Fluid Mechanics: Compressible Fluids, density function

[runningninja]

Homework Statement

"Suppose that a liquid has an appreciable compressibility. Its density therefore varies with depth and pressure. The density at the surface is ρ₀
(a) Show that the density varies with pressure according to ρ=ρ₀e^kp where P is gauge pressure at any depth and k is the compressibility, a constant.
(b) Find P as a function of depth y.

Homework Equations

ρ = m/V (Density definition)
k = (1/V)(ΔV/ΔP) (Compressiblity definition)
dP/dy = ρg (Pascal's Law)
My professor also hinted that I should replace any deltas with differentials, that I need to eliminate V, and to find dρ/dV (which is -m/V²

The Attempt at a Solution

From the second equation, I multiplied both sides by dP (after changing ΔV to dV and ΔP to dP) to obtain the following:
k dP = (1/V) dV
I then integrated both sides:
∫k dP (from P₀ to P) = ∫(1/V) dV (from 0 to V)
kP(gauge)=ln(V)
e^kP(gauge) = V
I get the feeling that I'm going in the correct direction, but haven't found a viable solution from this point. I fooled around with the density definition and Pascal's Law to no avail. Any assistance would be much appreciated.

 

[Chestermiller]

Good start! Your integration limits on V are incorrect. They should be from V₀ to V. Also, for a given amount of mass the product of density and volume is constant.

 

[runningninja]

Good start! Your integration limits on V are incorrect. They should be from V₀ to V. Also, for a given amount of mass the product of density and volume is constant.

When I intergrate on those limits I get kP(gauge)=ln(V)-ln(V0), which seems to be worse than before.
Also, if m = ρV is constant, how does that work with ρ and V both being varying functions?

 

[Chestermiller]

When I intergrate on those limits I get kP(gauge)=ln(V)-ln(V0), which seems to be worse than before.
Also, if m = ρV is constant, how does that work with ρ and V both being varying functions?

The compressibility condition should have a minus sign (the volume decreases with increasing pressure):

k dP = -(1/V) dV

$$\frac{V}{V_0}=\frac{\rho_0}{\rho}=e^{-kP}$$

 

[runningninja]

I understand where the v/v0 came from (the ln) but how did you derive p0/p? Did it come from another integral somewhere?
EDIT: nvm, I understand now. Thanks a bunch!