Question

A barometer having a cross-sectional area of \(1.00 \mathrm{~cm}^{2}\) at sea level measures a pressure of \(76.0 \mathrm{~cm}\) of mercury. The pressure exerted by this column of mercury is equal to the pressure exerted by all the air on \(1 \mathrm{~cm}^{2}\) of Earth's surface. Given that the density of mercury is \(13.6 \mathrm{~g} / \mathrm{mL},\) and the average radius of Earth is \(6371 \mathrm{~km}\), calculate the total mass of Earth's atmosphere in kilograms.

Short answer

The mass of Earth's atmosphere is approximately \(5.15 \times 10^{18} \mathrm{~kg}\).

Step-by-step solution

Determine the mass of the mercury column

The volume \( V \) of the mercury column can be calculated by the formula \( V = A \times h \) where \( A = 1 \mathrm{~cm}^{2} \) is the cross-sectional area and \( h = 76.0 \mathrm{~cm} \) is the height. Convert everything to mks units to have \( V = 1 \times 10^{-4} \mathrm{~m}^{2} \times 76.0 \times 10^{-2} \mathrm{~m} = 7.6 \times 10^{-6} \mathrm{~m}^{3} \). The mass \( m \) of this volume of mercury is calculated by the formula \( m = \rho \times V \) where \( \rho = 13.6 \mathrm{~g} / \mathrm{mL} = 13.6 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3} \) is the density. Therefore, \( m = 13.6 \times 10^{3} \mathrm{~kg/m}^{3} \times 7.6 \times 10^{-6} \mathrm{~m}^{3} = 0.103 kg \).

Calculate the Earth's total surface area

The total surface area \(A_e\) of the Earth can be calculated using the formula for the surface area of a sphere: \( A_e = 4\pi r^{2} \) where \( r = 6371 \mathrm{~km} = 6.371 \times 10^{6} \mathrm{~m} \) is the Earth's radius. Thus, \( A_e = 4\pi (6.371 \times 10^{6} \mathrm{~m})^{2} = 5.1 \times 10^{14} \mathrm{~m}^{2} \).

Calculate total mass of Earth's atmosphere

The mass of the Earth's atmosphere is given by the total force exerted by the atmospheric pressure divided by the acceleration due to gravity. However, each square meter of Earth’s surface is subject to the same force as that on our 1 centimeter square of mercury at sea level. Consequently, the total mass \( M \) of Earth’s atmosphere will be the mass of mercury times the total surface area of the Earth divided by the area of our mercury column. In this case, \( M = 0.103 \mathrm{~kg} \times 5.1 \times 10^{14}\mathrm{~m}^{2} / 1 \times 10^{-4} \mathrm{~m}^{2} = 5.15 \times 10^{18} \mathrm{~kg} \).

Key concepts

Barometric Pressure

Understanding barometric pressure is crucial when examining the Earth's atmosphere. Barometric pressure, also commonly called atmospheric pressure, is the force per unit area exerted by the weight of the air in the Earth's atmosphere. As you rise above sea level, this pressure decreases because the atmosphere becomes less dense.

At sea level, standard barometric pressure is often measured using a mercury barometer, in which the weight of the mercury column provides a direct readout of air pressure. This unit is typically expressed in millimeters or inches of mercury (mmHg or inHg). In our exercise, the pressure exerted by a 76.0 cm column of mercury at sea level is equal to the atmospheric pressure on 1 cm² of the Earth's surface. This classic experiment, originating from the work of Torricelli in the 1640s, gives us an elegant and historical method for visually demonstrating atmospheric pressure.

Mercury Density

Mercury is a unique element that plays a key role in measuring barometric pressure. It has a high density, which means that a relatively short column of mercury can balance the atmospheric pressure. This is due to its density of 13.6 g/mL, which is also expressed as 13.6 x 10³ kg/m³ when converted to the SI unit of kg/m³. Because of this high density, mercury is used in barometers as it provides a compact and accurate measure.

Significance of Mercury's Density in Calculations

In our textbook exercise, we used the known density of mercury to calculate the mass of the mercury in a barometer's column. Knowing the column's cross-sectional area and height allowed us to determine its volume, and then apply the formula m = ρ x V, where m is mass, ρ is density, and V is volume, to find the mass. That mass, in turn, was crucial to calculating the total mass of Earth's atmosphere.

Earth's Surface Area

The Earth's surface area is an immense topic not only in geography but also in calculations related to atmospheric studies. It's simply the total area covered by the surface of our planet. Our Earth is a sphere, and the formula we use to calculate its surface area is A_e = 4πr², where r is the radius of the Earth.

In the step by step solution presented, we used Earth’s average radius of approximately 6,371 kilometers, leading to a calculated surface area over 510 trillion square meters (5.1 x 10¹⁴ m²). This vast surface area when multiplied by the atmospheric pressure (as represented by the mass of the column of mercury in a barometer at sea level), gives us the ability to estimate the total mass of Earth's atmosphere.

Atmospheric Pressure

Atmospheric pressure is an essential concept in understanding not just weather, but also the principles that govern fluid mechanics and various scientific calculations. It's the pressure exerted by the weight of the atmosphere, which is composed of a mixture of gases. Standard atmospheric pressure at sea level is approximately 101,325 Pascals, which also equates to 760 mm Hg (millimeters of mercury) in a mercury barometer.

In our problem, by comparing the mass of the mercury column to the total mass of Earth's atmosphere, we are essentially scaling up the pressure measurement from a small, manageable size to the entirety of Earth's surface area. This correlation between a small sample area and the total surface area is what allows the mass and pressure to be extrapolated to find the zenith of our quest—the total mass of the Earth's atmosphere.