In state-of-the-art vacuum systems, pressures as low as 1.00 10-9 Pa are being attained. Calculate the number of molecules in a 1.80-m3 vessel at this pressure and a temperature of 31.0°C.
Homework Posting Tips
Please show your work. Tutors will not do your homework for you. Please show your work for any question that you are posting.
You seem to be having an identity crisis -- you've posted several physics questions under different names.
Please use the same name for your posts.
To calculate the number of molecules in a vessel at a given pressure and temperature, we can use the ideal gas law equation:
PV = nRT
Where:
P = pressure (in Pascals)
V = volume (in cubic meters)
n = number of moles of gas
R = ideal gas constant (8.314 J/(mol·K))
T = temperature (in Kelvin)
First, we need to convert the pressure from 1.00 × 10^(-9) Pa to the same unit as the ideal gas constant. The ideal gas constant is usually expressed in units of J/(mol·K), so we need to convert Pascals to atmospheric pressure. 1 atmosphere (atm) is roughly equal to 101325 Pascals.
1 atm = 101325 Pa
Therefore, the pressure in atm is:
P = (1.00 × 10^(-9) Pa) / (101325 Pa/atm) ≈ 9.87 × 10^(-15) atm
Next, let's convert the temperature from 31.0°C to Kelvin:
T = 31.0°C + 273.15 ≈ 304.15 K
Now we can rearrange the ideal gas law to solve for the number of moles (n):
n = PV / RT
n = (9.87 × 10^(-15) atm) * (1.80 m^3) / ((8.314 J/(mol·K)) * (304.15 K))
Simplifying the equation gives:
n ≈ 6.0 × 10^(-15) mol
Finally, we know that there are approximately 6.022 × 10^23 molecules in one mole of any substance (Avogadro's number), so we can find the number of molecules by multiplying the number of moles by Avogadro's number:
Number of molecules = (6.0 × 10^(-15) mol) * (6.022 × 10^23 molecules/mol)
Number of molecules ≈ 3.61 × 10^9 molecules
Therefore, at a pressure of 1.00 × 10^(-9) Pa and a temperature of 31.0°C, there are approximately 3.61 × 10^9 molecules in a 1.80-m^3 vessel.