How does volume affect microstates?
Entropy is a function of the volume. As the volume increases in a system, the entropy increases.
As we know,
#S = k_Blnt#
#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#
#((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#
#color(blue)barul|stackrel(" ")(" "((delP)/(delT))_V = alpha/kappa = ((delS)/(delV))_T" ")|#
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.