Question

For an advanced lab project you decide to look at the red line in the Balmer series. According to the Bohr Theory, this is a single line. However, when you examine it at high resolution, you find that it is a closely-spaced doublet. From your research, you determine that this line is the 3s to 2p transition in the hydrogen spectrum. When an electron is in the 2p subshell, its orbital motion creates a magnetic field and as a result, the atom's energy is slightly different depending on whether the electron is spin-up or spin-down in this field. The difference in energy between these two states is ΔE = 2μBB, where μB is the Bohr magneton and B is the magnetic field created by the orbiting electron. The figure below shows your conclusion regarding the energy levels and your measured values for the two wavelengths in the doublet are λa = 6.544550 ✕ 10−7 m and λb = 6.544750 ✕ 10−7 m. (Let h = 6.626069 ✕ 10−34 J · s, c = 2.997925 ✕ 108 m/s, and μB = 9.274009 ✕ 10−24 J/T.) Determine the magnitude of the internal magnetic field (in T) experienced by the electron. When doing calculations, express all quantities in scientific notation, when possible keep six places beyond the decimal, and round your answer off to at least three significant figures at the end.

Answer 1

1.000153 T

Step-by-step explanation:

The energy change ΔE = hc(1/λb - 1/λa)

= 6.626069 ✕ 10⁻³⁴ J · s 2.997925 × 10⁸ m/s(1/6.544750 × 10⁻⁷ m - 1/6.544550 × 10⁻⁷ m)

= 19.864457907 × 10⁻²⁶(1527942.2438 - 1527988.9374) = 19.864457907 × 10⁻²⁶(-46.6936)

= 927.543052 × 10⁻²⁶

= -9.275431× 10⁻²⁴ J.

This energy change ΔE = 2μBB. So the magnetic field, B is

B = ΔE/2μB where μB = 9.274009 ✕ 10⁻²⁴ J/T

B = -9.275431× 10⁻²⁴ J/9.274009 ✕ 10⁻²⁴ J/T = -1.000153 T

The magnitude of the magnetic field B = 1.000153 T