In state-of-the-art vacuum systems, pressures as low as 1.00 x 10^-9 Pa are being attained. Calculate the number of molecules in a 1.90-m^3 vessel at this pressure and a temperature of 35.0°C.
Okay. So I know I have to use the Ideal Gas Law, PV = nRT, I then transformed it into being PV/RT = n.
1.00 X 10^-9 is my P.
1.90 m^3 is my V.
308.15 K is my T. (35 degrees C + 273.15)
How do I solve for how many molecules there are? What is my R?
the number of molucles is n*avagradro's number
R? Pressure is Pa, volume in m^3, T i Kelvins... R has to be in Pa-M^3/mole.kelvins or 8.314
Okay so my equation was set up correctly and should read.. (1.00x10^-9)(1.90m^3)/(8.314x308.15) Correct? So my R would be 2.67724^-12?
To solve for the number of molecules, you can rearrange the ideal gas law equation as n = PV/RT, where n is the number of moles of gas.
The Ideal Gas Constant, R, has a value of 8.314 J/(mol·K).
Given:
Pressure (P) = 1.00 × 10^-9 Pa
Volume (V) = 1.90 m^3
Temperature (T) = 308.15 K
First, convert the pressure from Pascals (Pa) to atmospheres (atm):
1 atm = 101325 Pa
1 Pa = 1/(101325 atm)
So, P = (1.00 × 10^-9 Pa) × (1/(101325 atm/Pa)) = 9.869 × 10^-15 atm
Now substitute the values into the equation n = PV/RT
n = (9.869 × 10^-15 atm) × (1.90 m^3) / (8.314 J/(mol·K) × 308.15 K)
Simplify the units:
n = (9.869 × 10^-15) × (1.90) / (8.314 × 308.15) mol
Finally, multiply the number of moles by Avogadro's number, 6.022 × 10^23 molecules/mol, to find the number of molecules:
Number of molecules = (9.869 × 10^-15) × (1.90) / (8.314 × 308.15) mol × (6.022 × 10^23 molecules/mol)
Solving this expression will give you the number of molecules in the 1.90 m^3 vessel at the given pressure and temperature.
To solve for the number of molecules (n) using the Ideal Gas Law, you have correctly rearranged the equation as PV/RT = n.
However, you are missing the value of the ideal gas constant (R), which is needed for the calculation. The ideal gas constant (R) has a value of 8.314 J/(mol·K).
Now, let's plug in the values into the equation:
P = 1.00 x 10^-9 Pa
V = 1.90 m^3
T = 308.15 K
R = 8.314 J/(mol·K)
PV/RT = n
(1.00 x 10^-9 Pa) * (1.90 m^3) / (8.314 J/(mol·K) * 308.15 K) = n
Calculating this expression will give you the number of moles (n) of gas in the given volume.
To convert from moles to the number of molecules, you can use Avogadro's number, which is approximately 6.022 x 10^23 molecules/mol. Simply multiply the value of the calculated moles by Avogadro's number to obtain the number of molecules.
Let me know if you need further assistance with the calculation!