To tackle your question, we’ll break it down into two parts: first, calculating the volume occupied by 1.00 mol of an ideal gas at standard conditions, and then determining the Loschmidt number, which represents the number of molecules per cubic centimeter at those conditions.
Calculating the Volume of an Ideal Gas
At standard conditions, we use the Ideal Gas Law, which is expressed as:
PV = nRT
Where:
- P = pressure (in atmospheres)
- V = volume (in liters)
- n = number of moles of gas
- R = ideal gas constant (0.0821 L·atm/(K·mol))
- T = temperature (in Kelvin)
Given:
- P = 1.00 atm
- n = 1.00 mol
- R = 0.0821 L·atm/(K·mol)
- T = 273 K
Now, substituting these values into the Ideal Gas Law:
V = nRT / P
Plugging in the numbers:
V = (1.00 mol) × (0.0821 L·atm/(K·mol)) × (273 K) / (1.00 atm)
Calculating this gives:
V = 22.414 L
Thus, 1.00 mol of an ideal gas occupies approximately 22.414 liters at standard conditions.
Determining the Loschmidt Number
The Loschmidt number is defined as the number of molecules in one cubic centimeter of an ideal gas at standard conditions. To find this, we can use the ideal gas law again but rearranged to find the number of molecules.
First, we need to convert the volume from liters to cubic centimeters (cm³). Since 1 liter equals 1000 cm³, we have:
22.414 L = 22,414 cm³
Next, we know that 1 mole of any substance contains Avogadro's number of molecules, which is approximately:
N_A = 6.022 × 10²³ molecules/mol
Now, to find the number of molecules per cubic centimeter, we can use the formula:
Number of molecules per cm³ = (N_A / Volume in cm³)
Substituting the values:
Number of molecules per cm³ = (6.022 × 10²³ molecules/mol) / (22,414 cm³)
Calculating this gives:
Number of molecules per cm³ ≈ 2.68 × 10¹⁹
This confirms that the Loschmidt number at standard conditions is approximately 2.68 × 10¹⁹ molecules/cm³.
Summary
In summary, at standard conditions, 1.00 mol of an ideal gas occupies about 22.414 liters, and the Loschmidt number, representing the number of molecules per cubic centimeter, is approximately 2.68 × 10¹⁹. These calculations illustrate the relationship between pressure, volume, temperature, and the number of particles in a gas, which are fundamental concepts in understanding gas behavior in chemistry.
