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? \((\mathbf{b})\) What is the pressure, in atmospheres, on the body of a diver if he is \(21 \mathrm{ft}\) below the surface of the water when the atmospheric pressure is 742 torr?

Short answer

The height of a 1-iodododecane barometer column at atmospheric pressure of 749 torr is approximately \(8.48 \: \text{m}\). The pressure on a diver at 21 ft below the surface of the water when the atmospheric pressure is 742 torr is approximately \(1.596 \: \text{atm}\).

Step-by-step solution

(Part A: Convert atmospheric pressure to pascal)

: First, convert the given atmospheric pressure from torr to pascal using the conversion factor. \( P_\text{atm} = 749 \: \text{torr} \cdot \frac{133.33 \: \text{Pa}}{1 \: \text{torr}} = 99892.17 \: \text{Pa} \)

(Part A: Calculate the height of the 1-iodododecane column)

: Next, use the formula \(P = \rho g h\) to find the height (h) of the 1-iodododecane column. Rearranging for h, we get \( h = \frac{P_\text{atm}}{\rho_\text{ido} g} \) Density of 1-iodododecane, \(\rho_\text{ido} = 1.20 \: \text{g/mL} = 1200 \: \text{kg/m}^3\) Substitute the values: \(h = \frac{99892.17 \: \text{Pa}}{(1200 \: \text{kg/m}^3)(9.81 \: \text{m/s}^2)} = 8.48 \: \text{m}\) So, the height of the 1-iodododecane barometer column is \(8.48 \: \text{m}\).

(Part B: Convert feet to meters and torr to pascal)

: First, convert 21 feet to meters and atmospheric pressure to pascal. Depth of the diver, \(h = 21 \: \text{ft} \cdot \frac{1 \: \text{m}}{3.2808 \: \text{ft}} = 6.400 \: \text{m}\) Atmospheric pressure, \(P_\text{atm} = 742 \: \text{torr} \cdot \frac{133.33 \: \text{Pa}}{1 \: \text{torr}} = 98910.26 \: \text{Pa}\)

(Part B: Calculate the pressure exerted by the water column)

: Use the formula \(P = \rho_\text{w} g h\) to find the additional pressure exerted by the water column on the diver. Density of water, \(\rho_\text{w} = 1000 \: \text{kg/m}^3\) Pressure due to water, \(P_\text{w} = (1000 \: \text{kg/m}^3)(9.81 \: \text{m/s}^2)(6.400 \: \text{m}) = 62844 \: \text{Pa}\)

(Part B: Calculate the total pressure on the diver in atmospheres)

: Add the atmospheric and water pressure, then convert the total pressure to atmospheres. Total pressure, \(P_\text{total} = P_\text{atm} + P_\text{w} = 98910.26 \: \text{Pa} + 62844 \: \text{Pa} = 161754.26 \: \text{Pa}\) Convert to atmospheres, \(\frac{1 \: \text{atm}}{101325 \: \text{Pa}} \cdot P_\text{total} = \frac{161754.26 \: \text{Pa}}{101325 \: \text{Pa}} = 1.596 \: \text{atm}\) The pressure on the diver is \(1.596 \: \text{atm}\).

Key concepts

Barometer column height

One critical aspect of understanding barometers is determining the height of a column of fluid in the barometer, which is directly related to atmospheric pressure. The basic principle is that the weight of the column of liquid is balanced by the atmospheric pressure pushing on it from outside. This is why a denser liquid like mercury is typically used, due to its higher density allowing for manageable column heights.

For any liquid in a barometer, the equilibrium condition can be expressed as the hydrostatic pressure equation: \[ P = \rho g h \]where:

• \( P \) is the atmospheric pressure,
• \( \rho \) is the density of the liquid,
• \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2\)),
• \( h \) is the height of the liquid column.

In a typical scenario with 1-iodododecane as the barometer fluid, a given atmospheric pressure will support a liquid column height that can be calculated by rearranging the formula:\[ h = \frac{P}{\rho g} \]By substituting the known values—density of 1-iodododecane and atmospheric pressure—one can deduce the height of the column, which is an essential step in understanding how barometers work.

Atmospheric pressure conversion

Converting atmospheric pressure from one unit to another is a crucial practice in ensuring the accuracy of scientific calculations and experiments. Atmospheric pressure is commonly measured in several units, including torr, pascals (Pa), and atmospheres (atm), each suited to different scientific contexts.

Conversion is typically performed using standard conversion factors. For example, when converting from torr to pascals, the conversion factor is:1 torr \( = 133.33 \text{ Pa} \)
For pressures commonly measured in weather or laboratory contexts (expressed in atmospheres), it's essential to remember that:1 atm \( = 101325 \text{ Pa} \)

• Convert 749 torr to pascals: \(749 \, \text{torr} \cdot 133.33 \frac{\text{Pa}}{\text{torr}} \approx 99892.17 \, \text{Pa}\)
• Understanding these conversions is important for working accurately across different scales and systems.

Converting between these units allows scientists and engineers to quickly change the perspective of their measurements to suit different applications, whether it be a laboratory setup, atmospheric science, or engineering challenges.

Density and pressure relationship

Understanding the density and pressure relationship is vital when discussing fluids and barometric science. Density, defined as mass per unit volume, influences how pressure is exerted by a liquid column. Higher density liquids exert more pressure over the same height than less dense liquids, which is why mercury is such a favored choice in barometers due to its high density.

To explore this relationship, consider the formula used to determine the barometer column height:\[ P = \rho g h \]For a constant atmospheric pressure, the height of the liquid column, \( h \), varies inversely with the density \( \rho \), meaning:- A liquid with a higher density will have a shorter column.- Conversely, a lower density liquid will require a taller column to balance the same air pressure.
This relationship underscores the importance of selecting the appropriate liquid for barometers depending on context and measurement needs. For instance, with 1-iodododecane having a density of \( 1.20 \, \text{g/mL} \), larger column heights are necessary, as seen in the exercise, compared to denser mercury. Understanding these principles ensures accurate measurements and comprehension of fluid dynamics in various scientific and practical applications.