Hydrostatics Relation for Seawater Pressure with Temperature Variation

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SUMMARY

The discussion focuses on deriving the pressure as a function of depth in seawater using the hydrostatic relation and an empirical formula relating pressure and density. The formula provided is p/pa = (k+1)(d/da)^7-k, where pa is the surface pressure, da is the surface density, and k is a dimensionless constant. The hydrostatic relation dp/dz = -dg is utilized, with temperature expressed as T = T0 - kz and density as d = P/RT. The participant struggled with the complexity of the derivatives and sought clarification on the correct approach to incorporate depth into the equations.

PREREQUISITES
  • Understanding of hydrostatic pressure principles
  • Familiarity with empirical formulas in fluid mechanics
  • Knowledge of thermodynamic equations, specifically the ideal gas law
  • Basic calculus, particularly differentiation techniques
NEXT STEPS
  • Study the derivation of hydrostatic pressure equations in fluid mechanics
  • Learn about the implications of temperature variation on seawater density
  • Explore the application of the ideal gas law in varying conditions
  • Review techniques for simplifying complex derivatives in calculus
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Students in physics or engineering disciplines, particularly those focusing on fluid mechanics, thermodynamics, or environmental science, will benefit from this discussion.

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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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