From the ideal gas law, calculate the volume per helium atom in a litre of helium gas kept
at room temperature and atmospheric pressure.
Additionally, calculate the length of the side of a cube with this volume.
22.4 L/mole ... 1/22.4 mole/L
number of atoms ... 6.02E23 / 22.4
1 L = 1E3 cm^3
To calculate the volume per helium atom in a liter of helium gas, we need to use the ideal gas law equation. The ideal gas law is expressed as:
PV = nRT
Where:
P = Pressure
V = Volume
n = Number of moles of gas
R = Ideal gas constant
T = Temperature
Given that we have helium gas at room temperature and atmospheric pressure, we can assume:
P = 1 atm (atmospheric pressure is roughly 1 atm)
T = room temperature (around 298 K)
We also need to find the number of moles of helium gas. To do this, we can use Avogadro's number, which tells us the number of atoms or molecules in a mole. Avogadro's number is approximately 6.022 x 10^23 mol^-1.
Since we want to find the volume per helium atom, we need to determine the volume occupied by a single helium atom.
The formula for the volume of a cube is:
V = s^3
Where:
V = Volume
s = Length of one side of the cube
To find the length of the side of a cube with the volume we calculated, we can rearrange the formula:
s = V^(1/3)
Now let's calculate it step by step:
Step 1: Find the number of moles using the ideal gas law equation:
PV = nRT
Assuming P = 1 atm, V = 1 L, R = ideal gas constant = 0.0821 atm L/mol K, and T = 298 K, we get:
(1 atm)(1 L) = n (0.0821 atm L/mol K)(298 K)
Simplifying the equation, we find:
n = (1 L) / (0.0821 atm L/mol K)(298 K)
n = 0.0409 mol
Step 2: Calculate the number of helium atoms in one mole using Avogadro's number:
Number of atoms = (Number of moles) x (Avogadro's number)
Number of atoms = (0.0409 mol) x (6.022 x 10^23 mol^-1)
Number of atoms = 2.464 x 10^22 helium atoms
Step 3: Find the volume occupied by a single helium atom:
Volume per atom = (Volume of gas) / (Number of atoms)
Volume per atom = (1 L) / (2.464 x 10^22 atoms)
Volume per atom = 4.06 x 10^-23 L/atom
Step 4: Calculate the length of the side of a cube with this volume:
s = (Volume per atom)^(1/3)
s = (4.06 x 10^-23 L/atom)^(1/3)
s ≈ 1.6 x 10^-8 L^1/3/atom
Therefore, the volume per helium atom in a liter of helium gas kept at room temperature and atmospheric pressure is approximately 4.06 x 10^-23 L/atom, and the length of the side of a cube with this volume is approximately 1.6 x 10^-8 L^1/3/atom.