"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=ρ0ekp where P is gauge pressure at any depth and k is the compressibility, a constant.
(b) Find P as a function of depth y.
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)
ekP(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.