Justin W.

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Physical Chemistry

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Question:

For a quantum particle in a cube, the total translational energy is given by E_n = (h^2/(8ma^2)) x [n_x^2 + n_y^2 + n_z^2], where the n_i are the translational quantum states in the x, y, and z dimensions, and a is the length/width/height of the box. What is the degeneracy when n_x^2 + n_y^2 + n_z^2 = 6?

Justin W.

Answer:

The degeneracy can be thought of as the multiple different ways to satisfy the equation n_x^2 + n_y^2 + n_z^2 = 4, with different values of the n_i. This will tell us the different combination of eigenstates that sum to the same total energy of 6. Note that the values of the n_i are all positive integers (1, 2, 3, . . .) for a particle in a box. For this simple case, we simply enumerate the possibilities. We have (1)^2 + (1)^2 + (2)^2 = 6 [That is, n_x = 1, n_y = 1, n_z = 2] (1)^2 + (2)^2 + (1)^2 = 6 (2)^2 + (1)^2 + (1)^2 = 6 So the degeneracy is 3. For n_x = n_y = n_z = 1, the degeneracy would (obviously) be simply 1, because that is the only way for the sum of their squares to equal 3, when n is greater than or equal to 1.

Chemistry

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Question:

A chemical element has 11 protons, 10 neutrons, and 10 electrons. What is the identity of this element?

Justin W.

Answer:

An element is defined by the number of protons. Therefore, this must be some isotope of sodium, Na, according to the . Because there is one fewer electron than the number of protons, this is a sodium cation with a charge of +1. The mass number includes the number of protons plus neutrons, which is 11 + 10 = 21. The species is therefore 21Na+.

Calculus

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Question:

For the function f(x) = 2x^2 + 3x, determine the slope of the tangent line to f(x) at x = 4.

Justin W.

Answer:

The tangent line is simply "math speak" for a line with an instantaneous slope equal to the derivative of f(x) at x = 4. The derivative f'(x) = 4x + 3 by the power rule, so f'(4) = 4(4) + 3 = 19. The tangent line has the equation f(x) - f(4) = 19(x - 4). Since f(4) = 2(4)^2 + 3(4) = 44, the tangent line has the equation f(x) - 44 = 19(x - 4). In slope-intercept form this is f(x) = 19x - 32.

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