The populations of proton spins in the highest energy level of a sample in magnetic fields of 1.5T and 7.0 T are N' and N, respectively. The value of $\ln \frac{N^{\prime}}{N} \operatorname{in}(\gamma, \mathrm{h}, \mathrm{k}, \mathrm{T}$ are gyromagnetic ratio of the proton, Planck's constant, Boltzmann constant and temperature of the sample, respectively, assume that the partition functions for both systems can be approximated as 1)
(a)$5.5 \frac{\gamma \mathrm{h}}{k T}$
(b)$\frac{3}{14} \frac{\gamma \mathrm{h}}{k T}$
(c)$\frac{14}{3} \frac{\gamma \mathrm{h}}{k T}$
(d)$8.5 \frac{\gamma \mathrm{h}}{k T}$
Answer
Answer: A ✓ checked by 4AB · confidence medium
Explanation
Take the lower proton-spin level as the energy zero (the convention under which the partition function q ≈ 1). The upper (highest) level then lies at $\Delta E=\gamma\hbar B$ (h in the options stands for ħ).Population of the upper level: $N_{\text{upper}}=\dfrac{e^{-\gamma\hbar B/kT}}{q}\approx e^{-\gamma\hbar B/kT}$. So $N'=e^{-\gamma\hbar(1.5)/kT}$ and $N=e^{-\gamma\hbar(7.0)/kT}$.