Finding the mass of the earth's atmosphere

[deltabourne]

A barometer having a cross-sectional area of 1.00 cm^2 at sea level measures a pressure of 76.0 cm of mercury. The pressure exerted by this column of mercury is equal to the pressure exerted by all the air on 1 cm^2 of Earth's surface. Given that the density of mercury is 13.6 g/mL and the average radius of Earth is 6371 km, calculate the total mass of Earth's atmosphere in kilograms (Hint: The surface area of a sphere is 4pir^2)

I have no idea where to start on this one, and how to relate the mercury to find the mass of the atmosphere.. can anyone point me in the right direction?

 

[quantumdude]

I don't think you even need the density of mercury. You know the pressure and the area over which the pressure acts. Since P=F/A, you can determine the force that exerts the pressure (which is the weight of the air above the barometer). That gives you the mass of a column of air with a cross section of 1cm². To get the mass of the entire atmosphere, you need to use the formula for the area of a sphere that was given.

 

[M98Ranger]

To find the mass of the Earth's atmosphere, we can use the given information about the pressure exerted by a column of mercury and the density of mercury. The key concept to understand here is that the pressure exerted by the column of mercury is equal to the pressure exerted by the entire atmosphere on 1 cm^2 of Earth's surface. This means that we can use the pressure and density of mercury to calculate the pressure and density of the atmosphere.

First, let's convert the pressure measurement from centimeters of mercury to pascals (Pa). 1 cm of mercury is equal to 1333.22 Pa, so the pressure exerted by the atmosphere is 76.0 x 1333.22 = 101319.72 Pa.

Next, we can use the ideal gas law (PV = nRT) to calculate the density of the atmosphere. We know the pressure (101319.72 Pa) and the temperature (assumed to be 15°C or 288.15 K), and we can assume a volume of 1 cm^3 since we are working with 1 cm^2 of Earth's surface. Solving for density (n/V), we get 0.0012 mol/cm^3.

Now, we can use the density of mercury (13.6 g/mL) to find the mass of the atmosphere per unit volume. 0.0012 mol/cm^3 of air is equivalent to 0.0012 mol/cm^3 x 28.97 g/mol = 0.0347 g/cm^3. This means that for every cubic centimeter of Earth's surface, there is 0.0347 g of air.

To find the total mass of the atmosphere, we can use the surface area of a sphere (4πr^2) to calculate the total surface area of the Earth. Plugging in the average radius of the Earth (6371 km or 6,371,000,000 cm), we get a surface area of 5.1 x 10^14 cm^2.

Finally, we can multiply the mass per unit volume (0.0347 g/cm^3) by the total surface area (5.1 x 10^14 cm^2) to find the total mass of the atmosphere in grams. This comes out to be approximately 1.8 x 10^16 g.

To convert this to kilograms, we divide by