Problem 15 - Chemistry: The Central Science

Question

(a) How high in meters must a column of water be to exert a pressure equal to that of a 760 -mm column of mercury? The density of water is \(1.0 \mathrm{~g} / \mathrm{mL}\), whereas that of mercury is \(13.6 \mathrm{~g} / \mathrm{mL}\). (b) What is the pressure in atmospheres on the body of a diver if he is \(39 \mathrm{ft}\) below the surface of the water when atmospheric pressure at the surface is \(0.97\) atm?

Step-by-Step Solution

Answer

The height of the water column required to exert the same pressure as that of a 760-mm column of mercury is 10.31 m. The pressure on the diver at a depth of 39 ft below the surface of the water with an atmospheric pressure of 0.97 atm is 2.12 atm.

Step 1: a) Finding the height of the water column

To find the height of the water column that exerts equal pressure as a 760-mm column of mercury, we need to equate the pressures exerted by both columns. First, let's assign variables to the given information: Density of water, ρw = 1.0 g/mL = 1000 kg/m³ (converted to SI units) Density of mercury, ρm = 13.6 g/mL = 13600 kg/m³ (converted to SI units) Height of mercury column, hm = 760 mm = 0.76 m (converted to SI units) Gravity, g = 9.81 m/s² Height of water column, h = ? Now, we equate the pressures: ρw × g × h = ρm × g × hm 1000 × 9.81 × h = 13600 × 9.81 × 0.76 Now, let's solve for h: h = (13600 × 9.81 × 0.76) / (1000 × 9.81) = 10.31 m The height of the water column required to exert the same pressure as that of a 760-mm column of mercury is 10.31 m.

Step 2: b) Finding the pressure on the diver

To find the pressure on the diver, we first calculate the pressure due to the water column and then add the atmospheric pressure at the surface. Let's assign variables to the given information: Depth of diver, d = 39 ft = 11.887 m (converted to SI units) Pressure at surface, Psurface = 0.97 atm = 98,182.3 Pa (converted to SI units using 1 atm = 101,325 Pa) Density of water, ρw = 1000 kg/m³ Gravity, g = 9.81 m/s² Pressure on diver, P = ? Now, we calculate the pressure due to the water column: Pwater = ρw × g × d = 1000 × 9.81 × 11.887 = 116,733.27 Pa Now, let's add the atmospheric pressure at the surface to find the total pressure on the diver: P = Pwater + Psurface = 116,733.27 + 98,182.3 = 214,915.57 Pa Finally, to get the pressure in atmospheres, we can convert back to atmospheres: P = 214,915.57 Pa × (1 atm / 101,325 Pa) = 2.12 atm The pressure on the diver is 2.12 atm.

Key Concepts

Density of Liquids

In fluid mechanics, density is a key concept that often comes up when dealing with liquids. The density of a liquid is defined as the mass of the liquid divided by its volume. For example, water has a density of 1.0 g/mL, which is equivalent to 1000 kg/m³ in the SI unit system. On the other hand, mercury's density is significantly higher at 13.6 g/mL or 13,600 kg/m³. These values are crucial in calculating the pressure exerted by a liquid column. For instance, denser liquids exert more pressure at a given height compared to less dense ones. That's why a smaller height of mercury can exert the same pressure as a much taller column of water.

Pressure Conversion

Pressure conversion is an essential skill when dealing with different units of pressure. In fluid dynamics, pressure is typically measured in Pascals (Pa) in the SI system. However, other units such as atmospheres (atm), millimeters of mercury (mmHg), and Torr are common as well. For example, 1 atm is equivalent to 101,325 Pa, and 760 mmHg. These conversion factors enable us to switch between units depending on the requirements of the calculation, like when converting atmospheric pressure to Pascals for further use in calculations.

Atmospheric Pressure

Atmospheric pressure is the force exerted by the weight of the atmosphere above a given point. It is typically measured in atmospheres (atm), another common unit being Pascals (Pa). On the Earth's surface at sea level, the standard atmospheric pressure is approximately 1 atm or 101,325 Pa. When underwater, for example, the diver experiences pressure due to the water column above them plus the atmospheric pressure from above the water surface. These combined pressures determine the total force exerted on objects submerged in a liquid.

SI Units Conversion

Conversion to SI units is a fundamental process in physics problems to ensure consistency and accuracy. The International System of Units (SI) is a standardized system that helps scientists and engineers communicate findings without discrepancies. In fluid mechanics, you often need to convert between different units like feet to meters for depth, or grams per milliliter (g/mL) to kilograms per cubic meter (kg/m³) for density. For pressure, converting between atmospheric pressure (atm) to Pascals (Pa) is common by using the factor 1 atm = 101,325 Pa. These conversions ensure that all terms in the equations are compatible, allowing straightforward calculations and interpretations.