Exploring Force of Surface Tension: An Analysis Using Maxwell Relations

[mkbh_10]

Homework Statement

Discuss the variation of force of surface tension with the help of maxwell relations ?

Homework Equations

The Attempt at a Solution

It is a question from previous year question paper , my exams are going so i am asking for little help as i don't know how to connect the two as the books that i have don't mention it anywhere

 

[Mapes]

You can do this by writing the first law in differential form

$$dU=T\,dS-p\,dV+\mu\,dN$$

and adding a term for surface energy to let you set up Maxwell relations.

 

[mkbh_10]

i am still not getting it ?

 

[Mapes]

Surface tension adds an additional energy term $\gamma\,dA$ where $\gamma$ is the surface energy and $A$ is the area.

Maxwell relations arise because the equation I wrote above is really

$$dU=\left(\frac{\partial U}{\partial S}\right)_{V,N,A}dS+\left(\frac{\partial U}{\partial V}\right)_{S,N,A}dV+\left(\frac{\partial U}{\partial N}\right)_{S,V,A}dN+\left(\frac{\partial U}{\partial A}\right)_{S,V,N}dA$$

and we've assigned the variables $T$, $-p$, $\mu$, and $\gamma$ to the partial derivatives. Therefore

$$\left(\frac{\partial T}{\partial V}\right)=\left(\frac{\partial^2 U}{\partial S\,\partial V}\right)=\left(\frac{\partial^2 U}{\partial V\,\partial S}\right)=-\left(\frac{\partial p}{\partial S}\right)$$

You should be able to apply the same reasoning to differentials involving $\gamma$.