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 × 10^-9 m for different values of l (0, 1, 2), you correctly used the formula: E = (l^2 * h^2) / (2 * m * r^2).
Using the mass of an electron, m = 9.109 × 10^-31 kg, and the given radius, r = 2 × 10^-9 m, we can calculate the energies for each value of l:
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)
For l = 2:
E = (2^2 * h^2) / (2 * m * r^2)
Now, we need to calculate the values for the energies.
To calculate the relative populations of these levels at 300K, we need to use the Boltzmann distribution equation:
Relative population = exp(-E / (k * T))
where E is the energy, k is the Boltzmann constant (1.38 × 10^-23 J/K), and T is the temperature in Kelvin (300K).
For each level, plug in the respective energy into the equation and calculate the relative population.
For example, for the energy E when l = 0, the relative population is given by:
Relative population for l = 0 = exp(-E / (k * T))
Repeat these calculations for the energies obtained for l = 1 and l = 2, and you will have the relative populations for each level.
As for the "pictures" mentioned, you can represent the energy levels as vertical lines on a diagram. The height of each line corresponds to the energy level, and you can label each line with the respective value of l. This can help visualize and understand the concept of different energy levels.
Remember to use the correct values for the constants such as h (Planck's constant) and k (Boltzmann constant), and ensure that all units are consistent for accurate calculations.