SECOND LAW OF THERMODYNAMICS: FORMULAS, EQUATIONS, EXAMPLES - PHYSICAL - 2024

The second law of thermodynamics has several forms of expression. One of them states that no heat engine is capable of completely converting all the energy it absorbs into usable work (Kelvin-Planck formulation). Another way of stating it is to say that real processes occur in such a sense that the quality of energy is lower because entropy tends to increase.

This law, also known as the second principle of thermodynamics, has been expressed in different ways over time, from the early nineteenth century to the present, although its origins date back to the creation of the first steam engines in England., at the beginning of the 18th century.

Figure 1. When throwing the building blocks to the ground, it would be very surprising if they fell in order. Source: Pixabay.

But although it is expressed in many ways, the idea that matter tends to get disordered and that no process is 100% efficient, since losses will always exist.

All thermodynamic systems adhere to this principle, starting with the universe itself until the morning cup of coffee that waits quietly on the table exchanging heat with the environment.

The coffee cools as time passes, until it is in thermal equilibrium with the environment, so it would be very surprising if one day the opposite happened and the environment cooled while the coffee heated itself. It is unlikely to happen, some will say impossible, but it is enough to imagine it to get an idea of ​​the sense in which things happen spontaneously.

In another example, if we slide a book across the surface of a table, it will eventually stop, because its kinetic energy will be lost as heat due to friction.

The first and second laws of thermodynamics were established around 1850, thanks to scientists such as Lord Kelvin - creator of the term "thermodynamics" -, William Rankine - author of the first formal text on thermodynamics - and Rudolph Clausius.

Formulas and equations

Entropy - mentioned at the beginning - helps us to establish the sense in which things happen. Let's go back to the example of bodies in thermal contact.

When two objects at different temperatures come into contact and finally after a while reach thermal equilibrium, they are driven to it by the fact that the entropy reaches its maximum, when the temperature of both is the same.

Denoting entropy as S, the change in entropy ΔS of a system is given by:

The change in entropy ΔS indicates the degree of disorder in a system, but there is a restriction in the use of this equation: it is applicable only to reversible processes, that is, those in which the system can return to its original state without leaving trace of what happened-.

In irreversible processes, the second law of thermodynamics appears as follows:

Reversible and irreversible processes

The cup of coffee always gets cold and is a good example of an irreversible process, since it always occurs in only one direction. If you add cream to the coffee and shake, you will get a very nice combination, but no matter how much you shake again, you will not have the coffee and cream separately again, because stirring is irreversible.

Figure 2. Cup breakage is an irreversible process. Source: Pixabay.

Although most of the daily processes are irreversible, some are almost reversible. Reversibility is an idealization. For this to take place, the system must change very slowly, in such a way that at each point it is always in equilibrium. In this way it is possible to return it to a previous state without leaving a trace in the surroundings.

Processes that are fairly close to this ideal are more efficient, as they deliver a greater amount of work with less energy consumption.

The friction force is responsible for much of the irreversibility, because the heat generated by it is not the type of energy that is sought. In the book sliding across the table, frictional heat is energy that is not recovered.

Even if the book returns to its original position, the table will have been hot as a trace of coming and going on it.

Now look at an incandescent light bulb: most of the work done by the current through the filament is wasted in heat by the Joule effect. Only a small percentage is used to emit light. In both processes (book and light bulb), the entropy of the system has increased.

Applications

An ideal motor is one that is built using reversible processes and lacks friction that causes energy waste, converting almost all thermal energy into usable work.

We emphasize the word almost, because not even the ideal engine, which is Carnot's, is 100% efficient. The second law of thermodynamics takes care that this is not the case.

Carnot engine

The Carnot engine is the most efficient engine that can be devised. It operates between two temperature tanks in two isothermal processes - at constant temperature - and two adiabatic processes - without transfer of thermal energy.

The graphs called PV - pressure-volume diagrams - clarify the situation at a glance:

Figure 3. On the left the Carnot motor diagram and on the right the PV diagram. Source: Wikimedia Commons.

On the left, in figure 3 is the diagram of the Carnot engine C, which takes heat Q ₁ from the tank that is at temperature T ₁, converts that heat into work W and transfers the waste Q ₂ to the colder tank, which is at temperature T ₂.

Starting from A, the system expands until it reaches B, absorbing heat at the fixed temperature T ₁. In B, the system begins an adiabatic expansion in which no heat is gained or lost, to reach C.

In C another isothermal process begins: that of giving heat to the other colder thermal deposit that is at T ₂. As this happens, the system is compressed and reaches point D. There begins a second adiabatic process to return to the starting point A. In this way a cycle is completed.

The efficiency of the Carnot engine depends on the temperatures in Kelvin of the two thermal reservoirs:

Carnot's theorem states that this is the most efficient heat engine out there, but don't be too quick to buy it. Remember what we said about the reversibility of processes? They have to happen very, very slowly, so the power output of this machine is practically nil.

Human metabolism

Human beings need energy to keep all their systems working, therefore they behave like thermal machines that receive energy and transform it into mechanical energy to, for example, move.

The efficiency of the human body when doing work can be defined as the quotient between the mechanical power it can provide and the total energy input that comes with food.

Since the mean power P _m is work W done in a time interval Δt, it can be expressed as:

If ΔU / Δt is the rate at which energy is added, the body efficiency becomes:

Through numerous tests with volunteers, efficiencies of up to 17% have been achieved, delivering about 100 watts of power for several hours.

Of course, that will depend largely on the task that is done. Pedaling a bicycle has a slightly higher efficiency, around 19%, while repetitive tasks that include shovels, picks, and hoes have an efficiency as low as about 3%.

Examples

The second law of thermodynamics is implicit in all the processes that occur in the Universe. Entropy is always increasing, although in some systems it appears to decrease. For this to happen it has had to increase elsewhere, so that in the total balance it is positive.

- In learning there is entropy. There are people who learn things well and quickly, as well as being able to remember them easily later. It is said that they are people with low entropy learning, but surely they are less numerous than those with high entropy: those who find it more difficult to remember the things they study.

- A company with disorganized workers has more entropy than one in which workers carry out tasks in an orderly manner. It is clear that the latter will be more efficient than the former.

- Friction forces generate less efficiency in the operation of machinery, because they increase the amount of dissipated energy that cannot be used efficiently.

- Rolling a dice has a higher entropy than flipping a coin. After all, tossing a coin has only 2 possible outcomes, while tossing the die has 6. The more events that are likely, the more entropy there is.

Solved exercises

Exercise 1

A piston cylinder is filled with a mixture of liquid and water vapor at 300 K and 750 kJ of heat is transferred to the water by a constant pressure process. As a result, the liquid inside the cylinder vaporizes. Calculate the change in entropy in the process.

Figure 4. Figure for the resolved example 1. Source: F. Zapata.

Solution

The process described in the statement is carried out at constant pressure in a closed system, which does not undergo mass exchange.

Since it is a vaporization, during which the temperature does not change either (during phase changes the temperature is constant), the definition of entropy change given above can be applied and the temperature can go outside the integral:

ΔS = 750,000 J / 300 K = 2,500 J / K.

Since heat enters the system, the change in entropy is positive.

Exercise 2

A gas undergoes a pressure increase from 2.00 to 6.00 atmospheres (atm), maintaining a constant volume of 1.00 m ³, and then expanding at constant pressure until reaching a volume of 3.00 m ³. Finally it returns to its initial state. Calculate how much work is done in 1 cycle.

Figure 5. Thermodynamic process in a gas for example 2. Source: Serway -Vulle. Fundamentals of Physics.

Solution

It is a cyclical process in which the internal energy variation is zero, according to the first law of thermodynamics, therefore Q = W. In a PV (pressure - volume) diagram, the work done during a cyclical process is equivalent to the area enclosed by the curve. To give the results in the International System it is necessary to make a change of units in the pressure using the following conversion factor:

1 atm = 101.325 kPa = 101.325 Pa.

The area enclosed by the graph corresponds to that of a triangle whose base (3 - 1 m ³) = 2 m ³ and whose height is (6 - 2 atm) = 4 atm = 405,300 Pa

W _ABCA = ½ (2 m ³ x 405300 Pa) = 405300 J = 405.3 kJ.

Exercise 3

One of the most efficient machines ever built is said to be a coal-fired steam turbine on the Ohio River, which is used to power an electric generator operating between 1870 and 430 ° C.

Calculate: a) The maximum theoretical efficiency, b) The mechanical power that the machine delivers if it absorbs 1.40 x 10 ⁵ J of energy every second from the hot tank. The actual efficiency is known to be 42.0%.

Solution

a) The maximum efficiency is calculated with the equation given above:

To change the degrees centigrade to kelvin, just add 273.15 to the centigrade temperature:

Multiplying by 100% gives the maximum percentage efficiency, which is 67.2%

c) If the real efficiency is 42%, there is a maximum efficiency of 0.42.

The mechanical power delivered is: P = 0.42 x 1.40 x10 ⁵ J / s = 58800 W.

References

1. Bauer, W. 2011. Physics for Engineering and Sciences. Volume 1. Mc Graw Hill.
2. Cengel, Y. 2012. Thermodynamics. 7 ^ma Edition. McGraw Hill.
3. Figueroa, D. (2005). Series: Physics for Science and Engineering. Volume 4. Fluids and Thermodynamics. Edited by Douglas Figueroa (USB).
4. Knight, R. 2017. Physics for Scientists and Engineering: a Strategy Approach.
5. López, C. The First Law of Thermodynamics. Recovered from: culturacientifica.com.
6. Serway, R. 2011. Fundamentals of Physics. 9 ^na Cengage Learning.
7. Sevilla University. Thermal Machines. Recovered from: laplace.us.es