Effect of temperature on surface tension

[AmanWithoutAscarf]

Homework Statement

According to the Eötvös rule, the surface tension is a linear function of the temperature, but I cannot find any theoretical proof of the function, except for the following excersise.

Relevant Equations

$F=\sigma .L$

The translated version is:

Surface tension is temperature-dependent. Therefore, it is essential to specify the temperature when providing the surface tension value of an interface. Typically, surface tension decreases with increasing temperature and approaches zero at the critical temperature. This excersise will explore these concepts in detail.

1.

a) Use Carnot's theorem to find the variation of surface tension $σ$ with temperature $T$.

b) Calculate the temperature change of the liquid film during adiabatic expansion.

2.

Inside a soap bubble of radius $\displaystyle r_{0}$ contains air (ideal gas) at temperature $\displaystyle T_{0}$ and pressure $\displaystyle p_{0}$. The surface tension of the soap solution at this temperature is $\displaystyle \sigma _{0}$. The specific heat of formation of a unit of soap film surface in an isothermal process is $\displaystyle q_{0}$. Find the derivative of bubble radius with respect to temperature $\displaystyle \frac{dr}{dT}$ when $\displaystyle T_{0}$. The outside pressure remains constant.

I did research on the topic and the Eötvös rule, but most of the results are just qualitative explanations or experiment-based proofs of the temperature-dependent function of surface tension.

Can anyone give me some hints on how to prove that linear relation (using Carnot's theorem)? And the following questions, if possible, please.

 

[Steve4Physics]

Can anyone give me some hints on how to prove that linear relation (using Carnot's theorem)?

To quote from here: https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Thermodynamics_and_Statistical_Mechanics_(Nair)/03:_The_Second_Law_of_Thermodynamics/3.01:_Carnot_Cycle
“The processes involved in the Carnot cycle may refer to compression and expansion if the material is a gas ... But any other pair of thermodynamic variables will do as well. We can think of a Carnot cycle utilizing magnetization and magnetic field, or surface tension and area …”

So I’d guess you are meant to conceive of some design of heat-engine based on a liquid being heated/cooled so that surface tension and area cyclically change and the driving 'force' is surface tension. Presumably the efficiency can be shown to be some function of surface tension, then the dependence of surface tension on temperature should emerge.

Just a guess though.

 

[AmanWithoutAscarf]

So if I apply Carnot cylcle with the driven force from surface tension, the efficiency will be:
$$\eta =1-\frac{T_{2}}{T_{1}} =\frac{W}{Q_{H}} =\frac{\int _{S1}^{S2} \sigma ( T) .dS}{\int _{T1}^{T2} C.dT}$$
Suppose the system witnesses a minimal change in temperature $dT$ when it is stretched by $dS$. We have:
$$ \begin{array}{l}
T_{1} -T_{2} =dT\\
W=\sigma ( T) .dS\\
Q_{H} =C.dT
\end{array}$$
Substituting the variables, I think the result would be: $$\frac{\sigma ( T) .dS}{C.dT} =\frac{dT}{T} $$
but I don't know how to cancel out $dS$ and $dT$ to have a linear function of $\sigma ( T)$ T.T