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Chapter 5: Problem 106

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. (Hint: The surface area of a sphere is \(4 \pi r^{2},\) where \(r\) is the radius of the sphere.)

Short Answer

Expert verified
To calculate the total mass of Earth's atmosphere, you first calculate the weight of a column of mercury that the atmospheric pressure can support at sea level, then use this to find the atmospheric pressure. Next, calculate the Earth's total surface area and use this along with the pressure to find the total weight of the atmosphere. Finally, divide this weight by the gravitational acceleration to find the total mass of Earth's atmosphere.

Step by step solution

01

Calculate the weight of mercury column

The height of the mercury column that the atmospheric pressure can support is given as 76.0 cm. The weight of this column is equal to the volume multiplied by density and gravity. Since volume = area * height and area is given as \(1 cm^2\), you can compute: \n\nWeight = \( (1 cm^2 * 76.0 cm) * (13.6 g / cm^3) * (9.81 m / s^2) \) \n\nConvert each quantity to its equivalent in SI units before you do the calculation.
02

Derive atmospheric pressure

The weight calculated above is exerted on an area of 1 cm². So, use force divided by area to determine the pressure that this weight represents. The pressure represents the atmospheric pressure at sea level. \n\n Pressure = Weight / Area.
03

Find the Earth's surface area

You have to use the formula that gives the surface area of a sphere: \(4 \pi r^{2}\), where r is the radius (6371 km), but you need to convert it to meters before estimating the area.
04

Estimating the total weight of the atmosphere

Since the atmospheric pressure is the force applied per unit area, you can then estimate the total force (or weight - mass times gravity) that the atmosphere exerts on the entire surface of the Earth by multiplying the pressure obtained in step 2 by the area estimated in step 3.\n\n Total weight = Pressure * total area.
05

Calculate the mass of the Earth's atmosphere.

Finally, since weight equals mass times gravity, you can use this relationship to find the total mass of the atmosphere by dividing the total weight of the atmosphere by gravitational acceleration. Remember that weight is a force measured in Newtons, and gravity is approximately 9.81 m/s².\n\n Mass = Total Weight / Gravity.

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Most popular questions from this chapter

In 2012 , Felix Baumgartner jumped from a balloon roughly \(24 \mathrm{mi}\) above Earth, breaking the record for the highest skydive. He reached speeds of more than 700 miles per hour and became the first skydiver to exceed the speed of sound during free fall. The helium-filled plastic balloon used to carry Baumgartner to the edge of space was designed to expand to \(8.5 \times 10^{8} \mathrm{~L}\) in order to accommodate the low pressures at the altitude required to break the record. (a) Calculate the mass of helium in the balloon from the conditions at the time of the jump \((8.5 \times\) \(\left.10^{8} \mathrm{~L},-67.8^{\circ} \mathrm{C}, 0.027 \mathrm{mmHg}\right) .\) (b) Determine the volume of the helium in the balloon just before it was released, assuming a pressure of 1.0 atm and a temperature of \(23^{\circ} \mathrm{C}\).

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