A. ) A goodyear blimp typically contains 5400 m cubed of helium at an absolute pressure of 1.1*10^5 Pa. The temperature of the helium is 280 K. What is the mass (in kg) of the helium of the blimp?
B. ) Estimate the spacing between the centers of neighboring atoms in a piece of solid aluminum, based on knowledge of the density (2700 kg/m^3) and atommic mass (26.9815 u) of aluminum.
A. To find the mass of the helium in the blimp, we can use the ideal gas law equation. The ideal gas law equation states:
PV = nRT
Where:
P = pressure of the gas (in Pa)
V = volume of the gas (in m^3)
n = number of moles of gas
R = ideal gas constant (8.314 J/(mol·K))
T = temperature of the gas (in K)
First, we need to convert the pressure from Pa to atm. 1 atm is equal to 1.01325 × 10^5 Pa, so 1.1 × 10^5 Pa = 1.08613 atm.
Next, we need to calculate the number of moles of helium using the ideal gas law equation. Rearranging the equation, we get:
n = PV / RT
Substituting the values into the equation:
n = (1.08613 atm) * (5400 m^3) / (8.314 J/(mol·K)) * (280 K)
Simplifying the equation:
n ≈ 278.35 mol
Next, we need to convert moles to mass by multiplying the number of moles by the molar mass of helium. The molar mass of helium is approximately 4.0026 g/mol (grams per mole).
Mass of helium = n * molar mass of helium
Mass of helium = 278.35 mol * 4.0026 g/mol
Finally, convert grams to kilograms:
Mass of helium ≈ 1113.4 kg
Therefore, the mass of the helium in the blimp is approximately 1113.4 kg.
B. To estimate the spacing between the centers of neighboring aluminum atoms, we can use the relationship between density, atomic mass, and molar volume.
The volume of one mole of any substance is known as its molar volume. For aluminum, the molar mass is 26.9815 g/mol.
First, we need to convert the molar mass from grams to kilograms:
Molar mass of aluminum = 26.9815 g/mol = 0.0269815 kg/mol
Next, we can calculate the molar volume of aluminum using the known density and molar mass:
Molar volume = (mass of one mole of aluminum) / (density of aluminum)
Molar volume = 0.0269815 kg/mol / 2700 kg/m^3
Simplifying the equation:
Molar volume ≈ 1.00 × 10^(-8) m^3/mol
The molar volume represents the volume occupied by one mole of atoms. Since aluminum has a face-centered cubic crystal structure, each atom is surrounded by 12 neighboring atoms. Thus, the molar volume is also the volume per aluminum atom.
To find the spacing between the centers of neighboring aluminum atoms, we can calculate the cube root of the molar volume:
Spacing = (Molar volume)^(1/3)
Spacing = (1.00 × 10^(-8) m^3/mol)^(1/3)
Spacing ≈ 2.15 × 10^(-3) m
Therefore, the estimated spacing between the centers of neighboring aluminum atoms is approximately 2.15 × 10^(-3) m.