Solved thermodynamics exercise

Changing the size of a system

Exercise 7 · Lesson 3 — Fundamental Concepts

extensivity
intensivity
homogeneity
surface
dimensional analysis

Statement

An ordinary homogeneous system has mass $m$ , volume $V$ , particle number $N$ , internal energy $U$ , temperature $T$ and pressure $P$ . We assume here that $U$ is extensive, long-range interactions and edge effects being neglected. Its size is changed at fixed intensive state: $(m, V, N, U) \mapsto (λm, λV, λ N, λ U)$ , with $λ > 0$ , while $T$ and $P$ remain unchanged.

Determine the degree of homogeneity of $m / V$ , $V / N$ , $U / N$ , $P V$ , $U^{2} / V$ and $V^{1/3}$ . Which quantities are extensive, which are intensive, and which are neither?
Two expressions are proposed for an energy: $U_{1} = a N k_{B} T$ and $U_{2} = a N^{2} k_{B} T$ , where $a > 0$ is dimensionless and independent of the size of the system. Check their dimensions, then determine which one is consistent with the extensivity assumption. Explain why these two checks are distinct.
For a sphere of radius $R$ , express the area $A$ in terms of the volume $V$ . Determine the degrees of homogeneity of $A$ and $A / V$ when a single sphere is enlarged at fixed density.
Compare the area of a sphere of volume $8 V$ with the total area of eight separate spheres, each of volume $V$ . Explain why “enlarging an object” and “juxtaposing copies” do not give the same result for the area of its interfaces.

Hint

Replace each quantity by its value after the change of scale. The resulting factor must be

λ^{k}

for a quantity that is homogeneous of degree

k

. For the sphere, start from

V = 4 π R^{3} /3

and

A = 4 π R^{2}

Detailed solution

Solution

Question 1. Scale factors multiply in a product and divide in a quotient. Explicitly,

\frac{λm}{λV} = \frac{m}{V}, \frac{λV}{λ N} = \frac{V}{N}, \frac{λ U}{λ N} = \frac{U}{N} .

These three ratios are of degree $0$ and hence intensive. Next,

P (λV) = λ P V, \frac{( λ U ) ^{2}}{λV} = λ \frac{U ^{2}}{V}, (λV)^{1/3} = λ^{1/3} V^{1/3} .

Thus $P V$ and $U^{2} / V$ are of degree $1$ , and hence extensive in the sense of the given scaling law; $V^{1/3}$ is of degree $1/3$ , and is therefore neither extensive nor intensive. The degree of homogeneity says nothing about units: $P V$ is expressed in joules, but $U^{2} / V$ in $J^{2} m^{- 3}$ .

Question 2.

N

is a dimensionless number and

k_{B}

is expressed in

J K^{- 1}

. Both expressions therefore have the dimension of an energy. However,

U_{1} (λ N, T) = λ U_{1} (N, T), U_{2} (λ N, T) = λ^{2} U_{2} (N, T) .

Only $U_{1}$ is consistent with the assumed extensivity. A formula can therefore have the correct units and yet depend incorrectly on the size of the system. This does not rule out every non-extensive energy: it is the assumption made in this exercise that excludes $U_{2}$ .

Question 3. Solve for the radius, then substitute it into the area:

R = (\frac{3 V}{4 π})^{1/3}, A = 4 π (\frac{3 V}{4 π})^{2/3} = (36 π)^{1/3} V^{2/3} .

Hence $A (λV) = λ^{2/3} A (V)$ : the area is of degree $2/3$ . Dividing by $λV$ gives $A (λV) / (λV) = λ^{- 1/3} A (V) / V$ . The surface-to-volume ratio is of degree $- 1/3$ and decreases as the sphere grows.

Question 4. For the single sphere,

A (8 V) = 8^{2/3} A (V) = 4 A (V)

. The eight separate spheres, by contrast, have a total area

8 A (V)

. Their total area is therefore twice as large. Enlarging a compact object preserves its shape and reduces the relative weight of the interfaces. Juxtaposition preserves all the interfaces of the copies. The geometrical operation must be specified when studying surface contributions; these are precisely the contributions neglected in the extensive description of the bulk.