Hydrostatics Relation for Seawater Pressure with Temperature Variation

[srh]

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.

 

[teknodude]

Why do you need to differentiate with the hydrostatic relation?

 

[piercas]

I would first start by understanding the variables and their relationships in the given formula. The formula relates pressure and density for seawater with a constant temperature. This means that the temperature is not changing with depth, only the pressure and density are affected. The formula also includes a dimensionless constant, k, which may have a specific value or range of values depending on the characteristics of the seawater being studied.

Next, I would consider the hydrostatics relation, which states that the change in pressure with depth (dp/dz) is equal to the negative of the product of the density (d) and the acceleration due to gravity (g). This relationship holds true for any fluid, including seawater.

Using this information, I would approach the problem by first substituting the given formula for density (d) into the hydrostatics relation. This would give me an equation in terms of pressure (p) and the other variables. Then, I would use the given temperature variation equation, T=T0-kz, to substitute for temperature (T) in the hydrostatics relation, since we are looking for the pressure as a function of depth.

After making these substitutions, I would rearrange the equation to solve for pressure (p) as a function of depth (z). This may involve taking the derivative with respect to z, but it should not be too complicated since the variables are now in terms of z. Finally, I would check my solution by plugging in different values for depth to see if the resulting pressure values make sense.

In summary, the key to solving this problem is understanding the relationships between pressure, density, and temperature in seawater, and using the given equations to manipulate the hydrostatics relation to solve for pressure as a function of depth.