- #1
Jacob Nie
- 9
- 4
- Homework Statement
- There is an equation in Riley's Mathematical Methods that I am confused about:
Applying (4.43) to ##dS##, with variables ##V## and ##T##, we find
$$dU = T \ dS - P \ dV = T\left[ \left(\dfrac{\partial S}{\partial V}\right)_T \ dV + \left(\dfrac{\partial S}{\partial T}\right)_V \ dT\right] - P \ dV.$$
- Relevant Equations
- Eq 4.43:
$$ dU = \left(\dfrac{\partial U}{\partial X}\right)_Y \ dX + \left(\dfrac{\partial U}{\partial Y}\right)_X \ dY$$
What I don't understand is why ##dS## is expanded in only the two differentials ##dV## and ##dT.## Why doesn't it look more like:
$$dS = \left(\dfrac{\partial S}{\partial V}\right)_{T,P,U} \ dV + \left(\dfrac{\partial S}{\partial T}\right)_{V,P,U} \ dT + \left(\dfrac{\partial S}{\partial P}\right)_{V,T,U} \ dP + \left(\dfrac{\partial S}{\partial U}\right)_{V,T,P} \ dU$$
?
$$dS = \left(\dfrac{\partial S}{\partial V}\right)_{T,P,U} \ dV + \left(\dfrac{\partial S}{\partial T}\right)_{V,P,U} \ dT + \left(\dfrac{\partial S}{\partial P}\right)_{V,T,U} \ dP + \left(\dfrac{\partial S}{\partial U}\right)_{V,T,P} \ dU$$
?