Modern vacuum pumps make it easy to attain pressures of the order of 10−13 atm in the laboratory. consider a 1.0 cm3 volume of air and treat the air as an ideal gas. (a) at a pressure of 9.00 × 10−14 atm and an ordinary temperature of 273.15 k (0.0 ◦c), how many molecules are present in a volume of 1.00 cm3 ? (b) how many molecules would be present at the same temperature but at 1.00 atm instead? this is known as loschmidt’s number.

Since the gases behave ideally, we use the ideal gas equation:

PV = nRT
where
P is pressure
V is volume
n is the number of moles
R is the gas constant equal to 0.0821 L·atm/mol·K
T is the absolute temperature

a.) P = 9.00×10⁻¹⁴ atm; T = 273.15 K; V = 1 cm³ = 0.001 L

(9.00×10⁻¹⁴ atm)(0.001 L) = n(0.0821 L·atm/mol·K)(273.15 K)
n = 4.013×10⁻¹⁸ moles
Since there are 6.022×10²³ molecules in 1 mole,
Number of molecules = 2,416,847

b.) P = 1 atm; T = 273.15 K; V = 1 cm³ = 0.001 L

(1 atm)(0.001 L) = n(0.0821 L·atm/mol·K)(273.15 K)
n = 4.459×10⁻⁶ moles
Since there are 6.022×10²³ molecules in 1 mole,
Number of molecules = 2.685×10¹⁹