The best vacuum that can be attained in the laboratory corresponds to a pressure of about 10^-18 atm, or 1.01 × 10¹³ Pa How many molecules are there per cubic centimetre in such a vacuum at 22ºC?

To determine the number of molecules in a vacuum at a pressure of approximately 10^-18 atm, we can use the ideal gas law, which relates pressure, volume, temperature, and the number of molecules. The ideal gas law is expressed as PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin.

Converting Units

First, we need to convert the pressure from atmospheres to pascals (Pa) since the pressure is given as 1.01 × 10¹³ Pa. We also need to convert the temperature from degrees Celsius to Kelvin. The conversion from Celsius to Kelvin is done by adding 273.15:

• Temperature: 22ºC + 273.15 = 295.15 K
• Pressure: 1.01 × 10¹³ Pa (already in the correct unit)

Using the Ideal Gas Law

Next, we can rearrange the ideal gas law to find the number of moles (n):

n = PV / RT

Here, R (the ideal gas constant) is approximately 8.314 J/(mol·K). Now we can plug in the values:

n = (1.01 × 10¹³ Pa) × (1 m³) / (8.314 J/(mol·K) × 295.15 K)

Calculating this gives:

n ≈ (1.01 × 10¹³) / (8.314 × 295.15) ≈ 4.05 × 10¹⁰ moles

Calculating the Number of Molecules

To find the total number of molecules, we use Avogadro's number, which is approximately 6.022 × 10²³ molecules/mol:

Total molecules = n × Avogadro's number

Total molecules ≈ (4.05 × 10¹⁰ moles) × (6.022 × 10²³ molecules/mol)

This results in approximately:

Total molecules ≈ 2.44 × 10³⁴ molecules

Finding the Number of Molecules per Cubic Centimetre

Since we want the number of molecules per cubic centimetre, we need to convert cubic metres to cubic centimetres. There are 1 × 10⁶ cubic centimetres in a cubic metre:

Number of molecules per cm³ = Total molecules / (1 × 10⁶ cm³)

Number of molecules per cm³ ≈ (2.44 × 10³⁴) / (1 × 10⁶) ≈ 2.44 × 10²⁸ molecules/cm³

Summary

In a vacuum with a pressure of about 10^-18 atm at 22ºC, there are approximately 2.44 × 10²⁸ molecules per cubic centimetre. This calculation illustrates how even in a high vacuum, there are still a significant number of molecules present, albeit at extremely low densities.

Approved