Question

In an ultrahigh vacuum system, the pressure is measured to be 8.4 × 10−11 torr (where 1 torr = 133 Pa). The gas molecules have a molecular diameter of 2.2 × 10−10 m and the temperature is 310 K. Avogadro's number is 6.02214×1023 1/mol. Find the number of molecules in a volume of 0.87 m3 . Answer in units of molecules.

Answer 1

The number of molecules in the volume is
`N_v = 2.27109* 10^(12)` molecules

Step-by-step explanation:

From the question we are told that

The pressure of the ultrahigh vacuum is
`P = 8.4*10^(-11) torr = 8.4*10^(-11) * 133 = 1.1172 *10^(-8)Pa`

The molecular diameter of the gas molecules
`d = 2.2*10^(-10) m`

The temperature is
`T = 310 \ K`

Avogadro's number is
`N = 6.02214 *10^(23)\ l/mol`

The volume of the gas is
`V = 0.87 m^3`

From the ideal gas law[
`PV = nRT`] that the number of mole is mathematically represented as

`n = (PV)/(RT)`

Where R is the gas constant with a value
`R = 8.314\ J/mol`

Substituting values

`n = (1.1172 *10^(-8) * 0.87)/(8.314 * 310)`

`n = 3.771*10^(-12) \ mole`

The number of molecules is mathematically represented as

`N_v = n * N`

Substituting values

`N_v = 3.771*10^(-12) * 6.02214 *10^(23)`

`N_v = 2.27109* 10^(12)` molecules