# Why is Entropy Highest at Lowest Energy?

• avistein
In summary, the concept of entropy is closely related to the likelihood or probability of different arrangements of particles within a system. In thermal equilibrium, entropy is at its maximum because that is the most likely state for a system to be in. Additionally, entropy can only be defined for equilibrium states and not during a process, and it increases when energy is transferred between subsystems.
avistein
Why entropy is highest at lowest energy?
Entropy is disorderness.So low energy means low disorderness.But why it is highest?

can you quote the exact words from the book i don't quite understand what you mean

I mean to say that more entropy relate to more stability.So more entropy relate to less energy. But how? more disorderness will relate to more energy.

No such thing. Entropy and energy are different beasts. Completely.
Entropy tends to be maximum because disordered states are much, much more probable than ordered states. At any given energy.

Simple example: black and white balls in a box. Start of with all white balls to the left, all black to the right. Only one way to do that. Shake. Energy = how hard you shake. So average kinetic energy of the balls. Entropy = average color of e.g. left half of the box, or something like that to measure disorder.

Even with just a few balls, average color is 50% gray with very small deviations. Imagine the smallness of the deviations with NAvogadro balls !

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Entropy is not disorderness. In normal systems, entropy increases with energy. According to the third law of thermodynamics, the entropy goes to zero at zero temperature, so your information is totally flawed.

Can we help avi on class XI level ? My expose might not be what helps him best at this point.
avi: you touched upon a very important concept. Can you indicate what you picked up and where ?

avistein said:
Why entropy is highest at lowest energy?
Entropy is disorderness.So low energy means low disorderness.But why it is highest?

Do you mean energy or free energy?

ok.I mean to say that why entropy is highest at equilibrium?

avistein said:
ok.I mean to say that why entropy is highest at equilibrium?
Entropy is only defined for equilibrium states. Perhaps you mean: why does entropy increase overall when heat flows from a high temperature body to a lower temperature body?

The answer is rather simple: while the higher temperature body that loses heat experiences a decrease in entropy, the lower temperature body that gains heat experiences a bigger increase in entropy. This is because dS = dQ/T so a negative heat flow at high temperature has a smaller magnitude than a positive heat flow at low temperature. Since that is the way heat flows (out of the hot and into the cold), entropy will always increase as heat spreads out.

AM

I mean to say that why entropy is highest at equilibrium?
Another way to deal with this is to turn it around: an isolated system tends towards a situation of highest entropy. That's what we call equilibrium, because it stays there. On a macroscopic scale. On a microscopic scale small fluctuations occur in a dazzling tempo with correponding micromicroscopic fluctuations in entropy. Noticeable changes in entropy are so utterly unlikely that we can consider them absent.

I'm not all that happy with Andy's statement that entropy is only defined for equilibrium states.
The expression on Boltzmann's grave is more general than that.

BvU said:
I'm not all that happy with Andy's statement that entropy is only defined for equilibrium states.
The expression on Boltzmann's grave is more general than that.

You have to define a macrostate first. The entropy is proportional to the logarithm of the number of possible microstates that can exist within the body or system in question while it presents that same macrostate. If a body is not in equilibrium, how do you define the macrostate? (e.g. if temperature, pressure, or volume or a combination of these is undefined).

AM

This all goes over avi's head. No good. we need some class XI gentle introduction of the entropy concept. Without allowing him/her to cultivate misunderstandings that will bother him/her later on, like
I mean to say that more entropy relate to more stability.So more entropy relate to less energy. But how? more disorderness will relate to more energy

Avi ?

Entropy is highest in (thermal) equilibrium, because entropy always wants* to increase, and it only stops increasing when it reaches a local maximum. An equilibrium is a stationary state. If entropy is increasing, then the state isn't stationary. Therefore, if the state is in equilibrium, then entropy isn't increasing, which means it must have hit a local maximum.

*Entropy is overwhelmingly likely to increase whenever energy is transferred between subsystems, which is constantly happening.

Khashishi said:
An equilibrium is a stationary state. If entropy is increasing, then the state isn't stationary. ...

*Entropy is overwhelmingly likely to increase whenever energy is transferred between subsystems, which is constantly happening.
When we speak about entropy increasing we mean that there is a positive difference between the entropy of a system before and after the process (i.e. between two equilibrium states of the system). Unless the system is effectively in equilibrium during the process (ie. a reversible process) entropy is not defined during the process.

AM

avistein said:
ok.I mean to say that why entropy is highest at equilibrium?

From classical thermodynamics point of view , this is a postulate which can be tested experimentally.

From modern statistical mechanics point of view, this can be proven (provided that you postulate something else ).

avistein said:
Why entropy is highest at lowest energy?
Entropy is disorderness.So low energy means low disorderness.But why it is highest?

I used to get very confused by thinking of entropy as disorder. I would use the word dispersion (i.e. how close to uniformity things are distributed).

Let's say you have a closed bottle of gas of volume $V$, where there area total of $N$ atoms, with total energy $U$. There are many many ways that $N$ atoms with total energy $U$ can be arranged in a closed bottle of volume $V$. The entropy $S(U,V,N)$ of this gas is a measure of how many ways there are to arrange all the $N$ atoms so that the total energy will be $U$, and the total volume will be $V$.

One of the fundamental theories of thermodynamics states that each of these possible arrangements are equally likely. Because of this, we may say that a system always tends toward maximum entropy simply because that is (by far) the most likely state for the system to be in.

As far as why energy is minimized at maximum entropy, this is a mathematical statement of the following:
For a closed system,
At constant entropy $S$, the equilibrium state will be that of the minimum energy $U$.
This is saying the same thing as...
For a closed system,
At constant energy $U$, the equilibrium state will be that of the maximum entropy $S$

Anybody notice poor avi is out of the picture ?
Anybody notice entropy isn't even in the PF library ?

## 1. What is entropy at lowest energy?

Entropy at lowest energy refers to the state of a system at its minimum energy level, where the disorder or randomness of the system is at its lowest possible level. This is often referred to as the most stable state of a system.

## 2. How does entropy change at lowest energy?

Entropy does not change at lowest energy, as it is already at its minimum level. However, it can increase or decrease as the system moves away from this state.

## 3. What is the relationship between entropy and lowest energy?

The relationship between entropy and lowest energy is that entropy tends to increase as the system moves away from its lowest energy state. This is due to the fact that as the system becomes more disordered, it has more possible microstates, which leads to an increase in entropy.

## 4. Can entropy ever reach zero at lowest energy?

In theory, entropy can reach zero at lowest energy. However, this would require a perfectly ordered system with no possible microstates, which is not possible in the real world.

## 5. How does the concept of entropy at lowest energy relate to the laws of thermodynamics?

The concept of entropy at lowest energy relates to the second law of thermodynamics, which states that the total entropy of a closed system will always tend to increase over time. At lowest energy, the system is in its most stable state, and any changes would result in an increase in entropy, following the second law.

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