Define entropy.Calculate the rise of entropy in a device through reversible and also irreversible processes.Exordinary the expected fate of the world in entropic terms.Calculate the raising disorder of a mechanism.

You are watching: Find the amount of heat qh it would take to bring all of the ice to its melting point, 0∘c.

Figure 1. The ice in this drink is progressively melting. Ultimately the liquid will reach thermal equilibrium, as predicted by the second regulation of thermodynamics. (credit: Jon Sullivan,

Tbelow is yet another way of expushing the second law of thermodynamics. This version relates to a principle referred to as entropy. By studying it, we shall view that the directions connected via the second law—warm deliver from hot to cold, for example—are related to the tendency in nature for systems to come to be disordered and also for much less power to be obtainable for use as work-related. The entropy of a device can in truth be presented to be a measure of its disorder and of the unavailcapability of power to execute job-related.

Making Connections: Entropy, Energy, and Work

Respeak to that the simple meaning of energy is the ability to do occupational. Entropy is a meacertain of exactly how a lot energy is not easily accessible to carry out work. Although all creates of power are interconvertible, and also all can be offered to execute job-related, it is not always possible, even in principle, to transform the whole easily accessible energy into occupational. That unobtainable power is of interest in thermodynamics, bereason the field of thermodynamics emerged from efforts to convert warm to job-related.

We deserve to view exactly how entropy is identified by recalling our discussion of the Carnot engine. We provided that for a Carnot cycle, and also hence for any kind of reversible procedures,

displaystylefracQ_ extcQ_ exth=fracT_ extcT_ exth\.

Rearvarying terms yields

displaystylefracQ_ extcT_ extc=fracQ_ exthT_ exth\

for any type of reversible process. Qc and Qh are absolute values of the warm deliver at temperatures Tc and also Th, respectively. This ratio of fracQT\ is defined to be the readjust in entropy ΔS for a reversible process, DeltaS=left(fracQT ight)_ extrev\, where Q is the heat transport, which is positive for warmth deliver right into and negative for warmth transfer out of, and also T is the absolute temperature at which the reversible procedure takes location. The SI unit for entropy is joules per kelvin (J/K). If temperature alters throughout the procedure, then it is commonly a good approximation (for small alters in temperature) to take T to be the average temperature, avoiding the need to usage integral calculus to find ΔS.

The definition of ΔS is strictly valid only for reversible processes, such as supplied in a Carnot engine. However, we deserve to uncover ΔS specifically also for real, irreversible processes. The reason is that the entropy S of a mechanism, like interior energy U, depends just on the state of the system and not exactly how it reached that problem. Entropy is a property of state. Thus the readjust in entropy ΔS of a system between state 1 and also state 2 is the exact same no matter just how the readjust occurs. We just should find or imagine a reversible procedure that takes us from state 1 to state 2 and also calculate ΔS for that procedure. That will certainly be the change in entropy for any type of procedure going from state 1 to state 2. (See Figure 2.)

Figure 2. When a system goes from state 1 to state 2, its entropy alters by the exact same amount ΔS, whether a hypothetical reversible course is followed or a real irreversible path is taken.

Now let us take a look at the adjust in entropy of a Carnot engine and also its heat reservoirs for one complete cycle. The warm reservoir has a loss of entropy DeltaS_ exth=frac-Q_ exthT_ exth\, bereason heat transport occurs out of it (remember that once heat transfers out, then Q has actually an adverse sign). The cold reservoir has actually a acquire of entropyDeltaS_ extc=fracQ_ extcT_ extc\, bereason heat transport occurs right into it. (We assume the reservoirs are sufficiently big that their temperatures are continuous.) So the full change in entropy is ΔStot = ΔSh + ΔSc .

Hence, considering that we recognize that fracQ_ exthT_ exth=fracQ_ extcT_ extc\ for a Carnot engine, DeltaS_ exttot=fracQ_ exthT_ exth=fracQ_ extcT_ extc=0\.

This outcome, which has actually basic validity, implies that the total change in entropy for a mechanism in any reversible process is zero.

The entropy of assorted components of the mechanism may change, but the complete readjust is zero. In addition, the device does not affect the entropy of its surroundings, because warm move in between them does not happen. Thus the reversible procedure changes neither the full entropy of the device nor the entropy of its surroundings. Sometimes this is stated as follows: Reversible procedures perform not impact the full entropy of the world. Real procedures are not reversible, though, and also they carry out readjust full entropy. We deserve to, but, use hypothetical reversible procedures to determine the value of entropy in genuine, irreversible procedures. Example 1 illustprices this suggest.

Example 1. Entropy Increases in an Irreversible (Real) Process

Spontaneous warm deliver from hot to cold is an irreversible procedure. Calculate the complete readjust in entropy if 4000 J of warm transfer occurs from a hot reservoir at Th = 600 K(327ºC) to a cold reservoir at Tc = 250 K(−23ºC), assuming there is no temperature readjust in either 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 very same final state and, therefore, the same adjust in entropy is completed for the objects if reversible heat move procedures happen between the 2 objects whose temperatures are the same as the temperatures of the equivalent objects in the irreversible procedure.


How deserve to we calculate the readjust in entropy for an irreversible procedure as soon as ΔStot = ΔSh + ΔSc is valid just for reversible processes? Remember that the total adjust in entropy of the hot and also cold reservoirs will certainly be the exact same whether a reversible or irreversible process is involved in heat transfer from hot to cold. So we can calculate the readjust in entropy of the hot reservoir for a hypothetical reversible process in which 4000 J of heat carry occurs from it; then we carry out the same for a theoretical reversible process in which 4000 J of warm move occurs to the cold reservoir. This produces the very same alters in the warm and cold reservoirs that would happen if the heat carry were allowed to occur irreversibly between them, and so it additionally produces the same alters in entropy.


We now calculate the two alters in entropy making use of ΔStot = ΔSh + ΔSc. First, for the warm deliver from the hot reservoir,

displaystyleDeltaS_ exth=frac-Q_ exthT_ exth=frac-4000 ext J600 ext K=-6.67 ext J/K\

And for the cold reservoir,

displaystyleDeltaS_ extc=frac-Q_ extcT_ extc=frac4000 ext J250 ext K=16.0 ext J/K\

Hence the full is

eginarraylllDeltaS_ exttot&=&DeltaS_ exth+DeltaS_ extc\ ext &=&left(-6.67+16.0 ight) ext J/K\ ext &=&9.33 ext J/Kendarray\


Tright here is an increase in entropy for the mechanism of 2 warmth reservoirs undergoing this irreversible warmth move. We will view that this means tbelow is a loss of ability to perform work-related with this moved energy. Entropy has boosted, and power has actually come to be unobtainable to perform work-related.

It is reasonable that entropy rises for warm deliver from hot to cold. Since the readjust in entropy is fracQT\, tbelow is a larger readjust at lower temperatures. The decrease in entropy of the warm object is therefore much less than the rise in entropy of the cold object, producing an in its entirety boost, simply as in the previous example. This outcome is exceptionally general:

Tright here is a boost in entropy for any kind of system undergoing an irreversible process.

With respect to entropy, there are just 2 possibilities: entropy is continuous for a reversible process, and also it increases for an irreversible procedure. There is a fourth version of the second regulation of thermodynamics declared in regards to entropy:

The total entropy of a system either increases or remains constant in any kind of process; it never decreases.

For example, heat move cannot happen spontaneously from cold to warm, bereason entropy would decrease.

Entropy is incredibly various from energy. Entropy is not conserved but increases in all real procedures. Reversible procedures (such as in Carnot engines) are the procedures in which the a lot of warm transfer to work takes place and also are additionally the ones that store entropy consistent. Thus we are caused make a connection in between entropy and the availcapacity of energy to perform work-related.

Entropy and the Unavailcapacity of Energy to Do Work

What does a adjust in entropy intend, and why have to we be interested in it? One factor is that entropy is directly pertained to the fact that not all warm deliver have the right to be converted into job-related. Example 2 gives some indication of just how a boost in entropy outcomes in much less warmth transfer right into occupational.

Example 2. Less Work is Produced by a Given Heat Transfer When Entropy Change is Greater

Calculate the work-related output of a Carnot engine operating between temperatures of 600 K and 100 K for 4000 J of warmth move to the engine.Now suppose that the 4000 J of heat move occurs first from the 600 K reservoir to a 250 K reservoir (without doing any occupational, and also this produces the increase in entropy calculated above) before transporting right into a Carnot engine operating in between 250 K and also 100 K. What work-related output is produced? (See Figure 4.)

Figure 4. (a) A Carnot engine functioning at in between 600 K and also 100 K has 4000 J of warm transport and performs 3333 J of work. (b) The 4000 J of warm deliver occurs first irreversibly to a 250 K reservoir and also then goes into a Carnot engine. The increase in entropy resulted in by the warm carry to a cooler reservoir outcomes in a smaller work output of 2400 J. There is a permanent loss of 933 J of energy for the objective of doing job-related.


In both parts, we should initially calculate the Carnot efficiency and also then the work-related output.

Systems to Part 1

The Carnot efficiency is given by mathitEff_ extC=1-fracT_ extcT_ exth\.

Substituting the provided temperatures yields mathitEff_ extC=1-frac100 ext K600 ext K=0.833\.

Now the job-related output deserve to be calculated making use of the definition of effectiveness for any heat engine as offered by mathitEff=fracWQ_ exth\.

Solving for W and also substituting known terms gives

eginarraylllW&=&mathitEff_ extCQ_ exth\ ext &=&left(0.833 ight)left(4000 ext J ight)=3333 ext Jendarray\

Solution to Part 2

Similarly, mathitEffprime_ extC=1-fracT_ extcTprime_ extc=frac100 ext K250 ext K=0.600\

so that

eginarraylllW&=&mathitEffprime_ extCQ_ exth\ ext &=&left(0.600 ight)left(4000 ext J ight)=2400 ext Jendarray\


Tbelow is 933 J much less job-related from the same warm transport in the second process. This result is necessary. The very same heat transport right into 2 perfect engines produces different work-related outputs, bereason the entropy adjust differs in the two situations. In the second situation, entropy is better and also less work-related is created. Entropy is linked with the unavailcapability of power to carry out occupational.

When entropy rises, a specific amount of power becomes permanently unaccessible to perform work-related. The energy is not lost, yet its character is changed, so that some of it have the right to never be converted to doing work—that is, to an arranged pressure acting through a distance. For instance, in Example 2, 933 J less work was done after a rise in entropy of 9.33 J/K arisen in the 4000 J heat transport from the 600 K reservoir to the 250 K reservoir. It have the right to be presented that the amount of energy that becomes uneasily accessible for work-related is Wunavail = Δ⋅ T0, wbelow T0 is the lowest temperature made use of. In Example 2, Wunavail = (9.33 J/K)(100 K) = 933 J as found.

Heat Death of the Universe: An Overdose of Entropy

In the beforehand, energetic cosmos, all matter and also power were easily interchangeable and also identical in nature. Gravity played a critical role in the young world. Although it may have seemed disorderly, and therefore, superficially entropic, in reality, tbelow was massive potential energy accessible to do work—all the future energy in the cosmos.

As the cosmos matured, temperature differences emerged, which created even more opportunity for occupational. Stars are hotter than planets, for example, which are warmer than icy asteroids, which are warmer still than the vacuum of the area in between them.

Most of these are cooling dvery own from their generally violent births, at which time they were offered via energy of their own—nuclear energy in the situation of stars, volcanic energy on Planet and various other planets, and so on. Without added energy input, however, their days are numbered.

As entropy boosts, much less and also less energy in the world is easily accessible to execute work. On Planet, we still have great stores of energy such as fossil and also nuclear fuels; large-scale temperature distinctions, which deserve to carry out wind energy; geothermal energies due to differences in temperature in Earth’s layers; and tidal energies owing to our abundance of liquid water. As these are used, a particular fractivity of the power they contain have the right to never before be converted into doing work-related. Eventually, all fuels will be worn down, all temperatures will certainly equalize, and also it will certainly be impossible for warmth engines to feature, or for work-related to be done.

Entropy increases in a closed mechanism, such as the universe. But in parts of the world, for circumstances, in the Solar device, it is not a in your area closed system. Energy flows from the Sun to the planets, replenishing Earth’s stores of power. The Sun will certainly proceed to supply us with power for around another five billion years. We will certainly reap direct solar power, as well as side effects of solar energy, such as wind power and also biomass power from photosynthetic plants. The energy from the Sun will store our water at the liquid state, and also the Moon’s gravitational pull will continue to carry out tidal energy. But Earth’s geothermal power will certainly slowly run dvery own and also won’t be replenimelted.

But in regards to the universe, and also the incredibly long-term, extremely massive picture, the entropy of the cosmos is boosting, and also so the availcapacity of power to do work is constantly decreasing. Eventually, when all stars have passed away, all develops of potential power have actually been used, and also all temperatures have actually equalized (depending upon the mass of the cosmos, either at a really high temperature complying with a global contraction, or a really low one, just prior to all activity ceases) tbelow will certainly be no possibility of doing occupational.

Either way, the world is destined for thermodynamic equilibrium—maximum entropy. This is frequently dubbed the warm death of the universe, and will certainly suppose the finish of all task. However before, whether the universe contracts and also heats up, or proceeds to expand also and also cools down, the finish is not close to. Calculations of black holes indicate that entropy deserve to conveniently proceed for at leastern 10100 years.

Order to Disorder

Entropy is connected not only to the unavailcapacity of energy to perform work—it is likewise a measure of disorder. This concept was initially postulated by Ludwig Boltzmann in the 1800s. For example, melting a block of ice implies taking a extremely structured and orderly system of water molecules and also converting it into a disorderly liquid in which molecules have no resolved positions. (See Figure 5.) Tbelow is a large increase in entropy in the procedure, as viewed in the following example.

Figure 5. When ice melts, it becomes even more disordered and less structured. The systematic plan of molecules in a crystal framework is reinserted by a much more random and less orderly movement of molecules without solved places or orientations. Its entropy rises because warm move occurs right into it. Entropy is a measure of disorder.

Example 3. Entropy Associated via Disorder

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


As prior to, the adjust in entropy can be calculated from the interpretation of ΔS when we discover the energy Q necessary to melt the ice.


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

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

Now the change in entropy is positive, because warm transfer occurs into the ice to reason the phase change; thus,

displaystyleDeltaS=fracQT=frac3.34 imes10^5 ext JT\

T is the melting temperature of ice. That is, = 0ºC = 273 K. So the change in entropy is

eginarraylllDeltaS&=&frac3.34 imes10^5 ext J273 ext K\ ext &=&1.22 imes10^3 ext J/Kendarray\


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

In another easily imagined example, mean we mix equal masses of water initially at two different temperatures, say 20.0ºC and 40.0ºC. The outcome is water at an intermediate temperature of 30.0ºC. Three outcomes have resulted: entropy has raised, some energy has actually become unavailable to perform occupational, and the mechanism has come to be much less orderly. Let us think about each of these results.

First, entropy has boosted for the exact same reason that it did in Example 3. Mixing the 2 bodies of water has the exact same impact as warmth transfer from the hot one and the exact same warm move right into the cold one. The mixing decreases the entropy of the hot water however boosts the entropy of the cold water by a greater amount, producing an overall rise in entropy.

2nd, once the 2 masses of water are combined, tright here is just one temperature—you cannot run a heat engine via them. The power that can have been provided to run a warmth engine is now unobtainable to perform work-related.

Third, the mixture is much less orderly, or to use another term, less structured. Rather than having two masses at different temperatures and via various distributions of molecular speeds, we now have actually a solitary mass through a uniform temperature.

These 3 results—entropy, unavailability of energy, and also disorder—are not only associated however are in truth basically equivalent.

Life, Evolution, and also the 2nd Law of Thermodynamics

Some world misunderstand the second legislation of thermodynamics, stated in terms of entropy, to say that the procedure of the development of life violates this regulation. With time, facility organisms progressed from much much easier ancestors, representing a large decrease in entropy of the Earth’s biosphere. It is a reality that living organisms have progressed to be extremely structured, and also much reduced in entropy than the substances from which they prosper. But it is always feasible for the entropy of one component of the universe to decrease, provided the complete adjust in entropy of the universe rises. In equation develop, we have the right to compose this as ΔStot = ΔSsyst + ΔSenvir > 0.

Thus ΔSsyst deserve to be negative as lengthy as ΔSenvir is positive and also greater in magnitude.

How is it feasible for a mechanism to decrease its entropy? Energy carry is vital. If I pick up marbles that are scattered around the room and also put them right into a cup, my work-related has reduced the entropy of that mechanism. If I gather iron ore from the ground and transform it into steel and construct a bridge, my job-related has actually diminished the entropy of that device. Energy coming from the Sun deserve to decrease the entropy of local units on Earth—that is, ΔSsyst is negative. But the as a whole entropy of the remainder of the cosmos rises by a greater amount—that is, ΔSenvir is positive and better in magnitude. Thus, ΔStot = ΔSsyst + ΔSenvir > 0, and the second legislation of thermodynamics is not violated.

Every time a plant stores some solar power in the develop of chemical potential power, or an updraft of warmth air lifts a soaring bird, the Earth have the right to be viewed as a warm engine operating between a warm reservoir supplied by the Sun and a cold reservoir provided by dark external space—a warmth engine of high intricacy, resulting in local decreases in entropy as it provides part of the warm deliver from the Sun right into deep area. There is a huge full rise in entropy resulting from this enormous heat carry. A little part of this heat carry is stored in structured systems on Planet, creating much smaller sized neighborhood decreases in entropy. (See Figure 6.)

Figure 6. Earth’s entropy might decrease in the procedure of intercepting a small part of the warm carry from the Sun right into deep area. Entropy for the entire procedure increases significantly while Planet becomes more structured with living devices and also stored power in miscellaneous creates.

PhET Explorations: Reversible Reactions

Watch a reactivity continue over time. How does total energy affect a reaction rate? Vary temperature, barrier elevation, and potential energies. Record concentrations and also time in order to extract price coefficients. Do temperature dependent studies to extract Arrhenius parameters. This simulation is finest provided with teacher guidance because it presents an analogy of chemical reactions.


Click to downfill the simulation. Run using Java.

Section Summary

Entropy is the loss of power available to do work-related.Anvarious other create of the second legislation of thermodynamics states that the complete entropy of a device either increases or remains constant; it never decreases.Entropy is zero in a reversible process; it rises in an irreversible procedure.The ultimate fate of the cosmos is likely to be thermodynamic equilibrium, wbelow the global temperature is constant and no power is easily accessible to perform job-related.Entropy is likewise associated with the tendency toward disorder in a closed device.

Conceptual Questions

A woguy shuts her summer cottage up in September and also retransforms in June. No one has gotten in the cottage in the meantime. Exsimple what she is likely to uncover, in terms of the second legislation of thermodynamics.Consider a system via a specific energy content, from which we wish to extract as a lot work-related as possible. Should the system’s entropy be high or low? Is this orderly or disorderly? Structured or uniform? Explain briefly.Does a gas end up being even more orderly when it liquefies? Does its entropy change? If so, does the entropy rise or decrease? Exordinary your answer.Exordinary just how water’s entropy have the right to decrease when it freezes without violating the second legislation of thermodynamics. Specifically, describe what happens to the entropy of its surroundings.Is a uniform-temperature gas more or less orderly than one through numerous different temperatures? Which is more structured? In which have the right to warm deliver bring about occupational done without warmth transfer from an additional system?Give an example of a spontaneous procedure in which a mechanism becomes less ordered and also power becomes much less easily accessible to perform work-related. What happens to the system’s entropy in this process?What is the adjust in entropy in an adiabatic process? Does this indicate that adiabatic procedures are reversible? Can a process be exactly adiabatic for a macroscopic system?Does the entropy of a star boost or decrease as it radiates? Does the entropy of the space right into which it radiates (which has a temperature of around 3 K) rise or decrease? What does this do to the entropy of the universe?Exsimple why a building made of bricks has actually smaller entropy than the very same bricks in a disarranged pile. Do this by considering the variety of means that each could be created (the variety of microclaims in each macrostate).

Problems & Exercises

(a) On a winter day, a particular house loses 5.00 × 108 J of warm to the exterior (about 500,000 Btu). What is the total change in entropy due to this heat move alone, assuming an average indoor temperature of 21.0ºC and an average outdoor temperature of 5.00ºC? (b) This large readjust in entropy suggests a huge amount of energy has actually become unavailable to execute work-related. Where carry out we find more power once such energy is shed to us?On a hot summer day, 4.00 × 106 J of warm carry into a parked vehicle takes area, boosting its temperature from 35.0ºC to 45.0ºC. What is the rise in entropy of the vehicle due to this warm move alone?A warm rock ejected from a volcano’s lava fountain cools from 1100ºC to 40.0ºC, and also its entropy decreases by 950 J/K. How much warmth move occurs from the rock?When 1.60 × 105 J of warmth transfer occurs right into a meat pie initially at 20.0ºC, its entropy increases by 480 J/K. What is its last temperature?The Sun radiates power at the rate of 3.80 × 1026 W from its 5500ºC surchallenge right into dark empty space (a negligible fraction radiates onto Planet and also the other planets). The reliable temperature of deep space is −270ºC. (a) What is the rise in entropy in sooner or later as a result of this warm transfer? (b) How a lot job-related is made unavailable?(a) In reaching equilibrium, how a lot warmth transport occurs from 1.00 kg of water at 40.0ºC once it is placed in call via 1.00 kg of 20.0ºC water in getting to equilibrium? (b) What is the change in entropy as a result of this warm transfer? (c) How much occupational is made unaccessible, taking the lowest temperature to be 20.0ºC? Explicitly display exactly how you follow the actions in the Problem-Solving Strategies for Entropy.What is the decrease in entropy of 25.0 g of water that condenses on a bathroom mirror at a temperature of 35.0ºC, assuming no readjust in temperature and provided the latent heat of vaporization to be 2450 kJ/kg?Find the boost in entropy of 1.00 kg of liquid nitrogen that starts at its boiling temperature, boils, and warms to 20.0ºC at consistent pressure.A big electric power terminal geneprices 1000 MW of electrical energy with an performance of 35.0%. (a) Calculate the warm carry to the power station, Qh, in someday. (b) How much heat carry Qc occurs to the setting in one day? (c) If the heat transport in the cooling towers is from 35.0ºC water into the regional air mass, which boosts in temperature from 18.0ºC to 20.0ºC, what is the total rise in entropy as a result of this heat transfer? (d) How a lot power becomes unobtainable to carry out work bereason of this boost in entropy, assuming an 18.0ºC lowest temperature? (Part of Qc can be used to run heat engines or for sindicate heating the surroundings, yet it hardly ever is.)(a) How much heat carry occurs from 20.0 kg of 90.0ºC water placed in call through 20.0 kg of 10.0ºC water, producing a final temperature of 50.0ºC? (b) How a lot work-related might a Carnot engine do with this warmth carry, assuming it operates in between two reservoirs at consistent temperatures of 90.0ºC and 10.0ºC? (c) What increase in entropy is developed by mixing 20.0 kg of 90.0ºC water with 20.0 kg of 10.0ºC water? (d) Calculate the amount of occupational made uneasily accessible by this mixing making use of a low temperature of 10.0ºC, and compare it through the occupational done by the Carnot engine. Explicitly show exactly how you follow the procedures in the Problem-Solving Strategies for Entropy. (e) Discuss how everyday processes make significantly even more power unobtainable to perform job-related, as implied by this difficulty.

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entropy: a measurement of a system’s disorder and also its incapability to perform occupational in a system

adjust in entropy: the proportion of heat transport to temperature fracQT\

second law of thermodynamics declared in regards to entropy: the complete entropy of a device either rises or stays constant; it never before decreases

Selected Solutions to Problems & Exercises

1. (a) 9.78 × 104 J/K; (b) In order to acquire even more power, we should geneprice it from points within the house, choose a heat pump, human bodies, and also other appliances. As you recognize, we use a lot of power to keep our dwellings warm in the winter bereason of the loss of warm to the exterior.