# I really do not understand entropy - entropy increases wrt T

• barnflakes
In summary, the conversation discusses the relationship between heat energy, temperature, and entropy in a monatomic gas. While the equation for entropy suggests that increasing the temperature also increases the number of microstates, the speaker is struggling to understand how this applies to a gas with no internal degrees of freedom. However, the other participant provides an example with atoms and explains that adding energy can create more microstates and increase temperature.
barnflakes
Every explanation of this I have read has been extremely poor.

Imagine we have a MONATOMIC gas, with no internal degrees of freedom. The gas is confined to a box of volume V, and this volume is constant and is not allowed to increased upon adding heat energy.

We add an infinitesimal amount of heat energy to the box, delta Q. Now, there is an equation which tells me that the entropy just increased:

\delta S = \delta Q/T

However, let's think about the statistical mechanical definition of entropy, which is that the entropy is proportional to the number of microstates that the system can occupy for a given energy.

If the entropy increases, the number of microstates that give the same total energy must have increased.

I cannot for the life of me see how increasing the temperature increases the number of microstates of a monatomic gas.

The heated gas has no additional translational degrees of freedom compared with before, and it has no rotational or vibrational degrees of freedom since it is monatomic so that doesn't count either.

So where are these additional microstates coming from?

The higher the temperature the more ways you can reproduce the observed heat through motion.

Let a system with 3 atoms and only one of these have speed v0. The total energy is $$E_0=\frac{1}{2}mv_0^2$$ and, for easy to compute, let energy quantum step is e0 = E0/2.
With no energy addition, after some time we have these probabilities: $$(2e_0,0,0)\times3,\,(e_0,e_0,0)\times3 = 6 \text{states}$$
Adding energy e0 to the system we have: $$(3e_0,0,0)\times3,\,(2e_0,e_0,0)\times3,\,(e_0,e_0,e_0)\times3 =9 \text{states}$$
and so on. So, energy addition make more microstates. See that the new system have more energy, so more temperature. That is the meaning of T over the fraction. In high temperature the same energy addition make less entropy addition.
In some systems energy is fragment above, so energy addition over a limit makes less microstates, and for these cases the entropy move to less but the system is not alone.

## 1. What is entropy and how does it relate to temperature?

Entropy is a measure of the disorder or randomness in a system. The higher the temperature, the more energy is available to increase the disorder in the system, thus leading to an increase in entropy.

## 2. Why does entropy increase with temperature?

As temperature increases, the molecules in a system gain more kinetic energy and are able to move around and interact with each other more, leading to a greater degree of disorder and an increase in entropy.

## 3. How is entropy related to the second law of thermodynamics?

The second law of thermodynamics states that the total entropy of a closed system will always increase over time. This means that, in any natural process, the total amount of disorder or randomness in the system will increase, and this is reflected in the increase of entropy with temperature.

## 4. Does entropy always increase with temperature?

In most cases, yes. However, there are some exceptions, such as when a substance undergoes a phase change (e.g. from solid to liquid), where the increase in entropy is not directly related to temperature but rather to the change in the physical state of the substance.

## 5. How does entropy affect the behavior of a system?

The increase in entropy with temperature leads to a tendency for a system to move towards a state of maximum disorder. This can affect the behavior of the system, as it may lead to changes in pressure, volume, and other properties in order to increase the overall entropy of the system.

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