How many unique combinations of the quantum numbers l and 𝑚𝑙 are there when b. n = 4?

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n is the principal quantum number, and it determines the energy level of an electron in an atom. For n = 4, the possible values of the azimuthal quantum number l range from 0 to n - 1.

For each value of l, the magnetic quantum number ml can take integer values from -l to +l, including zero.

Calculate the number of possible ml values for each l value. For example, if l = 0, ml can only be 0, giving 1 combination.

Continue this process for l = 1, 2, and 3, calculating the number of ml values for each.

Sum the number of combinations for each l value to find the total number of unique combinations of l and ml when n = 4.

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Key Concepts

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Quantum Numbers

Quantum numbers are sets of numerical values that describe the unique quantum state of an electron in an atom. They include the principal quantum number (n), azimuthal quantum number (l), magnetic quantum number (ml), and spin quantum number (ms). Each quantum number provides specific information about the electron's energy level, shape of the orbital, orientation, and spin.

Azimuthal Quantum Number (l)

The azimuthal quantum number (l) determines the shape of an electron's orbital and can take on integer values from 0 to n-1, where n is the principal quantum number. For n = 4, l can be 0, 1, 2, or 3, corresponding to the s, p, d, and f orbitals, respectively. Each value of l defines a different type of orbital with distinct shapes.

Magnetic Quantum Number (ml)

The magnetic quantum number (ml) specifies the orientation of an orbital in space and can take on integer values ranging from -l to +l, including zero. For each value of l, there are 2l + 1 possible values of ml. For example, if l = 2 (d orbital), ml can be -2, -1, 0, +1, or +2, resulting in five unique orientations.

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Textbook Question

Using Heisenberg’s uncertainty principle, calculate the uncertainty in the position of b. a proton moving at a speed of (5.00±0.01) × 104 m/s. The mass of a proton is 1.673×10−27 kg.

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