A good vacuum pump on Earth can produce a vacuum with a pressure as low as 1.96 × 10-8 torr. How many molecules are present in each milliliter at a temperature of 23.0ºC?
At 1 atm = 760 Torr and 0^oC, 1 mole of any gas contains 6.02E23 particles of gas and occupies a volume of 22.4 Liters. This number of particles would be reduced by (1.96E-8/760)if at 0^C. At 23^oC = 296K the number of particles would be also be reduced by(273/296)because of temp expansion to the volume at 23^C. So, the number of Particles at 1.96E-8 Torr = (6.02E23/22.4)(1.96E-8/760)(273/296) particles => 6.39E11 particles (molecules).
To find the number of molecules present in each milliliter at a given pressure and temperature, we can use the Ideal Gas Law. The Ideal Gas Law states that the pressure of a gas is directly proportional to the number of molecules, the temperature, and the volume of the gas.
The formula is:
PV = nRT
Where:
P = pressure (in pascals)
V = volume (in cubic meters)
n = number of molecules (in moles)
R = gas constant (8.314 J/(mol·K))
T = temperature (in Kelvin)
First, we need to convert the pressure given in torr to pascals. 1 torr is equivalent to 133.322 pascals.
1.96 × 10^-8 torr * 133.322 pascals/torr = 2.61 × 10^-6 pascals
Next, we need to convert the temperature given in degrees Celsius to Kelvin. The conversion formula is: K = °C + 273.15
23.0ºC + 273.15 = 296.15 K
Now, we can plug the values into the Ideal Gas Law to find the number of molecules:
(2.61 × 10^-6 pascals) * V = n * (8.314 J/(mol·K)) * (296.15 K)
Since we want to find the number of molecules per milliliter, we need to convert the volume to cubic meters. 1 milliliter is equivalent to 1 × 10^-6 cubic meters.
(2.61 × 10^-6 pascals) * (1 × 10^-6 cubic meters) = n * (8.314 J/(mol·K)) * (296.15 K)
Simplifying,
2.61 = n * (8.314 J/(mol·K)) * (296.15 K)
n = 2.61 / ((8.314 J/(mol·K)) * (296.15 K))
Calculating the right side of the equation,
n ≈ 1.18 × 10^-4 mol
Finally, we need to convert the number of moles to the number of molecules. 1 mole is equivalent to 6.022 × 10^23 molecules.
n * (6.022 × 10^23 molecules/mol) ≈ 1.18 × 10^-4 mol * (6.022 × 10^23 molecules/mol)
≈ 7.1 × 10^19 molecules
Therefore, there are approximately 7.1 × 10^19 molecules present in each milliliter at a temperature of 23.0ºC.