given E, = l^2 h^2 / 2 m r^2 calculate energies for an electron localised to a circle of radius 2 x 10^-9 m for l = 0,1,2, draw pictures of three levels and calculate the relative populations of these levels at 300K
i did the first bit but came out with, 0, 6.0249 x 10-6 J mol-1 and 2.4099 x 10-5 J mol-1 are these right and how do i work out the second part and what kind of picture?? Confused
appreciate the help btw i used m = 9.109 x 10^-31 kg for the mass of an electron
To calculate the energies for an electron localized to a circle of radius 2 x 10^-9 m for different values of angular momentum quantum number (l), you have correctly used the formula:
E = (l^2 * h^2) / (2 * m * r^2)
where:
E = energy
l = angular momentum quantum number
h = Planck's constant (6.626 x 10^-34 J*s)
m = mass of an electron (9.109 x 10^-31 kg)
r = radius of the circle (2 x 10^-9 m)
For l = 0:
E = (0^2 * h^2) / (2 * m * r^2)
E = 0
For l = 1:
E = (1^2 * h^2) / (2 * m * r^2)
E = (1 * h^2) / (2 * m * r^2)
E = (1 * (6.626 x 10^-34 J*s)^2) / (2 * (9.109 x 10^-31 kg) * (2 x 10^-9 m)^2)
E ≈ 6.0249 x 10^-6 J
For l = 2:
E = (2^2 * h^2) / (2 * m * r^2)
E = (4 * h^2) / (2 * m * r^2)
E = (4 * (6.626 x 10^-34 J*s)^2) / (2 * (9.109 x 10^-31 kg) * (2 x 10^-9 m)^2)
E ≈ 2.4099 x 10^-5 J
So, the values you obtained for the energies are correct.
Now, let's move on to the second part of the question, which involves calculating the relative populations of these energy levels at 300K.
To calculate the relative populations, we can use the Boltzmann distribution formula:
P(l) = (exp(-E(l) / (k * T))) / Σ(exp(-E(i) / (k * T)))
where:
P(l) = relative population of the level with angular momentum quantum number (l)
E(l) = energy of the level with angular momentum quantum number (l)
k = Boltzmann constant (1.381 x 10^-23 J/K)
T = temperature in Kelvin
Σ = summation symbol
Given that the temperature is 300K, we can plug in the values and calculate the relative populations.
P(0) = (exp(-E(0) / (k * T))) / (exp(-E(0) / (k * T)) + exp(-E(1) / (k * T)) + exp(-E(2) / (k * T)))
P(1) = (exp(-E(1) / (k * T))) / (exp(-E(0) / (k * T)) + exp(-E(1) / (k * T)) + exp(-E(2) / (k * T)))
P(2) = (exp(-E(2) / (k * T))) / (exp(-E(0) / (k * T)) + exp(-E(1) / (k * T)) + exp(-E(2) / (k * T)))
Now we can substitute the values of E(0), E(1), E(2), k, and T into the formula to calculate the relative populations for each level.