CP A large number of hydrogen atoms in Is states are placed...

CP A large number of hydrogen atoms in I$s$ states are placed in an external magnetic field that is in the $+z$ -direction.Assume that the atoms are in thermal equilibrium at room temperature, $T=300 \mathrm{K}$ . According to the Maxwell-Boltzmann distribution (see Section $39.4 ),$ what is the ratio of the number of atoms in the $m_{s}=\frac{1}{2}$ state to the number in the $m_{s}=-\frac{1}{2}$ state when the magnetic-field magnitude is (a) $5.00 \times 10^{-5} \mathrm{T}$ (approximately the earth's field); (b) $0.500 \mathrm{T} ;(\mathrm{c}) 5.00 \mathrm{T}$ ?

Key Concepts

Maxwell-Boltzmann Distribution

This distribution describes how the populations of particles or atoms are spread among various energy states when the system is in thermal equilibrium. In the context of magnetic systems, it quantifies how the probability of finding an atom in a particular spin state depends on the energy difference between that state and others, scaled by the temperature. It is fundamental in calculating the relative numbers of atoms in different magnetic sublevels under an external magnetic field.

Zeeman Splitting

Zeeman splitting refers to the splitting of atomic energy levels into multiple components when subjected to an external magnetic field. This occurs because the magnetic dipole moments associated with electron spins interact with the magnetic field, causing shifts in energy that depend on the orientation of the magnetic moment relative to the field. This process is essential for analyzing the energy differences between spin states in problems involving magnetic fields.

Magnetic Energy of a Spin

The magnetic energy of a spin arises from the interaction between a particle’s magnetic moment and an external magnetic field. For a given spin state, the energy is determined by the product of the magnetic moment component along the field direction and the magnetic field strength. This concept is central to understanding how different spin states (such as m_s = +1/2 and m_s = -1/2) have different energies in a magnetic field, leading to a population imbalance as described by the Maxwell-Boltzmann distribution.

Thermal Equilibrium

Thermal equilibrium is the state in which all parts of a system have reached a uniform temperature and the energy distribution among the available states is stable. In the context of magnetic atoms, it means that the populations of the spin states are determined solely by their energy differences and the temperature, following the Maxwell-Boltzmann statistics.

Boltzmann Factor

The Boltzmann factor, expressed as exp(-?E/kT), quantifies the ratio of probabilities for occupancy of two energy states differing by an energy ?E at a temperature T. It is a key element in statistical mechanics and is used to compute the relative number of atoms in different spin states in a magnetic field, where even small energy differences lead to measurable differences in state populations under thermal conditions.

Transcript

00:01 We're told we have a hydrogen atom in the 1s state placed in an external magnetic field in the z hat direction.

00:06 The temperature t is 300 kelvin, and the spin quantum number, m sub s, is going to be plus or minus a half because we're dealing with electrons.

00:15 So we know the formula for potential energy, u, is equal to 2 .00232, muice of b, b, b, b times m sub s times the magnetic field b.

00:27 Well, m sub s here can be plus or minus a half.

00:30 So the difference between the potentials when mu plus here, u plus here is when m sub s is plus a half, and mu minus is when ms of s is minus a half, we subtract those from one another.

00:42 Since both terms are going to have this 2 .0023, mu, mu, sub b times m sub s, or excuse me, mu sub b, since both terms are going to have that, and both terms are also going to have the b.

00:53 We can pull that out.

00:54 So really you're subtracting the m sub -s terms.

00:57 So this says one -half minus negative one -half, which is equal to one.

01:02 So since the theory from the chapter says that the number of atoms in a given state, in a plus, divided by the number of atoms in the n -minus state, is equal to the exponential to the negative difference between the two energy states, which we have written above here, again, where one -half minus negative, one -half is one.

01:27 Divided by a bolt's constant k times the temperature t.

01:31 So plugging in mucebbb, being boers magnetron number of 9 .274 times 10 to the minus 24 joules per tesla, plugging in our value for the temperature, and then plugging in the value for the boltspin constant, k of 1 .38 times 10 to the minus 23 joules per kelvin, we get negative 4 .48 times 10 to the minus 3, inverse tesla times the magnetic field number or times the magnetic field b.

01:59 So for part a, let's switch back to black here...

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