For an electron in a hydrogen atom, the z component of the angular momentum has a maximum value of Lz = 4.22 x 10-34 J·s. Find the smallest possible value (algebraically) for the total energy (in electron volts) that this atom could have

To find the smallest possible value for the total energy of the electron in a hydrogen atom, we need to use the relationship between the angular momentum and the energy for a particle in a hydrogen atom.

The angular momentum of the electron is quantized and given by the equation:

L = √(l(l + 1)) * ħ

where L is the magnitude of the angular momentum, l is the orbital angular momentum quantum number, and ħ is the reduced Planck's constant (approximately 1.05 x 10^-34 J·s).

Given the maximum value of the z component of the angular momentum (Lz = 4.22 x 10^-34 J·s), we can find the orbital angular momentum quantum number (l) using the following equation:

Lz = mħ

where m is the magnetic quantum number. Since the maximum value is given, we have:

4.22 x 10^-34 J·s = m * 1.05 x 10^-34 J·s

Simplifying, we find:

m = 4.22 / 1.05 ≈ 4

Now, the total energy (E) of the electron in a hydrogen atom can be calculated using the equation:

E = -13.6 eV / n^2

where n is the principal quantum number.

The smallest possible value for the total energy occurs when n is maximized. To find this value, we use the equation:

l = n - 1

Since we have l = 4, we can substitute this into the equation to find:

4 = n - 1

n = 5

Now, substituting the value of n into the energy equation, we get:

E = -13.6 eV / (5^2)

E = -13.6 eV / 25

E ≈ -0.544 eV

Therefore, the smallest possible value (algebraically) for the total energy of the electron in the hydrogen atom is approximately -0.544 eV.