The atoms in a gas (gas constant R=8.31 J/mol-K) can be treated as classical particles if their de Broglie wavelength ƒÉ is much smaller than the average separation between the particles d. Consider monatomic Helium gas (mHec2 = 3727MeV), molar mass 4g/mol) at 1.0 atmosphere of pressure (1.0�~105) and at room temperature (T=293K).
a. Estimate d for this gas
b. Find the average de Broglie wavelength of the atoms in the gas ( KE = (3/2) KB T, KB = 8.617�~10-E5 eV/K
c. Find the pressure that would make d equal to ă from part b
a) The number density of He atoms under those conditions is
n = 6*10^23 atoms/24.4 liters = 2.4*10^19 atom/cm^3
The volume per atom is 1/n = 4*10^-20 cm^2 and the average distance between molecules is the cube root of that, or
3.5*10^-7 cm.
b. The average momentum of the He atoms is
p = sqrt(2mE) where E is the average kinetic energy,
E = (3/2) kT = 6.1*10^-14 erg. So,
p = sqrt(2*6.1*10^-14*4*1.67*10^-24)
= 9.0*10^-19 g cm/s
The de Broglie wavelength is
L = h/p = (6.6*10^-27 g cm^2/s)/(9*10^-19 g cm/s) = 7.3*10^-9 cm
This is about 50 times less than the spacing between atoms.
c) To make d equal to L, you have to decrease d by a factor of 50.
This means the density (and pressure) would have to increase by a factor of 50^3
a. To estimate the average separation between gas particles, we can use the ideal gas law which states that PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature. Rearranging the equation to solve for the volume V, we get V = (nRT)/P.
For monatomic Helium gas at 1 atmosphere of pressure and room temperature, we can calculate the number of moles n using the molar mass given. The number of moles n = 4g/4g/mol = 1 mol. Plugging in the values for n, R, P, and T, we have V = (1 mol * 8.31 J/mol-K * 293 K) / (1.0 x 10^5 Pa). Calculating this will give us the volume V.
Since we are dealing with a gas, we assume that the gas molecules are uniformly distributed throughout the volume. Therefore, we can assume that the volume V represents the total volume occupied by the gas particles.
Next, we need to estimate the effective shape of the gas particles. We can assume that they are spherical in shape. In this case, the average separation between particles, d, can be estimated using the formula d = (3V/4π)^(1/3). Plugging in the calculated volume V into this formula will give us an estimate for the average separation d.
b. The de Broglie wavelength λ is given by the equation λ = h/p, where h is the Planck's constant and p is the momentum of the particle. In classical mechanics, we can approximate the momentum using the kinetic energy KE of the particle, which is given by KE = (3/2) kT, where k is the Boltzmann constant and T is the temperature.
To calculate the average de Broglie wavelength, we need to find the average kinetic energy of the gas particles using the given temperature. Plugging in the values for the Boltzmann constant k and the temperature T, we can calculate the average kinetic energy KE.
Then, we can calculate the momentum by using the equation p = √(2mKE), where m is the mass of the gas particles.
Finally, we can calculate the average de Broglie wavelength λ by dividing the Planck's constant by the momentum p.
c. To find the pressure that would make d equal to λ, we need to set the two values equal to each other and solve for the pressure P. Using the equation d = λ = (h/p), we can substitute the expressions for d and λ obtained in parts a and b respectively.
Solving this equation for the pressure P will give us the value needed to make d equal to λ.