Define entropy.Calculate the boost of entropy in a system with reversible and irreversible processes.Explain the meant fate of the world in entropic terms.Calculate the raising disorder the a system.

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Figure 1. The ice in this drink is progressively melting. At some point the liquid will certainly reach thermal equilibrium, together predicted by the 2nd law the thermodynamics. (credit: Jon Sullivan,

There is however another way of expressing the second law that thermodynamics. This variation relates to a ide called entropy. By examining it, us shall view that the directions linked with the 2nd law—heat carry from hot to cold, for example—are regarded the tendency in nature for equipment to become disordered and for less energy to be accessible for usage as work. The entropy of a system can in reality be displayed to it is in a measure of that is disorder and of the unavailability of power to carry out work.

Making Connections: Entropy, Energy, and also Work

Recall the the simple an interpretation of power is the capacity to do work. Entropy is a measure of how much power is not available to carry out work. Back all creates of energy are interconvertible, and all deserve to be provided to perform work, it is not always possible, also in principle, to convert the entire available energy right into work. The unavailable power is of attention in thermodynamics, since the ar of thermodynamics emerged from efforts to convert heat come work.

We have the right to see how entropy is characterized by recalling our discussion of the Carnot engine. We noted that for a Carnot cycle, and also hence for any kind of reversible processes,


Rearranging state yields


for any kind of reversible process. Qc and also Qh space absolute values of the warmth transfer at temperature Tc and also Th, respectively. This ratio of \fracQT\\ is identified to it is in the change in entropy ΔS because that a reversible process, \DeltaS=\left(\fracQT\right)_\textrev\\, where Q is the heat transfer, i beg your pardon is hopeful for warmth transfer into and an adverse for heat transfer the end of, and T is the pure temperature at which the reversible procedure takes place. The SI unit for entropy is joules per kelvin (J/K). If temperature changes during the process, climate it is usually a an excellent approximation (for small changes in temperature) to take it T to be the mean temperature, avoiding the must use integral calculus to discover ΔS.

The definition of ΔS is strictly valid just for reversible processes, together as used in a Carnot engine. However, we can uncover ΔS precisely also for real, irreversible processes. The reason is that the entropy S the a system, prefer internal power U, depends only on the state the the system and not just how it got to that condition. Entropy is a residential or commercial property of state. For this reason the adjust in entropy ΔS the a system in between state 1 and state 2 is the very same no matter exactly how the readjust occurs. We simply need to find or imagine a reversible process that takes us from state 1 to state 2 and also calculate ΔS for the process. That will certainly be the readjust in entropy because that any process going indigenous state 1 to state 2. (See figure 2.)

Figure 2. When a mechanism goes native state 1 come state 2, its entropy changes by the very same amount ΔS, even if it is a theoretical reversible path is adhered to or a real irreversible route is taken.

Now let us take a look in ~ the adjust in entropy that a Carnot engine and its heat reservoirs because that one full cycle. The hot reservoir has a lose of entropy \DeltaS_\texth=\frac-Q_\texthT_\texth\\, since heat carry occurs the end of the (remember that when warmth transfers out, then Q has a negative sign). The cold reservoir has actually a acquire of entropy\DeltaS_\textc=\fracQ_\textcT_\textc\\, since heat transport occurs right into it. (We i think the reservoirs room sufficiently huge that your temperatures room constant.) for this reason the total change in entropy is ΔStot = ΔSh + ΔSc .

Thus, because we know that \fracQ_\texthT_\texth=\fracQ_\textcT_\textc\\ for a Carnot engine, \DeltaS_\texttot=\fracQ_\texthT_\texth=\fracQ_\textcT_\textc=0\\.

This result, which has general validity, method that the total adjust in entropy because that a mechanism in any kind of reversible procedure is zero.

The entropy of miscellaneous parts of the system might change, but the total readjust is zero. Furthermore, the device does not influence the entropy the its surroundings, since heat transfer between them does no occur. For this reason the reversible process changes neither the complete entropy that the system nor the entropy that its surroundings. Occasionally this is declared as follows: Reversible processes do not affect the total entropy that the universe. Real processes are not reversible, though, and also they do change total entropy. We can, however, use hypothetical reversible processes to determine the value of entropy in real, irreversible processes. Instance 1 illustrates this point.

Example 1. Entropy boosts in one Irreversible (Real) Process

Spontaneous warm transfer from hot to cold is one irreversible process. Calculation the total change in entropy if 4000 J of heat transfer wake up from a hot reservoir at Th = 600 K(327ºC) to a cold reservoir at Tc = 250 K(−23ºC), assuming over there is no temperature change in one of two people reservoir. (See figure 3.)

Figure 3. (a) Heat deliver from a warm object to a cold one is an irreversible process that produces an in its entirety increase in entropy. (b) The same last state and, thus, the same change in entropy is completed for the objects if reversible warm transfer processes occur in between the two objects who temperatures are the very same as the temperature of the corresponding objects in the irreversible process.


How have the right to we calculate the change in entropy for an irreversible process when ΔStot = ΔSh + ΔSc is valid just for reversible processes? Remember that the total adjust in entropy the the hot and cold reservoirs will certainly be the same whether a reversible or irreversible process is associated in warm transfer from warm to cold. So we deserve to calculate the adjust in entropy the the hot reservoir for a theoretical reversible procedure in which 4000 J of warm transfer wake up from it; climate we carry out the very same for a theoretical reversible procedure in i beg your pardon 4000 J of heat transfer occurs to the cold reservoir. This produces the same alters in the hot and also cold reservoirs the would occur if the heat transfer were permitted to take place irreversibly in between them, and so it also produces the same transforms in entropy.


We currently calculate the two changes in entropy using ΔStot = ΔSh + ΔSc. First, for the warm transfer native the warm reservoir,

\displaystyle\DeltaS_\texth=\frac-Q_\texthT_\texth=\frac-4000\text J600\text K=-6.67\text J/K\\

And for the cold reservoir,

\displaystyle\DeltaS_\textc=\frac-Q_\textcT_\textc=\frac4000\text J250\text K=16.0\text J/K\\

Thus the full is

\beginarraylll\DeltaS_\texttot&=&\DeltaS_\texth+\DeltaS_\textc\\\text &=&\left(-6.67+16.0\right)\text J/K\\\text &=&9.33\text J/K\endarray\\


There is one increase in entropy for the device of two heat reservoirs experience this irreversible heat transfer. We will watch that this method there is a ns of capability to do occupational with this transferred energy. Entropy has actually increased, and also energy has become unavailable to carry out work.

It is reasonable that entropy boosts for warmth transfer from warm to cold. Because the adjust in entropy is \fracQT\\, there is a larger adjust at reduced temperatures. The decrease in entropy that the warm object is therefore less than the boost in entropy the the cold object, developing an all at once increase, just as in the previous example. This an outcome is very general:

There is rise in entropy for any system undergoing an irreversible process.

With respect come entropy, there are only two possibilities: entropy is consistent for a reversible process, and also it rises for an irreversible process. There is a fourth version the the second law the thermodynamics proclaimed in regards to entropy:

The complete entropy of a device either boosts or remains constant in any process; it never decreases.

For example, warmth transfer cannot take place spontaneously native cold come hot, since entropy would decrease.

Entropy is really different from energy. Entropy is not conserved however increases in all genuine processes. Reversible processes (such as in Carnot engines) are the procedures in i beg your pardon the most heat transfer to work-related takes place and are also the persons that save entropy constant. For this reason we are brought about make a connection between entropy and the ease of access of power to do work.

Entropy and the Unavailability of power to do Work

What walk a adjust in entropy mean, and also why must we it is in interested in it? One reason is that entropy is directly related come the fact that not all warmth transfer have the right to be converted right into work. Example 2 offers some point out of how an increase in entropy outcomes in less warm transfer into work.

Example 2. Much less Work is created by a offered Heat Transfer once Entropy readjust is Greater

Calculate the job-related output of a Carnot engine operating in between temperatures that 600 K and 100 K because that 4000 J of warm transfer come the engine.Now expect that the 4000 J of heat transfer occurs an initial from the 600 K reservoir to a 250 K reservoir (without doing any kind of work, and this produce the increase in entropy calculated above) before transferring right into a Carnot engine operating in between 250 K and also 100 K. What occupational output is produced? (See number 4.)

Figure 4. (a) A Carnot engine working at in between 600 K and also 100 K has actually 4000 J of warmth transfer and performs 3333 J of work. (b) The 4000 J of warmth transfer occurs very first irreversibly come a 250 K reservoir and also then goes into a Carnot engine. The boost in entropy resulted in by the warm transfer come a chillier reservoir outcomes in a smaller occupational output of 2400 J. Over there is a permanent loss the 933 J of power for the function of act work.


In both parts, us must an initial calculate the Carnot efficiency and also then the job-related output.

Solution to part 1

The Carnot effectiveness is provided by \mathitEff_\textC=1-\fracT_\textcT_\texth\\.

Substituting the provided temperatures yields \mathitEff_\textC=1-\frac100\text K600\text K=0.833\\.

Now the occupational output deserve to be calculated utilizing the an interpretation of performance for any kind of heat engine as given by \mathitEff=\fracWQ_\texth\\.

Solving because that W and also substituting recognized terms gives

\beginarraylllW&=&\mathitEff_\textCQ_\texth\\\text &=&\left(0.833\right)\left(4000\text J\right)=3333\text J\endarray\\

Solution to part 2

Similarly, \mathitEff\prime_\textC=1-\fracT_\textcT\prime_\textc=\frac100\text K250\text K=0.600\\

so that

\beginarraylllW&=&\mathitEff\prime_\textCQ_\texth\\\text &=&\left(0.600\right)\left(4000\text J\right)=2400\text J\endarray\\


There is 933 J much less work from the same warmth transfer in the second process. This an outcome is important. The same heat transfer into two perfect engines produces different work outputs, due to the fact that the entropy change differs in the 2 cases. In the 2nd case, entropy is greater and less work-related is produced. Entropy is connected with the unavailability of power to do work.

When entropy increases, a details amount of energy becomes permanently have not to perform work. The power is no lost, however its character is changed, therefore that several of it can never be convert to law work—that is, to an organized force acting with a distance. Because that instance, in example 2, 933 J less work to be done after boost in entropy of 9.33 J/K occurred in the 4000 J warmth transfer native the 600 K reservoir come the 250 K reservoir. It have the right to be displayed that the lot of power that i do not care unavailable for job-related is Wunavail = Δ⋅ T0, where T0 is the shortest temperature utilized. In example 2, Wunavail = (9.33 J/K)(100 K) = 933 J as found.

Heat fatality of the Universe: an Overdose the Entropy

In the early, energetic universe, all matter and energy were conveniently interchangeable and also identical in nature. Heaviness played a an essential role in the young universe. Although that may have seemed disorderly, and also therefore, superficially entropic, in fact, over there was massive potential energy easily accessible to perform work—all the future energy in the universe.

As the universe matured, temperature differences arose, i m sorry created much more opportunity for work. Stars are hotter 보다 planets, for example, which space warmer 보다 icy asteroids, which are warmer still 보다 the vacuum the the an are between them.

Most the these are cooling under from their generally violent births, in ~ which time castle were listed with energy of your own—nuclear power in the situation of stars, volcanic energy on Earth and other planets, and so on. Without extr energy input, however, their days are numbered.

As entropy increases, less and less power in the world is obtainable to execute work. Top top Earth, us still have good stores of power such together fossil and nuclear fuels; large-scale temperature differences, which can provide wind energy; geothermal energies early out to distinctions in temperature in earth layers; and also tidal energies owing to our abundance of fluid water. As these room used, a certain portion of the energy they contain have the right to never be converted right into doing work. Eventually, every fuels will be exhausted, every temperatures will equalize, and also it will certainly be impossible for heat engines come function, or for work to it is in done.

Entropy increases in a close up door system, such as the universe. Yet in components of the universe, for instance, in the Solar system, it is not a locally closed system. Energy flows indigenous the sunlight to the planets, replenishing Earth’s stores of energy. The sunlight will continue to supply us with energy for about another 5 billion years. We will certainly enjoy straight solar energy, and also side effects of solar energy, such as wind power and also biomass power from photosynthetic plants. The energy from the sun will save our water at the fluid state, and the Moon’s gravitational pull will proceed to provide tidal energy. Yet Earth’s geothermal power will slowly run down and also won’t be replenished.

But in terms of the universe, and also the really long-term, very large-scale picture, the entropy of the world is increasing, and also so the accessibility of power to do work-related is continually decreasing. Eventually, once all stars have died, all forms of potential power have to be utilized, and also all temperatures have equalized (depending top top the fixed of the universe, either at a very high temperature following a global contraction, or a really low one, just before all activity ceases) there will be no possibility of act work.

Either way, the universe is destined for thermodynamic equilibrium—maximum entropy. This is often referred to as the heat fatality of the universe, and also will typical the end of all activity. However, even if it is the cosmos contracts and also heats up, or continues to expand and also cools down, the finish is not near. Calculations of black color holes suggest that entropy have the right to easily proceed for at least 10100 years.

Order to Disorder

Entropy is connected not only to the unavailability of power to do work—it is likewise a measure of disorder. This concept was initially postulated by Ludwig Boltzmann in the 1800s. Because that example, melting a block that ice method taking a very structured and also orderly system of water molecules and also converting it right into a disorderly fluid in i m sorry molecules have no resolved positions. (See figure 5.) over there is a big increase in entropy in the process, as watched in the following example.

Figure 5. Once ice melts, the becomes much more disordered and less structured. The systematic arrangement of molecules in a crystal structure is changed by a more random and less orderly activity of molecules without fixed areas or orientations. That is entropy increases due to the fact that heat carry occurs into it. Entropy is a measure up of disorder.

Example 3. Entropy associated with Disorder

Find the rise in entropy of 1.00 kg that ice originally at 0º C the is melted to kind water at 0º C.


As before, the readjust in entropy have the right to be calculated native the an interpretation of ΔS once we discover the power Q necessary to melt the ice.


The change in entropy is defined as: \DeltaS=\fracQT\\.

Here Q is the warmth transfer important to melt 1.00 kg the ice and also is given by QmLf, where m is the mass and Lf is the latent warm of fusion. Lf = 334 kJ/kg for water, therefore that Q = (1.00 kg)(334 kJ/kg) = 3.34 × 105 J.

Now the change in entropy is positive, due to the fact that heat transfer occurs right into the ice cream to cause the step change; thus,

\displaystyle\DeltaS=\fracQT=\frac3.34\times10^5\text JT\\

T is the melt temperature of ice. The is, = 0ºC = 273 K. For this reason the adjust in entropy is

\beginarraylll\DeltaS&=&\frac3.34\times10^5\text J273\text K\\\text &=&1.22\times10^3\text J/K\endarray\\


This is a far-ranging increase in entropy accompanying boost in disorder.

In an additional easily imagined example, suppose we mix same masses of water originally at two various temperatures, say 20.0ºC and also 40.0ºC. The result is water in ~ an intermediate temperature the 30.0ºC. 3 outcomes have actually resulted: entropy has increased, some power has come to be unavailable to execute work, and also the system has come to be less orderly. Let us think around each of these results.

First, entropy has actually increased because that the same factor that that did in Example 3. Mixing the 2 bodies the water has the same effect as heat transfer native the warm one and also the same warmth transfer right into the cold one. The mix decreases the entropy of the warm water yet increases the entropy that the cold water through a greater amount, producing an in its entirety increase in entropy.

Second, as soon as the two masses the water room mixed, over there is just one temperature—you cannot operation a heat engine v them. The power that might have been used to operation a warm engine is now unavailable to execute work.

Third, the mixture is much less orderly, or to use one more term, much less structured. Fairly than having actually two masses at different temperatures and with different distributions of molecule speeds, us now have actually a single mass through a uniform temperature.

These three results—entropy, unavailability that energy, and disorder—are not only related but are in fact basically equivalent.

Life, Evolution, and the second Law that Thermodynamics

Some civilization misunderstand the second law that thermodynamics, proclaimed in regards to entropy, come say the the process of the development of life violates this law. Over time, complex organisms advanced from much easier ancestors, representing a huge decrease in entropy of the earth biosphere. It is a reality that life organisms have developed to be very structured, and also much lower in entropy 보다 the substances indigenous which lock grow. But it is always feasible for the entropy the one part of the world to decrease, noted the total readjust in entropy the the universe increases. In equation form, we deserve to write this as ΔStot = ΔSsyst + ΔSenvir > 0.

Thus ΔSsyst deserve to be an adverse as long as ΔSenvir is positive and greater in magnitude.

How is it feasible for a mechanism to diminish its entropy? energy transfer is necessary. If I choose up marbles that room scattered around the room and put them right into a cup, my occupational has decreased the entropy of the system. If ns gather iron ore native the ground and also convert it right into steel and build a bridge, my work-related has diminished the entropy of that system. Power coming native the Sun can decrease the entropy of neighborhood systems on Earth—that is, ΔSsyst is negative. However the overall entropy of the rest of the universe boosts by a higher amount—that is, ΔSenvir is positive and also greater in magnitude. Thus, ΔStot = ΔSsyst + ΔSenvir > 0, and also the 2nd law that thermodynamics is not violated.

Every time a plant stores part solar power in the kind of chemical potential energy, or an updraft of heat air lifts a soaring bird, the planet can be viewed as a warmth engine operating between a warm reservoir gave by the Sun and a cold reservoir supplied by dark outer space—a warmth engine of high complexity, resulting in local reduce in entropy together it uses part of the warm transfer native the Sun right into deep space. There is a large total increase in entropy result from this massive heat transfer. A small component of this warmth transfer is stored in structured solution on Earth, developing much smaller regional decreases in entropy. (See figure 6.)

Figure 6. Earth entropy might decrease in the process of intercepting a small component of the heat transfer indigenous the Sun into deep space. Entropy for the entire process increases greatly while earth becomes more structured through living systems and stored power in miscellaneous forms.

PhET Explorations: Reversible Reactions

Watch a reaction continue over time. How does complete energy impact a reaction rate? vary temperature, obstacle height, and also potential energies. Document concentrations and also time in order come extract rate coefficients. Carry out temperature dependent studies to extract Arrhenius parameters. This simulation is best used v teacher guidance since it gift an analogy of chemical reactions.


Click to download the simulation. Run utilizing Java.

Section Summary

Entropy is the ns of energy easily accessible to perform work.Another form of the second law that thermodynamics states that the total entropy the a device either increases or stays constant; it never decreases.Entropy is zero in a reversible process; it boosts in one irreversible process.The can be fried fate the the cosmos is likely to it is in thermodynamic equilibrium, wherein the global temperature is continuous and no power is accessible to do work.Entropy is likewise associated v the propensity toward disorder in a close up door system.

Conceptual Questions

A mrs shuts she summer cottage up in September and returns in June. No one has entered the head in the meantime. Explain what she is likely to find, in terms of the second law of thermodynamics.Consider a system with a certain energy content, indigenous which us wish to extract as lot work as possible. Need to the system’s entropy it is in high or low? Is this orderly or disorderly? Structured or uniform? define briefly.Does a gas become much more orderly as soon as it liquefies? walk its entropy change? If so, go the entropy boost or decrease? describe your answer.Explain exactly how water’s entropy have the right to decrease as soon as it freezes there is no violating the 2nd law of thermodynamics. Specifically, describe what wake up to the entropy that its surroundings.Is a uniform-temperature gas much more or much less orderly 보다 one v several different temperatures? i m sorry is much more structured? In which have the right to heat transfer an outcome in job-related done without warm transfer from another system?Give an instance of a spontaneous process in i beg your pardon a device becomes much less ordered and also energy becomes less obtainable to execute work. What happens to the system’s entropy in this process?What is the change in entropy in an adiabatic process? walk this indicate that adiabatic procedures are reversible? deserve to a process be exactly adiabatic for a macroscopic system?Does the entropy that a star increase or decrease as it radiates? walk the entropy that the space into which that radiates (which has a temperature of around 3 K) increase or decrease? What go this perform to the entropy the the universe?Explain why a building made the bricks has smaller entropy 보다 the same bricks in a disorganized pile. Do this by considering the variety of ways the each could be developed (the variety of microstates in each macrostate).

Problems & Exercises

(a) On a winter day, a details house loser 5.00 × 108 J of warmth to the external (about 500,000 Btu). What is the total readjust in entropy as result of this warmth transfer alone, assuming an median indoor temperature of 21.0ºC and an average outdoor temperature of 5.00ºC? (b) This big change in entropy means a huge amount of power has end up being unavailable to perform work. Where execute we find much more energy once such power is shed to us?On a warm summer day, 4.00 × 106 J of warm transfer right into a parked automobile takes place, boosting its temperature from 35.0ºC to 45.0ºC. What is the increase in entropy of the car due to this warmth transfer alone?A hot rock ejected from a volcano’s lava spring cools indigenous 1100ºC come 40.0ºC, and also its entropy to reduce by 950 J/K. How much heat transfer occurs from the rock?When 1.60 × 105 J of warmth transfer occurs into a meat pie originally at 20.0ºC, its entropy increases by 480 J/K. What is its final temperature?The sunlight radiates power at the price of 3.80 × 1026 W from its 5500ºC surface into dark empty room (a negligible fraction radiates onto Earth and the other planets). The reliable temperature of deep room is −270ºC. (a) What is the rise in entropy in one day because of this warmth transfer? (b) How much work is make unavailable?(a) In getting to equilibrium, how much heat transfer occurs from 1.00 kg of water in ~ 40.0ºC when it is put in call with 1.00 kg of 20.0ºC water in getting to equilibrium? (b) What is the readjust in entropy because of this heat transfer? (c) How lot work is make unavailable, acquisition the shortest temperature to it is in 20.0ºC? Explicitly display how you follow the measures in the Problem-Solving techniques for Entropy.What is the to decrease in entropy of 25.0 g of water that condenses ~ above a bathroom winter at a temperature of 35.0ºC, assuming no adjust in temperature and also given the latent heat of vaporization to it is in 2450 kJ/kg?Find the rise in entropy of 1.00 kg of liquid nitrogen the starts at its boiling temperature, boils, and warms to 20.0ºC at consistent pressure.A large electrical power station generates 1000 MW of electricity with an performance of 35.0%. (a) Calculate the heat transfer come the strength station, Qh, in one day. (b) How much warm transfer Qc occurs to the setting in one day? (c) If the warmth transfer in the cooling towers is from 35.0ºC water right into the neighborhood air mass, which increases in temperature indigenous 18.0ºC to 20.0ºC, what is the total increase in entropy due to this heat transfer? (d) How much power becomes do not have to execute work therefore increase in entropy, suspect an 18.0ºC shortest temperature? (Part the Qc could be used to operate heat engines or for merely heating the surroundings, but it rarely is.)(a) How much warmth transfer occurs from 20.0 kg that 90.0ºC water inserted in contact with 20.0 kg of 10.0ºC water, creating a final temperature that 50.0ºC? (b) How much work could a Carnot engine do with this warmth transfer, assuming the operates between two reservoirs at constant temperatures of 90.0ºC and 10.0ºC? (c) What increase in entropy is produced by mixing 20.0 kg that 90.0ºC water v 20.0 kg the 10.0ºC water? (d) Calculate the lot of work made unavailable by this mixing using a short temperature that 10.0ºC, and compare it through the occupational done through the Carnot engine. Explicitly show how you monitor the procedures in the Problem-Solving tactics for Entropy. (e) Discuss how everyday processes make increasingly much more energy have not to do work, as implied by this problem.

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entropy: a measure up of a system’s disorder and also its i can not qualify to do work-related in a system

change in entropy: the ratio of warmth transfer to temperature \fracQT\\

second legislation of thermodynamics stated in regards to entropy: the total entropy of a system either rises or stays constant; it never ever decreases

Selected options to Problems & Exercises

1. (a) 9.78 × 104 J/K; (b) In order to gain more energy, we have to generate it from points within the house, prefer a warmth pump, person bodies, and other appliances. Together you know, we usage a lot of energy to keep our houses warm in the winter since of the lose of warm to the outside.