Challenger Deep in the Marianas Trench of the Pacific Ocean is the deepest known spot in the Earth's oceans, at 10.922 km below sea level. Taking density of seawater at atmospheric pressure (p0 = 101.3 kPa) to be 1024 kg/m3 and its bulk modulus to be B(p) = B0 + 6.67(p − p0), with B0 = 2.19 109 Pa, calculate the pressure and the density of the seawater at the bottom of Challenger Deep. Disregard variations in water temperature and salinity with depth.
Pressure at the bottom of Challenger Deep:
P = B0 + 6.67(p - p0) = 2.19 x 10^9 + 6.67(10.922 x 10^3 - 101.3) = 1.09 x 10^11 Pa
Density of seawater at the bottom of Challenger Deep:
Density = 1024 kg/m3
To calculate the pressure and density of the seawater at the bottom of Challenger Deep, we can use the hydrostatic pressure equation and the given bulk modulus equation.
Step 1: Calculate the pressure at the bottom of Challenger Deep.
Using the hydrostatic pressure equation:
p - p0 = B(p0) * (ρ - ρ0)
Where:
p is the pressure at the bottom of Challenger Deep
p0 is the atmospheric pressure (101.3 kPa)
B(p0) is the bulk modulus at atmospheric pressure (2.19 x 10^9 Pa)
ρ is the density of the seawater at the bottom of Challenger Deep (what we want to find)
ρ0 is the density of seawater at atmospheric pressure (1024 kg/m^3)
Let's substitute the known values into the equation and solve for p:
p - 101.3 = (2.19 x 10^9) * (ρ - 1024)
Step 2: Calculate the density at the bottom of Challenger Deep.
We can rearrange the equation to solve for ρ:
ρ = (p - 101.3) / (2.19 x 10^9) + 1024
Now we can substitute the calculated value of p to find the density at the bottom of Challenger Deep.
Let's calculate:
ρ = (10.922 x 10^3 - 101.3) / (2.19 x 10^9) + 1024
Upon evaluating the expression, the pressure at the bottom of Challenger Deep is approximately 1.108 GPa (gigapascal), and the density of the seawater at the bottom is approximately 1033.96 kg/m^3.
To calculate the pressure and density of seawater at the bottom of Challenger Deep, we can make use of the given information and the equation for bulk modulus.
1. Calculate the pressure at the bottom of Challenger Deep:
The pressure at any depth can be calculated using the formula: P = P0 + ρgh, where P is the pressure, P0 is atmospheric pressure, ρ is the density of seawater, g is the acceleration due to gravity, and h is the depth.
Given:
P0 = 101.3 kPa = 101,300 Pa
ρ = 1024 kg/m³
g = 9.8 m/s²
h = 10.922 km = 10,922 m
Substitute the values into the formula:
P = 101,300 Pa + (1024 kg/m³)(9.8 m/s²)(10,922 m)
P = 101,300 Pa + 108,048,704 Pa
P ≈ 108,149,004 Pa
Therefore, the pressure at the bottom of Challenger Deep is approximately 108,149,004 Pa.
2. Calculate the density of seawater at the bottom of Challenger Deep:
The bulk modulus equation given is: B(p) = B0 + 6.67(p - p0)
At the bottom of Challenger Deep, the pressure is denoted as p, and we want to find the corresponding density ρ.
Substitute B(p), B0, p, and p0 into the equation:
B(p) = B0 + 6.67(p - p0)
B(p) = 2.19 * 10^9 Pa + 6.67(p - 101,300 Pa)
Since we know that B(p) = ρ * g, where g is the acceleration due to gravity:
ρ * g = 2.19 * 10^9 Pa + 6.67(p - 101,300 Pa)
Substitute the known values:
1024 kg/m³ * 9.8 m/s² = 2.19 * 10^9 Pa + 6.67(p - 101,300 Pa)
Simplify the equation to solve for p:
10035.2 kg/(ms²) = 2.19 * 10^9 Pa + 6.67p - 668,041 Pa
10035.2 kg/(ms²) - 2.19 * 10^9 Pa + 668,041 Pa = 6.67p
10035.2 kg/(ms²) + 666,000,959 Pa = 6.67p
p ≈ 1,502,227,279 Pa
Therefore, the pressure at the bottom of Challenger Deep is approximately 1,502,227,279 Pa.
3. The density of the seawater at the bottom of Challenger Deep is the same as the given value, ρ = 1024 kg/m³.
Therefore, the pressure at the bottom of Challenger Deep is approximately 1,502,227,279 Pa, and the density of the seawater is approximately 1024 kg/m³.