In the Challenger Deep of the Marianas Trench, the depth of seawater is 10.9 km and the pressure is 1.16×10^8 Pa (about 1.15×10^3 atm). (a) If a cubic meter of water is taken from the surface to this depth, what is the change in its volume? (Normal atmospheric pressure is about 1.0×10^5 Pa. Assume that k for seawater is the same as the freshwater value given in Table 11.2.)

Verified step by step guidance

1. Identify the bulk modulus (k) of seawater from the given table or assume it to be approximately the same as that of freshwater if not specified. The bulk modulus is a measure of a substance's resistance to uniform compression.
2. Calculate the change in pressure (ΔP) experienced by the water as it is taken from the surface to the depth of 10.9 km. This can be done by subtracting the normal atmospheric pressure at the surface from the pressure at the depth.
3. Use the formula for bulk modulus, k = -ΔP / (ΔV/V), where ΔV is the change in volume and V is the original volume. Rearrange this formula to solve for ΔV, the change in volume: ΔV = -ΔP * V / k.
4. Substitute the values for ΔP, V (1 cubic meter), and k into the equation to find ΔV. Remember to keep track of the units to ensure they are consistent.
5. Interpret the result for ΔV. A negative value indicates a decrease in volume, which is expected when an object is subjected to an increase in pressure.

Key Concepts

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. It increases with depth in a fluid, calculated using the formula P = ρgh, where P is pressure, ρ is the fluid density, g is the acceleration due to gravity, and h is the depth. In the context of the Challenger Deep, the immense depth results in significantly high pressure on the water.

Bulk Modulus

The bulk modulus (k) is a measure of a substance's resistance to uniform compression. It quantifies how much a material's volume decreases under pressure, defined as the ratio of pressure increase to the relative decrease in volume. For seawater, the bulk modulus is essential for calculating the change in volume when subjected to the high pressures found at great depths.

Volume Change Calculation

The change in volume of a substance under pressure can be calculated using the formula ΔV = -V0(P - P0)/k, where ΔV is the change in volume, V0 is the original volume, P is the final pressure, P0 is the initial pressure, and k is the bulk modulus. This relationship allows us to determine how much a cubic meter of seawater will compress when taken from the surface to the depths of the ocean.