To find the length of the edge of each cube that represents the average spacing between molecules in an ideal gas at 27°C and 1.00 atm pressure, we can use the ideal gas law and some basic geometry. Let's break this down step by step.
Understanding the Ideal Gas Law
The ideal gas law is expressed as:
PV = nRT
Where:
- P = pressure (in atm)
- V = volume (in liters)
- n = number of moles of gas
- R = ideal gas constant (0.0821 L·atm/(K·mol))
- T = temperature (in Kelvin)
Converting Temperature
First, we need to convert the temperature from Celsius to Kelvin:
T(K) = T(°C) + 273.15
So, for 27°C:
T = 27 + 273.15 = 300.15 K
Calculating the Volume of One Mole of Gas
Next, we can rearrange the ideal gas law to find the volume occupied by one mole of gas:
V = nRT/P
For one mole of gas (n = 1), substituting the values:
V = (1 mol) * (0.0821 L·atm/(K·mol)) * (300.15 K) / (1 atm)
Calculating this gives:
V ≈ 24.45 L
Finding the Volume of a Single Molecule
Now, we need to determine how many molecules are in one mole of gas. Avogadro's number tells us that:
N = 6.022 x 10²³ molecules/mol
To find the volume occupied by a single molecule, we divide the total volume by Avogadro's number:
Volume per molecule = V / N
Volume per molecule ≈ 24.45 L / (6.022 x 10²³ molecules/mol)
Calculating this gives:
Volume per molecule ≈ 4.07 x 10⁻²³ L
Converting this to cubic centimeters (1 L = 1000 cm³):
Volume per molecule ≈ 4.07 x 10⁻²³ L * 1000 cm³/L ≈ 4.07 x 10⁻²⁰ cm³
Calculating the Edge Length of the Cube
Since we are imagining each molecule as the center of a cube, we can find the edge length of the cube by taking the cube root of the volume per molecule:
Edge length = (Volume per molecule)^(1/3)
Edge length ≈ (4.07 x 10⁻²⁰ cm³)^(1/3)
Calculating this gives:
Edge length ≈ 3.44 x 10⁻⁷ cm
Final Thoughts
Thus, the length of the edge of each cube, representing the average spacing between adjacent gas molecules at 27°C and 1.00 atm, is approximately 3.44 x 10⁻⁷ cm, or 3.44 nanometers. This small size illustrates just how sparse gas molecules are in a typical environment, despite the large volume they occupy collectively.
