to state two by adiabatic compression and return to state one using isothermal compression. For each leg the flow of work and or heat is opposite what it is for the normal or forward cycle. The heat Q-hot that was extracted from the hot reservoir in the forward cycle is now delivered heat and vice versa for the cold reservoir. To illustrate we start with the gas hot and maximally compressed. We remove the hot reservoir and insulate the system. The gas is allowed to expand adiabatically converting internal energy to work done on the environment until the gas temperature drops to T-cold. The cold reservoir is attached and the gas expands isothermally doing work on the environment and extracting heat from the cold reservoir. The cold reservoir is removed and the system is insulated. Now adiabatic compression converts work into internal energy until the gas temperature reaches T-hot. The hot reservoir is attached and isothermal compression converts work into heat which is delivered to the hot reservoir. Here is a schematic representation of the Carnot cycle run in the forward direction. Heat Q-hot is extracted from the hot reservoir. Some of this energy is converted into work W. The remainder is delivered as heat to the cold reservoir. By the first law Q-hot equals W plus Q-cold. One way to interpret this process is to say that a heat engine uses the natural tendency of heat to flow from hot to cold in order to generate mechanical work. For the cycle run in the reverse direction, the directions of energy flow are reversed. Work W is done on the system with the result that heat Q-cold is extracted from the cold reservoir and heat Q-hot is delivered to the hot reservoir. By the first law Q-hot equals Q-cold plus W. An interpretation of this process is to say that we have used mechanical work to make heat do something it does not do spontaneously, flow from cold to hot. We are quote pumping heat from the cold reservoir to the hot reservoir. Accordingly we can call this process a heat pump. This extremely useful system is the basis of refrigeration. The reverse directions of energy flow for these two cycles has the consequence that if we run both cycles in succession, everything returns to its original state. Each cycle completely reverses the effects of the other. So we say that the Carnot cycle is quote reversible. Although the first law tells us that energy is conserved, specific forms of energy are not. The heat Q-hot extracted from the hot reservoir is not equal to the heat Q-cold delivered to the cold reservoir. In fact, the first is greater than the second by the amount of work produced by the cycle. However, looking at the expressions for these heat values, we see that they are very similar, differing only in their respective temperature factors. Therefore, Q-cold over T-cold minus Q-hot over T-hot equals zero because both terms equal NK natural log X2 over X1. This result hints that something corresponding to transferred heat divided by temperature may be conserved in the Carnot cycle. This might be telling us that an amount of something equal to Q-cold over T-cold is delivered to the cold reservoir and this is equal to the same amount of something Q-hot over T-hot extracted from the hot reservoir. And maybe this conservation is characteristic of a reversible process. Now let's consider Carnot's principle, the claim that no heat engine is more efficient than the Carnot cycle. Let's suppose that we have found a quote supercycle that is more efficient than the Carnot cycle. Then we could do the following. Use a reversed Carnot cycle to deliver heat Q-hot to a hot reservoir at the expense of work W. In the process heat Q-cold is extracted from a cold reservoir. Then use our supercycle to extract the heat Q-hot from the hot reservoir and produce work W-prime. Recall that the work W we put into the reversed Carnot cycle to deliver heat Q-hot to the hot reservoir equals the work we would get out of a forward Carnot cycle for the same input heat. Since the supercycle is more efficient than the Carnot cycle it must produce more work from heat Q-hot. So W-prime must be greater than W. Then by the first law the heat delivered to the cold reservoir by the supercycle Q-cold-prime must be less than the heat Q-cold extracted by the reversed Carnot cycle. The net effect of this entire process has been to extract heat Q-cold-Q-cold-prime from the cold reservoir and produce work W-prime-W. We can get rid of the hot reservoir and simply direct the heat output of the reversed Carnot cycle into the supercycle. The heat output by the supercycle can be directed to the input of the reversed Carnot cycle leaving only the difference Q-cold-Q-cold-prime to be extracted from the cold reservoir. And the work W required for the reversed Carnot cycle can be taken from the greater work W-prime output by the supercycle leaving W-prime-W as the net useful work generated by the process. What we have, then, is precisely an energy from thin air machine that extracts heat from a single heat reservoir and generates work. This demonstrates the equivalence of Carnot's principle with Planck's statement of the second law.