Question

(a) The compound 1-iodododecane is a nonvolatile liquid with a density of 1.20 \(\mathrm{g} / \mathrm{mL}\) . The density of mercury is 13.6 \(\mathrm{g} / \mathrm{mL} .\) What do you predict for the height of a barometer column based on 1 -iodododecane, when the atmospheric pressure is 749 torr? (b) What is the pressure, in atmospheres, on the body of a diver if he is 21 ft below the surface of the water when the atmospheric pressure is 742 torr?

Short answer

The height of the 1-iodododecane barometer column is approximately 8.48 meters, and the pressure on the diver's body is approximately 1.6 atmospheres.

Step-by-step solution

a) Finding the height of the 1-iodododecane barometer column

The formula for hydrostatic pressure is: Pressure = Density × Gravity × Height We are given the atmospheric pressure (749 torr) and the density of 1-iodododecane (1.20 g/mL). We will first convert the pressure to pascals and the density to kg/m^3, then find the height of the column. 1 atm = 101325 Pa and 1 atm = 760 torr Density conversion: \(1.20 \, \frac{g}{mL} × \frac{1000 \, mL}{L} × \frac{1 \, L}{0.001 \, m^{3}} = 1200 \, \frac{kg}{m^{3}}\) Pressure conversion: \(749 \, \frac{torr}{atm} × \frac{101325 \, Pa}{760 \, torr} = 100338.5 \, Pa\) Now, using the formula for hydrostatic pressure, we can find the height: Height = \( \frac{Pressure}{Density × Gravity} = \frac{100338.5 \, Pa}{1200 \, \frac{kg}{m^{3}} \times 9.81 \, \frac{m}{s^2}} \approx 8.48 \, m \) The height of the 1-iodododecane barometer column is approximately 8.48 meters.

b) Pressure on the diver's body

To find the pressure on the diver, we need to find the hydrostatic pressure at the given depth and add it to the atmospheric pressure. 1. Convert depth to meters: 21 ft × 0.3048 = 6.4 m 2. Density of water: 1000 kg/m³ 3. Convert atmospheric pressure to pascals: \(742 \, \frac{torr}{atm} \times \frac{101325 \, Pa}{760 \, torr} = 99834.08 \, Pa\) Hydrostatic pressure at 6.4 meters depth: \( Pressure_{water} = Density_{water} \times Gravity \times Depth = 1000 \, \frac{kg}{m^{3}} \times 9.81 \frac{m}{s^2} \times 6.4 \, m = 62825.6 \, Pa \) Total pressure on the diver's body: \( Pressure_{total} = Pressure_{water} + Pressure_{atmospheric} = 62825.6 \, Pa + 99834.08 \, Pa = 162659.68 \, Pa \) Convert the total pressure to atmospheres: \( Pressure_{total} = \frac{162659.68 \, Pa}{101325 \, \frac{Pa}{atm}} \approx 1.6 \, atm \) The pressure on the diver's body is approximately 1.6 atmospheres.

Key concepts

Barometer

A barometer is an instrument used to measure atmospheric pressure. It functions on the principle that the pressure exerted by the atmosphere can support a column of liquid, such as mercury or water. In the case of a traditional mercury barometer, the height of the mercury column changes as the atmospheric pressure varies. If we use a different liquid with a lower density, like 1-iodododecane, the column will be higher for the same atmospheric pressure, since it takes more volume of a lighter liquid to exert the same pressure.

As noted in our example, to determine the height of a barometer using 1-iodododecane, we use the hydrostatic pressure calculation but adjusted for the specific density of the liquid in question. The height is found to be around 8.48 meters, which is significantly taller than a mercury barometer's height because mercury has a much higher density. Understanding this helps clarify why mercury is commonly used—it provides a more practical column height for measurement.

Density Conversions

Density conversions are critical when working with formulas in physics and engineering that require consistent units. In our example, the density of 1-iodododecane needs to be converted from grams per milliliter (g/mL) to kilograms per cubic meter (kg/m³) to use it in hydrostatic pressure equations. The conversion is done by multiplying the given density by 1000 to convert grams to kilograms and by dividing by 0.001 to convert milliliters to cubic meters.

It's important to remember that density is a measure of how much mass is contained in a given volume. When converting units, maintaining the correct mass-to-volume ratio is essential. This allows us to accurately relate density to the hydrostatic pressure formula: Pressure = Density × Gravity × Height. Being comfortable with density conversions allows for accurate and effective problem-solving across various applications.

Pressure Calculations

Pressure calculations are a fundamental aspect of fluid mechanics, and understanding them is key when studying hydrostatics and barometry. As seen in our example, we need to convert the atmospheric pressure from torr to pascals, a conversion necessary because the standard formulas for pressure are typically expressed in the SI unit of pascals (Pa).

The formula used for pressure calculation is simple: Pressure = Density × Gravity × Height. This formula tells us that pressure is proportional to the density of the fluid, the acceleration due to gravity, and the height of the fluid column. In both parts of our example, this formula allows us to calculate the height of a barometer column with a given density of liquid under atmospheric pressure and the pressure exerted on a diver under water.

Buoyancy and Diving

Buoyancy affects divers significantly—it's the upward force exerted by a fluid that opposes the weight of an object submerged in it. When diving, a person experiences both the atmospheric pressure and the pressure from the water above them. The deeper a diver goes, the greater the water pressure, adding to the total pressure they experience. This relates directly to the hydrostatic pressure concept where the diver's pressure is the sum of atmospheric pressure and hydrostatic pressure due to the water's density.

Diving also reminds us of Archimedes' principle, which states that the buoyant force on a submerged object is equal to the weight of the fluid it displaces. This principle helps explain how divers can control their descent and ascent by adjusting their buoyancy—with the aid of buoyancy control devices, divers manipulate their volume and therefore the amount of water displaced, allowing them to hover, sink, or rise in the water. Properly understanding buoyancy is crucial for safety in diving.