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How did the second law of thermodynamics change the world?

In May 1959, physicist and novelist CP Snow gave a speech entitled "Two Cultures", which caused widespread controversy. Snow believed that the sciences and the humanities had lost touch, making it very difficult to solve some of the world's problems. We see the same today with climate change denial and attacks on evolution. Snow was particularly displeased with what he saw as falling educational standards saying:

I have been to such gatherings many times: By the standards of traditional culture, these people are considered highly educated people. Once or twice I got so irritated that I asked how many of them could describe the second law of thermodynamics (aka the law of entropy). Almost no one can.

Thermodynamics means the kinetics of heat. Heat can flow, it can flow from one location to another, and move from one object to another. Fourier wrote down the first important model of heat flow and did some math. But the main reason scientists are interested in heat flow is a novel and lucrative technology: the steam engine.


Around 50 B.C., the Roman architect and engineer Vitruvius described a machine called a gyrosphere in his On Architecture, and a century later Greek mathematicians and engineers built a gyrosphere. ball. It's a hollow sphere with some water in it, and two tubes protruding from it, bent at an angle, as shown in the picture below. Heating the sphere turns the water into steam, which escapes through the end of the tube, and the reaction force spins the sphere. It was the first steam engine, and it proved that steam engines really do work.


At the age of 26 Watt discovered that steam could be a form of power. But actual steam power came much earlier. The discovery of steam power is generally credited to Italian engineer and architect Giovanni Branca, whose 1629 "Machines" includes 63 woodcuts of mechanisms. One of the paintings shows a paddle wheel that spins on its axis when steam from the pipes collides with the blades. Blanca speculates that the machine may have been used for milling flour, lifting water, and chopping wood, but it was probably never built. Blanca's steam engine was a conceptual machine, like da Vinci's flying machine.


Later, steam engines performed various industrial tasks, most commonly pumping water from mines. When the upper layer of mineral resources has been exploited, investors need to dig deeper into the ground, and they will inevitably encounter groundwater. At this time, either close the well and give up or pump out the groundwater. Investors are clearly unwilling to give up valuable mineral resources. Therefore, they urgently need a kind of equipment (machine) to complete the task of pumping water. Engineers turned their current attention to the steam engine, the study of which created a new branch of physics thermodynamics. Thermodynamics reveals everything from gases to the structure of the entire universe; it applies not only to inorganic matter in physics and chemistry, but may also apply to the complex processes of life. It is the law of conservation of energy in thermodynamics that breaks the illusion of perpetual motion machines.

One of these laws, the first law of thermodynamics, revealed a kind of energy related to heat and extended the law of conservation of energy to the realm of heat engines. Another study showed that certain methods of heat exchange that do not conflict with the conservation of energy are impossible because they must create order out of disorder. This is the second law of thermodynamics.

Thermodynamics is the mathematical physics of gases. It explains how the interaction of gas molecules produces the macroscopic signatures of temperature and pressure. Classical thermodynamics does not involve molecules (few scientists believed it at the time). Later, the gas law was further explained, based on a simple mathematical model that explicitly involved molecules. Gas molecules are thought of as tiny spheres that bounce off each other like perfectly elastic billiard balls, with no loss of energy in the collision. Although the molecules are not spherical, this model proved to be very valid. Known as the kinetic theory of gases, it proved the existence of molecules experimentally.


The early gas laws were developed intermittently over a period of nearly 50 years by the Irish physicist and chemist Robert Boyle, the French mathematician Jacques Alexandre, and the French physicist and chemist Guy Lu sack. In 1834, the French engineer and physicist Clapeyron combined all these laws into one, the ideal gas law, which we now write as

p is pressure, V is volume, T is temperature, and R is a constant. This equation states that pressure times volume is directly proportional to temperature. Later, physicists did a lot of research on many different gases to confirm the ideal gas law. The term "ideal" arises because real gases do not obey this law in all cases. But the ideal gas assumption is good enough for designing steam engines.


Thermodynamics is encapsulated in many more general laws and does not depend on the precise form of the gas laws. However, it does require that such laws exist, since temperature, pressure, and volume are not independent of each other. There must be some connection between them, but that doesn't matter.


The first law of thermodynamics derives from the mechanical law of conservation of energy. In classical mechanics, there are two distinct kinds of energy, kinetic energy, and potential energy. Neither energy is conserved. Newton's second law of motion states that changes in these two quantities cancel each other out, so the total energy does not change during motion.


However, this is not the whole of the conservation law. If you push a book that is placed on a table, its potential energy will not change if the table is horizontal. But its speed has changed, and it will soon come to a halt. So its kinetic energy starts from a non-zero initial value and then drops to zero. The total energy is thus also reduced, so energy is not conserved. where did it go Why did the book stop? According to Newton's first law, the book should continue to move unless an external force acts on it. This force is the friction between the book and the table. But what is friction?

There are some slight bumps on the rough surface of the book. These will touch parts of the table that are also slightly bumpy. They rub against each other creating a force, so the book slows down and loses energy. So where did the energy go? Maybe conservation laws don't apply at all. Or, this energy is still lurking somewhere. This is what the first law of thermodynamics tells us: The energy that "disappears" appears in the form of heat. Since the days of drilling fires, humans have known that friction generates heat. The first law of thermodynamics states that heat is a form of energy and that energy is conserved in thermodynamic processes.


The first law of thermodynamics limits the efficiency of a heat engine, the kinetic energy that can be obtained is always less than the energy input in the form of heat. It turns out that there is a theoretical limit to how efficiently a heat engine can convert thermal energy into kinetic energy, and only a portion of that energy can be converted into kinetic energy. The second law of thermodynamics turns this fact into a general principle, as we'll see later. In 1824, Carnot discovered this limitation in a simple model of how a steam engine works: the Carnot cycle.

To understand the Carnot cycle, it is important to distinguish between heat and temperature. In classical thermodynamics, neither concept is straightforward. Temperature is a property of fluids, but heat is only a measure of energy transfer between fluids, not an inherent property of the fluid state. In kinetic theory, the temperature of the fluid is the average kinetic energy of the molecules, and the heat transferred between the fluids is the change in the total kinetic energy of the molecules. Heat is a bit like potential energy in the sense that it is defined relative to an arbitrary reference height; this introduces an arbitrary constant, so the potential energy of an object is not uniquely defined. In short, heat is only meaningful when it is transferred, and temperature is a state. The two are linked, heat transfer is only possible if the temperatures are different, this is often called the zeroth law of thermodynamics because it logically precedes the first law.


Temperature can be measured with a thermometer, which uses the principle of the expansion of a fluid caused by an increase in temperature. Heat can be measured in terms of its relationship to temperature. In a standard test fluid, a 1-degree increase in the temperature of 1 gram of fluid (such as water) corresponds to a fixed increase in calorific content. This quantity is called the specific heat of the liquid. Note that an increase in heat is a change, not a state, as defined by the definition of heat.


We can think of the Carnot cycle as a cylinder with a movable piston at one end. This cycle has four steps:

  • The gas is heated so rapidly that there is no time for the temperature to change, and the gas expands, doing work on the piston.

  • Allowing the gas to expand further reduces the pressure and the gas cools.

  • Rapidly compress the gas to keep its temperature constant. The piston is now doing work on the gas.

  • Let the gas expand further, increasing the pressure. The gas returns to its original temperature.

Carnot's theorem proves that, in principle, the Carnot cycle is the most efficient way to convert heat into work. This places a severe limit on the efficiency of any heat engine, especially a steam engine.



In the relationship between gas pressure and volume, the Carnot cycle is shown in the figure below. German physicist and mathematician Rudolf Clausius found an easier way to visualize a Carnot cycle, as shown below (right). The two axes are temperature and a new fundamental quantity, entropy. In this coordinate, the cycle becomes a rectangle, and the amount of work done is the area of ​​the rectangle.

Entropy is like heat in that it's defined in terms of changes in state, not the state itself. Suppose a fluid change from some initial state to a new state. Then the entropy difference between these two states is the total change in heat divided by temperature. The transformation of entropy S can be expressed by the differential equation dS = dq / T. Entropy change is the change in heat per unit temperature. With the definition of entropy, the second law of thermodynamics is very simple. It shows that in any thermodynamic process, the entropy of an isolated system always increases, and the symbol is expressed as dS≥0 .


Classical thermodynamics is phenomenological, it describes what can be measured, but it is not based on any theory of related processes. Daniel Bernoulli first proposed the kinetic theory of gases in 1738. This theory provides a physical explanation for pressure, temperature, gas laws, and mysterious entropy. The basic idea, that gases consist of large numbers of identical molecules bouncing around in the air, occasionally colliding with each other, was much debated at the time. Since the molecules are small but have a non-zero size, occasionally two molecules collide. The kinetic theory of gases makes a simplifying assumption that the collisions between molecules are perfectly elastic so that no energy is lost during the collision.


When Bernoulli first proposed this model, the law of conservation of energy had not yet been established, and perfect elasticity seemed unlikely. The theory gradually gained support from a handful of scientists who came up with their own versions, incorporating all sorts of new ideas. German chemist and physicist August Kronig postulated that molecules cannot rotate. A year later, Clausius, one of the important founders of kinetic theory, canceled this simplification and proposed a key concept of the theory, namely, the mean free path of molecules, that is, between successive collisions, the mean free path of molecules Moving distance.


Both Kronig and Clausius derived the ideal gas law from kinetic theory. The three key variables are volume, pressure, and temperature. The volume is determined by the container, and the boundary conditions affect the behavior of the gas but are not characteristic of the gas itself. Pressure is the average force exerted by gas molecules when they collide with the container walls. It depends on how many molecules are in the container, and how fast they are moving. The temperature depends on the speed at which the gas molecules are moving, which is proportional to the average kinetic energy of the molecules.

Deriving Boyle's law (a special case of the ideal gas constant temperature law) is particularly straightforward. At a fixed temperature, the distribution of velocities doesn't change, so the pressure is determined by the number of molecules hitting the wall. If the volume is reduced, the number of molecules per cubic unit of space increases, and so does the chance of any molecule hitting a wall. The smaller the volume, the denser the gas, and the more molecules hit the walls. So Boyle's law has a deeper theoretical basis, based on molecular theory.



Inspired by Clausius, Maxwell wrote down the probability formula (based on a normal distribution) for a molecule to move at a given velocity, putting kinetic theory on a mathematical basis. Maxwell's formulas were the first laws of physics based on probability. Austrian physicist Ludwig Boltzmann then came up with the same formula, now known as the Maxwell-Boltzmann distribution. Boltzmann reinterpreted thermodynamics with the theory of gas dynamics, establishing what is now known as statistical mechanics. In particular, he proposed a new interpretation of entropy, linking thermodynamic concepts to the statistical properties of gas molecules.


Traditional thermodynamic quantities such as temperature, pressure, heat, and entropy refer to the macroscopic properties of gas molecules. However, macroscopic gases are made of many molecules spinning and colliding with each other. Boltzmann distinguished between the macroscopic and microscopic states of a system. Using this, he showed that entropy, a macroscopic state, can be interpreted as a statistical characteristic of a microscopic state. The equation is expressed as

Here S is the entropy of the system, W is the number of distinct microstates, and k is a constant known as Boltzmann's constant, which has a value of 1.38 × 10^(−23) joules per Kelvin. It is this axiom that explains entropy as a disorder. Ordered macrostates correspond to fewer microstates (W_1) than disordered macrostates (W_2).


Boltzmann's ideas were not widely accepted. On a technical level, thermodynamics is plagued by intractable conceptual problems. One is the exact meaning of "microstate". The position and velocity of a molecule are continuous variables that can take on an infinite number of values, but Boltzmann needs a finite number of microstates to count how many there are, and then take the logarithm. Therefore, these variables must be " coarse-grained " in some way, by breaking up the continuous interval of possible values ​​into a finite number of very small intervals. Another question that is more philosophical in nature is the arrow of time — the conflict between the time-reversible dynamics of a microstate dictated by entropy increases and the one-way time of a macrostate. These two issues are related, as we will see shortly.


However, the biggest obstacle to the acceptance of the theory is that matter is made of extremely small particles (atoms). This concept dates back to ancient Greece, but even around 1900, most physicists did not believe that matter was made of atoms. So they don't believe in molecules either, and gas theory based on molecules is obviously nonsense. Maxwell, Boltzmann, and other pioneers of motion theory were convinced that molecules and atoms were real, but to skeptics, atomic theory was just a convenient way to picture matter. Atoms have never been observed, so there is no scientific evidence for their existence. Molecules, specific combinations of atoms, are also controversial. Although atomic theory fits various experimental data in chemistry, it does not prove the existence of atoms.


What finally convinced the naysayers was the use of kinetic theory to predict Brownian motion. This effect was discovered by Scottish botanist Robert Brown. He pioneered the use of the microscope and discovered the existence of the cell nucleus, which is now considered the repository of the cell's genetic information. In 1827, Brown looked at pollen grains in liquid through a microscope, and he found smaller particles ejected from the pollen. The tiny particles swam around in a random fashion, and at first, Brown wondered if they were some kind of tiny life form. However, experiments have shown that particles from nonliving matter have the same effect. At the time, no one knew what caused this result. We now know that the particles ejected from pollen are organelles, tiny subsystems in cells that have specific functions. We interpret their random walks as evidence for the theory that matter is made of atoms.



The connection between atoms comes from the mathematical model of Brownian motion, which first appeared in the statistical studies of the Danish astronomer Torvald Thiele in 1880. Einstein came up with a physical explanation for Brownian motion: Particles floating in a fluid randomly bump into other particles, imparting tiny forces to them. On this basis, Einstein used mathematical models to make quantitative predictions of the statistics of Brownian motion, which were confirmed by Jean-Baptiste Perrin (1908-1909). Boltzmann committed suicide in 1906, just as the scientific community was beginning to realize that his theories were correct.


Entropy, together with the Boltzmann formula, provides an excellent model for many studies. It explains why heat engines can only achieve a certain level of efficiency. This applies not only to Victorian steam engines but also to modern car engines. Engine design is one of the practical areas of the laws of thermodynamics. Power generation is another application. In a coal, natural gas, or nuclear power plant, the initial production is heating. The heat creates steam, which drives a turbine. Turbines follow Faraday's principle of converting kinetic energy into electrical energy.


Thus, the laws of thermodynamics are the basis for many things we take for granted. Interpreting entropy as "disorder" helps us understand these laws and gain an intuitive feel for their physical basis. In some cases, however, interpreting entropy as a disorder seems to lead to paradoxes. This is a more philosophical area of ​​discussion, and it's fascinating.


The arrow of time is one of the deepest mysteries in physics. Time seems to flow in a certain direction. However, logically and mathematically, it seems that time can be turned back, and a lot of science fiction takes advantage of this. So why can't time turn back? At first glance, thermodynamics offers a simple explanation for the arrow of time: it is the direction in which entropy increases. The thermodynamic process is irreversible, such as oxygen and nitrogen will automatically mix, but not automatically do not separate.


However, there is a conundrum here, because any classical mechanical system, such as molecules in a room, is time-reversible. In the mathematical equation, if at a certain moment, the velocities of all particles are reversed at the same time, then the system will follow its trajectory, going backward in time. So why do we never see a broken egg automatically become whole?



The usual thermodynamic answer is that a broken egg is more disordered than a whole egg, with increased entropy, which is how time flows. Yet another explanation, the difference between entropy increase and time reversibility comes from the initial conditions, not the equations. The equations of molecular motion are time-reversible, but the initial conditions are not. The most important distinction here is the difference between the symmetry of an equation and the symmetry of its solution. The equations of the colliding molecules have time-reversal symmetry. From the time reversibility of the equation, it can be deduced at most that there must be another solution, that is, the time reversibility of the first solution. If Xiaoming throws the ball to Xiaohua, the time reversal solution is Xiaohua throwing the ball to Xiaoming. Likewise, since the equations of mechanics allow for a vase to fall to the ground and break into a thousand pieces, they must also allow for a solution in which a thousand glass shards mysteriously come together to assemble a complete vase.


But we've never seen broken vases recover by themselves. This is also a question about boundary conditions (initial conditions). The initial conditions of the vase-breaking experiment are easy to realize, and the experimental equipment is easy to obtain. In contrast, vase assembly experiments require extremely precise control of countless individual molecules, without any interference.


The mathematical calculation of entropy obscures details at these very small scales. It makes vibration disappear without increasing; it makes friction turn into heat, but it cannot turn heat into friction. The difference between the second law of thermodynamics and microscopic reversibility comes from the coarse-grained assumption. These assumptions implicitly specify an arrow of time: large-scale perturbations are allowed to disappear below appreciable levels over time, but small-scale perturbations are not allowed to follow time reversals.


How does a chicken create an ordered egg if entropy is always increasing? A common explanation is that living systems somehow borrow order from their environment and compensate for the disorder by making the environment more disordered than it would otherwise be. This extra order amounts to negative entropy, which chickens can use to hatch their eggs without violating the second law.

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