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 l, we can use the given equation E = (l^2 h^2) / (2 m r^2). Let's calculate the energies for l = 0, 1, and 2 using the mass of an electron (m = 9.109 x 10^-31 kg):
1. For l = 0:
E = (0^2 * h^2) / (2 * m * r^2)
= 0
2. For l = 1:
E = (1^2 * h^2) / (2 * m * r^2)
= (1 * h^2) / (2 * m * r^2)
First, we need the value of Planck's constant (h). It is approximately equal to 6.626 x 10^-34 J·s.
Plugging in the values:
E = (1 * (6.626 x 10^-34 J·s)^2) / (2 * (9.109 x 10^-31 kg) * (2 x 10^-9 m)^2)
= 6.0249 x 10^-6 J
3. For l = 2:
E = (2^2 * h^2) / (2 * m * r^2)
= (4 * h^2) / (2 * m * r^2)
Plugging in the values:
E = (4 * (6.626 x 10^-34 J·s)^2) / (2 * (9.109 x 10^-31 kg) * (2 x 10^-9 m)^2)
= 2.4099 x 10^-5 J
Now, for the second part of the question regarding the relative populations of these energy levels at 300K, we need to use the Boltzmann distribution formula:
P(l) = exp(-E(l) / (k*T)) / Σ[exp(-E(i) / (k*T))]
where P(l) represents the relative population of the level with energy E(l), k is the Boltzmann constant (approximately 1.381 x 10^-23 J/K), and T is the temperature in Kelvin.
Let's calculate the relative populations for the three levels at 300K:
1. For l = 0:
P(0) = exp(-E(0) / (k*T)) / [exp(-E(0) / (k*T)) + exp(-E(1) / (k*T)) + exp(-E(2) / (k*T))]
= exp(-0 / (1.381 x 10^-23 J/K * 300 K)) / [exp(-0 / (1.381 x 10^-23 J/K * 300 K)) + exp(-6.0249 x 10^-6 J / (1.381 x 10^-23 J/K * 300 K)) + exp(-2.4099 x 10^-5 J / (1.381 x 10^-23 J/K * 300 K))]
2. For l = 1:
P(1) = exp(-E(1) / (k*T)) / [exp(-E(0) / (k*T)) + exp(-E(1) / (k*T)) + exp(-E(2) / (k*T))]
3. For l = 2:
P(2) = exp(-E(2) / (k*T)) / [exp(-E(0) / (k*T)) + exp(-E(1) / (k*T)) + exp(-E(2) / (k*T))]
Now, evaluate these equations using the given values and you will obtain the relative populations of the three levels at 300K.