Please help with #7. Thanks What fraction of particles is in the ground state at

Question

Please help with #7. Thanks What fraction of particles is in the ground state at T = 300K? The nucleus of a hydrogen atom, a proton, has a magnetiemoment. In a magnetiefield, the proton has two states of different energy: spin up and spin down. This is the basis of proton NMR. The relative populations can be assumed to be given by the Boltzmann distribution, where the difference in energy between the two states is Delta epsilon = g m,u B, g = 2.79 for protons, and mu = 5.05 Times 10^-24 JT^-1. For a 300 MHz NMR instrument, B = 7T. Compute the relative population difference, |N_+ - N_-|/(N_+ + N_-), at room temperature for a 300 MHz machine. Describe how the population difference changes with temperature. What is the partition function?

Explanation / Answer

1. Nb / Na = e-[Eb - Ea] / kT
= e - gI mN B / kT =e^-2.79*5.05*10^-24*7/1.381x10-23 J/K*300 e - 1.9x10-4 = 0.9998
(Na-Nb)/(Na+Nb) = (1-0.9998) / (1+0.9998) = 9.4x10-5

2. With increase in temperature, molecules gain energy and jumps to higher energy levels thus the Boltzmann equilibrium shifts towards right side i.e., the population of higher energy states increases

3. Partition function tells about the statistical properties of a system under thermodynamic equilibrium. Partition functions are functions of the thermodynamic state variables, such as the temperature and volume. Most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives.

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