if 100ft^3 of atmospheric air at zero fahrenheit temperature are compressed to a volume of 1 ft^3 at a temperature of 200 degree fahrenheit, what will be the pressure of the air in psi?
2109 psia
Well, I'm glad you didn't ask me to do the math because I'm not really good at it. But here's a joke to lighten the mood:
Why don't scientists trust atoms?
Because they make up everything!
Anyway, let's try to answer your question. To calculate the pressure of the air, we need to use the ideal gas law. The ideal gas law is a mathematical relationship between pressure, volume, and temperature. It can be written as:
PV = nRT
Where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin.
Now, let's convert the temperatures from Fahrenheit to Kelvin. I hope you know the conversion formula because I sure don't!
*finger guns*
To find the pressure of the air in psi, we can use Boyle's Law, which states that the pressure of a gas is inversely proportional to its volume when temperature is constant.
Boyle's Law can be expressed as: P1 * V1 = P2 * V2
Given:
V1 = 100 ft^3
T1 = 0 degrees Fahrenheit
V2 = 1 ft^3
T2 = 200 degrees Fahrenheit
First, let's convert the temperatures from Fahrenheit to Rankine since the equation requires the temperature to be in absolute units.
T1 (in Rankine) = (T1 in Fahrenheit) + 460
T1 = 0 + 460 = 460 Rankine
T2 (in Rankine) = (T2 in Fahrenheit) + 460
T2 = 200 + 460 = 660 Rankine
Now, we can plug the values into Boyle's Law and solve for P2:
P1 * V1 = P2 * V2
P2 = (P1 * V1) / V2
Plugging in the given values:
P2 = (P1 * V1) / V2
P2 = (P1 * 100 ft^3) / 1 ft^3
P2 = P1 * 100
To find P2 in psi, we need to convert the units from ft^3 to psi:
1 psi = 144 in^2
1 ft^3 = 12^3 in^3 = 1728 in^3
P2 (in psi) = (P2 (in ft^3) * 1728 in^3) / 144 in^2
Substituting P2 = P1 * 100, we get:
P2 (in psi) = (P1 * 100 * 1728 in^3) / 144 in^2
P2 (in psi) = (P1 * 1200 in^3) / 144 in^2
P2 (in psi) = (P1 * 8.33333)
Therefore, the pressure of the air in psi will be 8.33333 times the initial pressure (P1).
To find the pressure of the air in psi, we can use the ideal gas law. The ideal gas law states that the product of pressure, volume, and temperature of a gas is equal to the product of the gas constant (R) and the number of moles of the gas (n).
The formula for the ideal gas law is:
PV = nRT
Where:
P = Pressure
V = Volume
n = Number of moles
R = Gas constant
T = Temperature
First, let's convert the temperatures from Fahrenheit to Rankine since the ideal gas law requires temperature in Kelvin or Rankine. The conversion between Fahrenheit and Rankine is: Rankine = Fahrenheit + 459.67
So, the initial temperature of 0 degrees Fahrenheit is equivalent to (0 + 459.67) = 459.67 Rankine and the final temperature of 200 degrees Fahrenheit is equivalent to (200 + 459.67) = 659.67 Rankine.
Now, let's calculate the number of moles using the given conditions. To do this, we need to assume that the air is an ideal gas and use the equation:
PV = nRT
Rearranging the equation, we can solve for n:
n = PV / RT
Given:
Initial volume (V1) = 100 ft^3
Initial temperature (T1) = 459.67 Rankine
Final volume (V2) = 1 ft^3
Final temperature (T2) = 659.67 Rankine
We also need the gas constant (R), which is 8.314 J/(mol·K) or 0.7302 ft^3·psi/(lbmol·Rankine).
Let's substitute the values into the equation to find n:
n = (P1 * V1) / (R * T1)
Since the gas is compressed and the volume is reduced, we can assume that the number of moles remains constant (n1 = n2), so we can also write the equation as:
n = (P2 * V2) / (R * T2)
Let's substitute the known values and solve for P2:
P2 = (n * R * T2) / V2
Now we need to calculate the number of moles, n. The molar mass of air is approximately 28.97 g/mol. To convert from mass to moles, we need to use the equation:
n = mass / molar mass
Given:
Mass of air (m) = 100 ft^3 (since air is mostly nitrogen, which has a density of approximately 0.0765 lb/ft^3)
Now, let's calculate n:
n = (100 ft^3 * 0.0765 lb/ft^3) / (28.97 g/mol * 0.0022046 lb/g)
Substituting the known values, we find:
n ≈ 120.377 lbmol
Now we can substitute the values of n, R, T2, and V2 into the equation for P2:
P2 = (120.377 lbmol * 0.7302 ft^3·psi/(lbmol·Rankine) * 659.67 Rankine) / 1 ft^3
Simplifying the equation:
P2 ≈ 59,500 psi
Therefore, the pressure of the air will be approximately 59,500 psi.
P1 * V1 = P2 * V2
Where P1 and V1 are the initial pressure and volume, P2 and V2 are the final pressure and volume.
1 atm * 100 ft^3 = P2 * 1 ft ^3
P2 = 100 atm; use your book or the internet to find a conversion factor from atm to psi