# How does volume affect microstates?

##### 1 Answer

**Entropy** is a function of the volume. As the volume increases in a system, the entropy increases.

As we know,

#S = k_Blnt# where

#t# is the most probable distribution of microstates in a system.

Hence, the number of microstates **increases** due to increasing volume.

To show this, we verify that

From the **Helmholtz Maxwell Relation**,

#dA = -SdT - PdV#

Since

#((delS)/(delV))_T = ((delP)/(delT))_V#

Volume must depend on pressure and temperature. If we write the total differential of the volume as a function of

#dV = ((delV)/(delT))_PdT + ((delV)/(delP))_TdP#

By definition, the isothermal compressibility factor is

#kappa = -1/V ((delV)/(delP))_T#

and the coefficient of thermal expansion is:

#alpha = 1/V ((delV)/(delT))_P#

As a result, we replace those terms to get:

#dV = ValphadT - VkappadP#

Now we divide by

#cancel(((delV)/(delT))_V)^(0) = Valpha cancel(((delT)/(delT))_V)^(1) - Vkappa ((delP)/(delT))_V#

Therefore:

#color(blue)barul|stackrel(" ")(" "((delP)/(delT))_V = alpha/kappa = ((delS)/(delV))_T" ")|#

Since **must be positive** for **liquids** and **solids**.

For gases, we consider merely that increasing temperature increases particle motion, and thus the force colliding with the rigid container walls. By definition, that leads to increasing gas pressure.

Therefore, **positive** for **gases** as well.