How Does de Broglie Wavelength Affect Particle Behavior in Helium Gas?

[mateomy]

The atoms in a gas can be treated as classical particles if their de Broglie wavelength is much smaller than the average separation between the particles. Compare the average de Broglie wavelength and the average separation between the atoms in a container of monatomic helium gas at 1.00 atm pressure and at room temperature (20 degrees C). At what temperature and pressure would you expect quantum effects to become important.I've calculated the de Broglie wavelength but I can't get the spacing between the atoms. Am I forgetting something from Chemistry? Can anyone give me a hint? I feel like I need a volume but clearly it isn't in the given info.

Thanks.

 

[TSny]

Hello. My hint would be "ideal gas law". You don't need volume, but you do need volume per particle.

 

[mateomy]

Awesome. I thought it had something to do with that. Thanks a lot.

 

[DJSedna]

I figured instead of putting this question into a new thread, I'd simply necro this one.

I'm currently stuck on this problem staring at a blank page. I understand that the De Broglie Wavelength formula is necessary here as well as the Ideal Gas Law, but I must be missing something. Can anyone give me a hint just to get me on track with this problem?

Thanks in advance.

 

[paranormal]

The de Broglie wavelength is given by the equation λ = h/mv, where h is Planck's constant, m is the mass of the particle, and v is its velocity. In the case of atoms in a gas, their velocity can be approximated by the average speed of the gas molecules, which is given by the root mean square velocity, vrms = √(3RT/M), where R is the gas constant, T is the temperature in Kelvin, and M is the molar mass of the gas.

For monatomic helium gas at 1.00 atm pressure and 20 degrees C (293 K), the molar mass is 4.003 g/mol. Plugging in these values, we get vrms = √(3*8.314 J/mol*K*293 K / 0.004003 kg/mol) = 1404 m/s.

Now, to calculate the average separation between the atoms, we can use the ideal gas law, PV = nRT, where P is the pressure, V is the volume, n is the number of moles, and R is the gas constant. Rearranging for volume, we get V = nRT/P. Assuming a container of 1 mole of monatomic helium gas, we get V = (1 mol * 8.314 J/mol*K * 293 K) / (1.00 atm * 101325 Pa/atm) = 2.44 x 10^-3 m^3.

The volume of the container can also be calculated by taking the average separation between the atoms and multiplying it by the number of atoms in the gas. The number of atoms can be calculated by dividing the mass of the gas (1 mole * 4.003 g/mol = 4.003 g) by the molar mass (4.003 g/mol), giving us 6.02 x 10^23 atoms. Therefore, the average separation between the atoms is V/n = (2.44 x 10^-3 m^3) / (6.02 x 10^23 atoms) = 4.06 x 10^-27 m.

Comparing this to the calculated de Broglie wavelength, which is given by λ = h/mv = (6.626 x 10^-34 J*s) / (4.003 x 10^-3 kg * 1404 m/s) = 1.49