Hydrostatics Relation for Seawater Pressure with Temperature Variation

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The discussion focuses on deriving a pressure function for seawater based on an empirical formula relating pressure and density while keeping temperature constant. The hydrostatic relation dp/dz = -dg is introduced, with density expressed as a function of pressure and temperature. The user struggles with substituting density and temperature into the equation, leading to complications in differentiation. Clarification is sought on the necessity of differentiating with respect to depth (z) in the hydrostatic context. The conversation emphasizes the challenges in applying the hydrostatic relation to the given empirical formula.
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Homework Statement


An empirical formula relating pressure and density for seawater w/ temperature constant is:
p/pa = (k+1)(d/da)^7-k
pa - pressure condition on the surface
da - density condition on the surface
k - dimensionless constant
Using this formula in the hydrostatics relation, determine the pressure as a function of depth.


Homework Equations


Hydrostatic relation: dp/dz = -dg where d is density
T=T0-kz
d=P/RT

The Attempt at a Solution


I tried to replace d w/ P/RT and then T w/ T0-kz to get the equation in terms of z. When I tried to take the derivative it was getting very complicated and I couldn't get it to work. I think I'm just approaching the problem incorrectly. I was wondering exactly how to start this problem. I know in order to take the derivative w/ respect to z there needs to be a z in the equation.
 
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Why do you need to differentiate with the hydrostatic relation?
 

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