Calculation of Change in Density of Water at a Depth of 400m

Given:
Density of water at surface (ρ₀) = 1030 kg/m³
Depth below the surface (h) = 400 m
Bulk modulus of water (K) = 2 × 10^9 N/m³

Formula:
The change in density (Δρ) can be calculated using the formula:

Δρ = -ρ₀ * (h/K)

where:
Δρ = Change in density
ρ₀ = Density at the surface
h = Depth below the surface
K = Bulk modulus of water

Calculation:
Substituting the given values into the formula, we have:

Δρ = -1030 kg/m³ * (400 m / 2 × 10^9 N/m³)

Simplifying the expression:

Δρ = -1030 kg/m³ * (2 × 10^(-4) m²/N)

Δρ = -206 × 10^(-4) kg/m³

Δρ = -0.0206 kg/m³

Explanation:
The change in density of water at a depth of 400 m below the surface is approximately -0.0206 kg/m³. This indicates that the density of water decreases as we go deeper into the lake.

The negative sign in the result indicates a decrease in density. This is because the pressure increases with depth, and as a result, the water molecules become more closely packed, leading to an increase in density. However, the compressibility of water also comes into play.

The bulk modulus of water represents its resistance to compression. As the depth increases, the pressure also increases, causing the water to compress slightly. This compression counteracts the increase in density due to increased pressure.

Therefore, the net effect is a decrease in density with depth. The change in density is directly proportional to the depth and inversely proportional to the bulk modulus. In this case, the change in density is relatively small due to the high bulk modulus of water.

It is important to note that this calculation assumes constant temperature and neglects other factors such as temperature variations, salinity, and dissolved substances, which can also affect the density of water in a lake.