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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?

Shane Macguire , 10 Years ago
Grade upto college level
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Askiitians Tutor Team

Last Activity: 2 Days ago

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.

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