Question

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?

Short answer

The height of the barometer column based on 1-iodododecane when the atmospheric pressure is 749 torr is approximately 8.52 meters.

Step-by-step solution

Convert atmospheric pressure to pascals

To use the formula for barometric pressure, we need to first convert the given atmospheric pressure of 749 torr to pascals (Pa). Using the conversion factor 1 torr = 133.322 Pa, we can compute the atmospheric pressure in pascals: Atmospheric pressure (Pa) = 749 torr * (133.322 Pa / 1 torr) = 99852.6 Pa

Calculate the height of the mercury column

We can calculate the height of the mercury column, h_Hg, using the barometric pressure formula: Atmospheric pressure = density_Hg * h_Hg * g where density_Hg is the density of mercury (13.6 g/mL), g is the acceleration due to gravity (9.81 m/s^2), and h_Hg is the height of the mercury column. We need to convert the density from g/mL to kg/m^3 as follows: density_Hg (kg/m^3) = 13.6 g/mL * (1 kg / 1000 g) * (1000^3 mL / 1^3 m^3) = 13600 kg/m^3 Now, we can solve for h_Hg: h_Hg = Atmospheric pressure / (density_Hg * g) = 99852.6 Pa / (13600 kg/m^3 * 9.81 m/s^2) = 0.755 m

Calculate the height of the 1-iodododecane column

The ratio of the heights of the 1-iodododecane and mercury columns is the reciprocal of the ratio of their densities. We'll first convert the density of 1-iodododecane to kg/m^3: density_1iodododecane (kg/m^3) = 1.20 g/mL * (1 kg / 1000 g) * (1000^3 mL / 1^3 m^3) = 1200 kg/m^3 Now we can compute the height of the 1-iodododecane column, h_1iodododecane, using the ratio of the heights: h_1iodododecane / h_Hg = density_Hg / density_1iodododecane h_1iodododecane = h_Hg * (density_Hg / density_1iodododecane) = 0.755 m * (13600 kg/m^3 / 1200 kg/m^3) = 8.52 m The height of the barometer column based on 1-iodododecane when the atmospheric pressure is 749 torr is approximately 8.52 meters.

Key concepts

Atmospheric Pressure

Atmospheric pressure is the force per unit area exerted against a surface by the weight of the air above that surface in the atmosphere of Earth. It's a critical factor in various scientific fields, including meteorology and aviation, and is typically measured in units such as pascals (Pa) or torr. At sea level, standard atmospheric pressure is 101,325 Pa, also known as one atmosphere (1 atm).

This concept is closely related to barometric measurements, where a barometer reads the atmospheric pressure and allows for predictions of weather changes. In essence, it's the weight of a column of air from the top of the atmosphere to the surface. Thus, understanding the pressure exerted by the atmosphere is essential for various calculations in physics and engineering, including the determination of barometer height.

Density Conversion

Density is the mass per unit volume of a substance. Often density is expressed in units such as grams per milliliter (g/mL) or kilograms per cubic meter (kg/m³). Converting density units is crucial when working with equations that require consistent units. For example, in the calculation of barometric pressure, we convert mercury's density from g/mL to kg/m³ to align with the standard units in the barometric formula.

Understanding how to perform these conversions is vital as it ensures accuracy in scientific calculations. The conversion factor comes from the basic relationship where 1 g/mL equals 1000 kg/m³, considering that 1 kg is 1000 g and 1 mL is 1 cubic centimeter (cc), which is a thousandth of a cubic meter.

Acceleration due to Gravity

The acceleration due to gravity, denoted as 'g', is the acceleration that the Earth imparts to objects on its surface due to gravitational force. Standard gravity is approximately 9.81 m/s². This value is a vector, meaning it has both magnitude and direction, which points towards the center of the Earth.

Gravity's acceleration plays a vital role in barometric pressure calculations, as it affects the height of the fluid column in the barometer. It's one of the constants in the barometric formula used to calculate the height of a liquid in response to atmospheric pressure. Understanding this acceleration is essential not only in atmospheric pressure calculations but also in nearly all aspects of classical mechanics.

Barometer Height Determination

A barometer is an instrument that measures atmospheric pressure. The height of the barometer's fluid column, often mercury or water, is inversely proportional to atmospheric pressure. A higher barometric height indicates lower atmospheric pressure and vice versa.

To determine the height of a barometer column, we use the formula that relates atmospheric pressure to the density of the fluid, the height of the column (h), and the acceleration due to gravity (g). In specific calculations such as the provided exercise, the height of a barometer column using a fluid different from mercury can be predicted by understanding the principles of density and pressure. The calculation involves using the mercury column height determined by standard barometric pressure and then adjusting it based on the density of the new fluid (1-iodododecane in this case).