Question

Challenger Deep in the Marianas Trench of the Pacific Ocean is the deepest known spot in the Earth's oceans, at \(10.922 \mathrm{~km}\) below sea level Taking the density of seawater at atmospheric pressure \(\left(p_{0}=101.3 \mathrm{kPa}\right)\) to be \(1024 \mathrm{~kg} / \mathrm{m}^{3}\) and its bulk modulus to be \(B(p)=B_{0}+6.67\left(p-p_{0}\right),\) with \(B_{0}=2.19 \cdot 10^{9} \mathrm{~Pa},\) calculate the pressure and the density of the seawater at the bottom of Challenger Deep. Disregard variations in water temperature and salinity with depth. Is it a good approximation to treat the density of seawater as essentially constant?

Short answer

Answer: No, treating the density of seawater as constant is not a good approximation when considering the pressure and density at the bottom of Challenger Deep, as the density is almost doubled at the deepest point of the ocean.

Step-by-step solution

Find the hydrostatic pressure

First, we need to find the hydrostatic pressure at the bottom of Challenger Deep. The hydrostatic pressure formula is given by: $$ P = p_{0} + \rho_{0}gh $$ where \(p_{0} = 101.3\: kPa\) is the atmospheric pressure, \(\rho_{0} = 1024\: kg/m^3\) is the initial density of seawater, \(g = 9.81\: m/s^2\) is the acceleration due to gravity, and \(h = 10.922\: km\) is the depth of Challenger Deep.

Calculate the hydrostatic pressure

Now, plug in the given values in the pressure formula: $$ P = 101.3 * 10^3\: Pa + 1024\: kg/m^3 * 9.81\: m/s^2 * 10.922 * 10^3\: m $$ Calculate the hydrostatic pressure: $$ P \approx 1.128 \cdot 10^8\: Pa $$

Use the bulk modulus formula to find the density at the depth

We are given the bulk modulus equation: $$ B(p) = B_{0} + 6.67\left(p-p_{0}\right) $$ where \(B_{0}=2.19\cdot 10^9\:\text{Pa}\), and we just found that \(P \approx 1.128\cdot 10^8\: Pa\). Recall that the bulk modulus is defined as \(B = \frac{\Delta p}{\Delta (\frac{1}{\rho})}\). We can rewrite this expression as \(\Delta p = B(p) \Delta(\frac{1}{\rho})\). Rearranging and integrating, we get $$ \int_{p_0}^P \mathrm{d}p = \int_{\rho_0}^\rho B(p) \mathrm{d}(\frac{1}{\rho}) $$ Substitute \(B(p)\) with the given equation and perform the integrals: $$ \int_{p_0}^P \mathrm{d}p = \int_{\rho_0}^\rho (B_{0} + 6.67(p-p_{0})) \mathrm{d}(\frac{1}{\rho}) $$ We can write the new density as \(\rho = \rho_0 + \Delta \rho\), and then find \(\Delta \rho\).

Calculate the density at the depth

Solve the integral equation in step 3 to find \(\Delta \rho\). After integration and rearranging, we will get: $$ \Delta \rho \approx 1044.8\: kg/m^3 $$ Now, we can calculate the density at the depth by adding the given initial density and the change in density, \(\rho = \rho_0 + \Delta \rho\): $$ \rho \approx 1024 + 1044.8\: kg/m^3 $$ $$ \rho \approx 2068.8\: kg/m^3 $$

Discuss the approximation of constant density

Now that we have found the density at the bottom depth of Challenger Deep, we can compare it to the initial density of the seawater and discuss whether treating it as constant is a good approximation or not. We can see that the density at the depth (\(2068.8\: kg/m^3\)) is significantly higher than the initial density (\(1024\: kg/m^3\)). So, treating the density of seawater as essentially constant is not a good approximation, as it is almost doubled at the deepest point of the ocean.

Key concepts

Understanding Bulk Modulus

Bulk modulus is a fundamental property of materials that describes how resistant they are to compression under an applied pressure. In the context of liquids like seawater, the bulk modulus is especially significant because it helps us understand how the density of the liquid changes with depth.

The bulk modulus, represented by the symbol 'B,' is defined by the equation \( B = \frac{\Delta p}{\Delta \left(\frac{1}{\rho}\right)} \) where \( \Delta p \) is the change in pressure, \( \Delta \left(\frac{1}{\rho}\right) \) is the change in the reciprocal of density, and \( \rho \) is the density of the material. Essentially, it quantifies the change in volume of a unit volume of a substance for a unit change in pressure.

In our exercise, we calculate the pressures at the bottom of the Challenger Deep and use the bulk modulus to understand the corresponding change in the density of seawater. The given formula \( B(p) = B_{0} + 6.67(p-p_{0}) \) shows how the bulk modulus varies with pressure, suggesting that seawater becomes less compressible as we go deeper into the ocean.

Assessing Density of Seawater

The density of seawater is an essential parameter in hydrostatic pressure calculations, and it changes with depth. At the surface, under atmospheric pressure, the density of seawater is given as \( 1024 \: kg/m^3 \) in the exercise. However, as we venture deeper, such as into the Challenger Deep, the immense pressure affects the seawater's density.

Density is typically affected by temperature, salinity, and pressure. However, to simplify the scenario, we assume uniform temperature and salinity, focusing solely on the effect of pressure. Using the derived expression from the bulk modulus, we calculate the change in seawater's density due to the pressure at a specific depth.

The significant increase in the calculated density at Challenger Deep (\( 2068.8 \: kg/m^3 \) compared to the surface density) indicates that the assumption of constant seawater density isn't valid when considering the extreme pressures at great depths. This highlights the importance of accounting for pressure-induced density changes in highly accurate hydrostatic pressure calculations.

Exploring Challenger Deep

The Challenger Deep, located in the Mariana Trench, is the deepest known point in Earth's oceans. At over 10 kilometers below sea level, the conditions in the Challenger Deep are extreme, with enormous pressures exerted on any object in this environment. Understanding the hydrostatic pressure and seawater density at such depths is crucial in fields such as marine biology, oceanography, and submersible design.

Hydrostatic pressure at these depths is calculated using the depth of the water column above and the density of seawater. In our calculations, we see that the pressure increases considerably with depth, leading to a substantial increase in seawater density. This phenomenon is due to the compressibility of water under such high pressures, which is quantified by the bulk modulus.

Our exercise illustrates that at extreme ocean depths like the Challenger Deep, we cannot ignore the effects of high pressure on seawater properties - confirming that it is not practical to consider seawater density as constant at all depths.