Question

At the surface of a freshwater lake the air pressure is 1.0 atm. At what depth under water in the lake is the water pressure 4.0 atm?

Short answer

Answer: Approximately 51.6 meters.

Step-by-step solution

Recall the hydrostatic pressure formula

The hydrostatic pressure formula is given by \(P = \rho gh\), where \(P\) is the pressure at a certain depth, \(\rho\) is the density of the fluid, \(g\) is the acceleration due to gravity, and \(h\) is the depth below the surface.

Define the variables and given values

We know the following values: - Atmospheric pressure, \(P_{atm} = 1 \, atm\) - Water pressure we want to find the depth for, \(P_{water} = 4 \, atm\) - Total pressure at the depth, \(P_{total} = P_{atm} + P_{water}\) - Density of freshwater, \(\rho = 1000 \, kg/m^3\) - Acceleration due to gravity, \(g = 9.81 \, m/s^2\) - Depth underwater, \(h\): This is what we want to find.

Calculate the total pressure at the depth

First, we need to find the total pressure at the depth we want to find. This is the sum of the atmospheric pressure and the water pressure at the depth. So, \(P_{total} = P_{atm} + P_{water} = 1 \, atm + 4 \, atm = 5 \, atm\). To convert this pressure to Pascals (Pa), we need to multiply by the conversion factor of \(1 \, atm = 101325 \, Pa\). Therefore, \(P_{total} = 5 \, atm \times 101325 \, Pa/atm = 506625 \, Pa\).

Rearrange the hydrostatic pressure formula and solve for depth

Now, we can use the formula \(P_{total} = \rho gh\) to find the depth \(h\). We want to solve for \(h\), so we can rearrange this equation as follows: \(h = \frac{P_{total}}{\rho g}\) Now, plug in the values we know: \(h = \frac{506625 \, Pa}{1000 \, kg/m^3 \times 9.81 \, m/s^2} \approx 51.6 \, m\)

Interpret the result

So, the depth under the surface of the freshwater lake at which the water pressure is 4.0 atm is approximately 51.6 meters.