Calculate the height of the atmosphere at the point where the pressure is one tenth of atmospheric pressure. Assume a constant temperature of 491.76 °R and assume that the ideal gas law applies. For atmospheric pressure on the surface of the earth, use 14.7 psia. Show your answer in miles. Note that you need the molecular weight of air which can be found in Page 40 of Himmelblau and Riggs (29g⁄(g mole)). In order for your units to be consistent, you need a conversion factor (gc) used in the American Engineering System which can be found in Page 21 of Himmelblau and Riggs

Question

Calculate the height of the atmosphere at the point where the pressure is one tenth of atmospheric pressure. Assume a constant temperature of 491.76 °R and assume that the ideal gas law applies. For atmospheric pressure on the surface of the earth, use 14.7 psia. Show your answer in miles. Note that you need the molecular weight of air which can be found in Page 40 of Himmelblau and Riggs (29g⁄(g mole)). In order for your units to be consistent, you need a conversion factor (gc) used in the American Engineering System which can be found in Page 21 of Himmelblau and Riggs

Answer 1

S Calculate the height of the atmosphere at the point where the pressure is one tenth of atmospheric pressure. Assume a constant temperature of 491.76 °R and assume that the ideal gas law applies. For atmospheric pressure on the surface of the earth, use 14.7 psia. Show your answer in miles. Note that you need the molecular weight of air which can be found in Page 40 of Himmelblau and Riggs (29g⁄(g mole)). In order for your units to be consistent, you need a conversion factor (gc) used in the American Engineering System which can be found in Page 21 of Himmelblau and Riggs.

Answer 2

To calculate the height of the atmosphere at the point where the pressure is one tenth of atmospheric pressure, we can use the ideal gas law equation:

PV = nRT

Where:
P = pressure
V = volume
n = number of moles of gas
R = gas constant
T = temperature

To solve for the height, we need to find the change in pressure which is equal to one tenth of the atmospheric pressure. Let's denote this as P1.

P1 = 14.7 psia / 10 = 1.47 psia

Now, let's convert the pressure from psia to Pascals. 1 psia is equal to 6894.76 Pascals.

P1 = 1.47 psia * 6894.76 Pa/psia ≈ 10150.794 Pa

Next, we need to find the molecular weight of air. According to the given information, we can find it on Page 40 of Himmelblau and Riggs, which is 29g/(g mole). Let's denote this as MW.

MW = 29 g/(g mole)

Now, we need to convert the temperature from Rankine (°R) to Kelvin (K). The conversion factor is 5/9.

T = 491.76 °R * 5/9 ≈ 273.2 K

Next, we can calculate the number of moles of gas (n) using the ideal gas law equation:

n = P1 * V / (R * T)

Since we are trying to find the height and the volume (V) changes with height, we can assume that the volume remains constant to simplify the calculation.

Now, we can rearrange the equation to solve for V:

V = (n * R * T) / P1

To find the height, we need to divide V by the cross-section area of the atmosphere. Assuming a spherical shape, the cross-section area can be calculated as:

A = 4πr^2

Let's denote the radius of the Earth as R_e and the height (h) as the variable we are solving for.

We can calculate the volume (V) as:

V = h * A

Now, combining the equations:

h * A = (n * R * T) / P1

h * 4πR_e^2 = (n * R * T) / P1

h = [(n * R * T) / P1] / (4πR_e^2)

Now, let's substitute the values into the equation:

h = [(n * R * T) / P1] / (4πR_e^2)

h = [(n * R * T) / P1] / (4π * (3959 miles * 5280 feet/mile)^2)

Now, let's plug in the values for n, R, T, P1, and R_e:

h = [(n * R * T) / P1] / (4π * (3959 * 5280)^2)

h = [(n * R * T) / P1] / (4π * (3959 * 5280)^2)

h = [(1 mole * 8.314 J/(mol·K) * 273.2 K) / 10150.794 Pa] / (4π * (3959 * 5280)^2)

Now, let's simplify the equation:

h = [(8.314 J/(mol·K) * 273.2 K) / 10150.794 Pa] / (4π * (3959 * 5280)^2)

h = (2266.1548 J / 10150.794 Pa) / (4π * (3959 * 5280)^2)

h ≈ 6.04 x 10^-10 J/Pa / (4π * (3959 * 5280)^2)

Now, let's calculate this value:

h ≈ 6.04 x 10^-10 J/Pa / (4π * (3959 * 5280)^2)

h ≈ 6.04 x 10^-10 / (4π * 3.30626 x 10^10)

Now, divide the number:

h ≈ 1.46 x 10^-21 J/(4π * 3.30626 x 10^10)

Finally, divide this number by gc to convert the units to miles. According to the given information, the value of gc can be found on Page 21 of Himmelblau and Riggs. Let's denote this as gc.

gc = gc (conversion factor)

h ≈ (1.46 x 10^-21 J/(4π * 3.30626 x 10^10)) / gc

Now, using the specific value of gc from the provided source, you can calculate the final result.

Please consult the specific values and formulas mentioned in the Himmelblau and Riggs book to obtain accurate values for gc and perform the final conversion to miles.