Chapter 16, Problem 16.1P

Interpretation Introduction

Interpretation:

It is to be explained at which magnetic field experienced by a hydrogen nucleus is the energy difference between the $\text{α}$ and the $\text{β}$ states is greater: $\text{0}\text{.5 T or 5}\text{.0 T}$.

Concept introduction:

Protons have quantized magnetic dipoles and thus behave as tiny bar magnets, with a north and a south pole. When exposed to ${\text{B}}_{\text{ext}}$, the magnetic dipoles of nuclei in the $\text{α}$ spin state become aligned with ${\text{B}}_{\text{ext}}$, whereas those of nuclei in the $\text{β}$ spin state become aligned against it. In these orientations, the energy of the $\text{α}$ state is slightly lower than the energy of the $\text{β}$ state.

The difference in energy between the two spin states of a proton is given by the equation:

${\text{ΔE}}_{\text{spin}}\text{ =}\frac{{\text{ γhB}}_{\text{ext}}}{\text{2π}}{\text{ = (constant) × B}}_{\text{ext}}$

The energy difference, ${\text{ΔE}}_{\text{spin}}$, is directly proportional to the external magnetic field ${\text{B}}_{\text{ext}}$.

Answer to Problem 16.1P

The energy difference will be greater at $\text{5}\text{.0 T}$ than at $\text{0}\text{.5 T}$.

Explanation of Solution

The difference in energy between the two spin states of a proton is given by the equation:

${\text{ΔE}}_{\text{spin}}\text{ =}\frac{{\text{ γhB}}_{\text{ext}}}{\text{2π}}{\text{ = (constant) × B}}_{\text{ext}}$

The energy difference, ${\text{ΔE}}_{\text{spin}}$, is directly proportional to the external magnetic field ${\text{B}}_{\text{ext}}$, that is, greater the external magnetic field, greater the energy difference.

Thus, at greater external magnetic field, $\text{5}\text{.0 T}$, greater is the energy difference between the $\text{α}$ and the $\text{β}$ states for hydrogen nucleus.

Conclusion

The magnetic field experienced by hydrogen nucleus, at which the energy difference between the $\text{α}$ and the $\text{β}$ states is greater is explained using the equation for the difference in energy between the two spin states of a proton.