I really do not understand entropy - entropy increases wrt T

[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?

 

[thierrykauf]

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

 

[theodoros.mihos]

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.