Calculate the deBroglie wavelength of a^ 87Rb atom that has been laser cooled to 120 µK. (Assume that the kinetic energy is .3/2kBT)
To calculate the deBroglie wavelength of a particle, you need to know its mass and velocity. In this case, we are given the temperature of the atom and the expression for the kinetic energy. We can use this information to find the velocity of the atom and then calculate the deBroglie wavelength.
Step 1: Calculate the velocity of the atom using the expression for kinetic energy.
Given that the kinetic energy (KE) is given by KE = (3/2) kBT, where kB is the Boltzmann constant and T is the temperature in Kelvin, we need to convert the given temperature to Kelvin.
120 µK is equal to 0.00012 K.
Now, we can calculate the velocity of the atom using the expression for kinetic energy:
KE = (1/2) mv^2, where m is the mass of the atom and v is the velocity.
Since the atom is given as ^87Rb (rubidium-87), we need to find the mass of the atom from the periodic table. The molar mass of ^87Rb is approximately 86.91 g/mol. We need to convert this to kg.
Molar mass of ^87Rb = 86.91 g/mol
Mass of one atom = 86.91 g/mol ÷ Avogadro's constant
≈ 1.44 x 10^-25 kg
Using the given kinetic energy equation:
(3/2) kBT = (1/2) mv^2
We rearrange the equation to solve for v:
v = √[(3kBT)/m]
Step 2: Calculate the deBroglie wavelength.
The deBroglie wavelength (λ) of a particle is given by the equation:
λ = h/mv
where h is the Planck's constant.
The value of Planck's constant (h) is approximately 6.626 x 10^-34 J·s.
With the calculated velocity (v) and the mass of the ^87Rb atom (m), we can now calculate the deBroglie wavelength.
λ = (6.626 x 10^-34 J·s) / (m * v)
Now, substituting the values into the equation, we can calculate the deBroglie wavelength.