What is the difference in volume (due only to pressure changes, not temperature or other factors) between 1000 kg of water at the surface (assume 4 degrees C) of the ocean and the same mass at the deepest known depth, 8.00 km? (Mariand Trench; assume also 4 degrees C.)
The formula and the bulk modulus (E) value you need are given here:
http://www.engineeringtoolbox.com/bulk-modulus-elasticity-d_585.html
Use density and E values for sea water
To find the difference in volume between the water at the surface and the deepest known depth, we need to consider the change in pressure.
First, let's determine the pressure at the surface and the deepest point. The pressure in a fluid increases with depth due to the weight of the overlying fluid.
The pressure at the surface of the ocean can be approximated using the formula:
P(surface) = P(atm) + ρ * g * h
Where:
P(surface) is the pressure at the surface (unknown).
P(atm) is the atmospheric pressure (assumed to be standard atmospheric pressure, which is approximately 101325 Pa).
ρ is the density of water (assumed to be constant at approximately 1000 kg/m³).
g is the acceleration due to gravity (approximately 9.81 m/s²).
h is the depth (0 m at the surface).
Similarly, the pressure at the deepest known depth can be calculated using:
P(deepest) = P(atm) + ρ * g * h
Where:
P(deepest) is the pressure at the deepest known depth (unknown).
h is the depth (8000 m at the deepest known depth).
Since the temperature is assumed to be constant at 4 degrees Celsius, the density (ρ) of water will also remain constant.
Now, we can calculate the difference in pressure between the two points:
ΔP = P(deepest) - P(surface)
Once we have the pressure difference, we can use the relationship between pressure and volume for an incompressible fluid, which is:
ΔV/V = -ΔP/P
Where:
ΔV is the change in volume (unknown).
V is the initial volume (mass divided by density).
ΔP is the change in pressure (calculated previously).
P is the initial pressure (calculated previously).
Finally, we can substitute the known values into the equation to find the change in volume (ΔV), given that the mass is 1000 kg.
This approach allows us to consider the effect of pressure changes on the volume of water without considering other factors like temperature.