The thermodynamic basis for surface tension starts from consideration of the Helmholtz free energy (per unit mass) is \( f \) defined by
\[
f = e - T s
\]
\(e\) is the system’s internal energy, the internal energy (aka, thermal energy) \(e\) is a manifestation of the random molecular motion of the system’s constituents. Thermodynamic property \(s\) is known as entropy.
\(T\) represents the absolute temperature of the system, typically measured in Kelvin (K)
For a reversible isothermal change, the work done on the system increases the free energy \( f \)
\[
df = de - T ds - s dT
\]
where the last term is zero for an isothermal change
Then from Property Relations \(\boxed{
T ds = de + p\,dv, \quad \text{or} \quad T ds = dh - v\,dp \quad \text{(Gibbs relation)}}
\),
\[
df = -p dv = \text{the work done on the system}
\]
these relations suggest that surface tension decreases with increasing temperature
For an interface of area \( A \), separating two fluids of densities \( \rho_1 \) and \( \rho_2 \) with volumes \( V_1 \) and \( V_2 \), and with a surface tension coefficient \( \sigma \) corresponding to free energy per unit area, the total (Helmholtz) free energy \( F \) of the system can be written as
\[
F = \rho_1 V_1 f_1 + \rho_2 V_2 f_2 + A \sigma
\]