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Chapter 6: Q. 6.22P. (page 234)

In most paramagnetic materials, the individual magnetic particles have more than two independent states (orientations). The number of independent states depends on the particle's angular momentum “quantum number” j, which must be a multiple of 1/2. For j = 1/2 there are just two independent states, as discussed in the text above and in Section 3.3. More generally, the allowed values of the z component of a particle's magnetic moment are

Short Answer

Expert verified

The expression of Newton's law of motion is as follows,

Here is the mass of the object and is the acceleration of the object.

Free body diagram of the Rocket is as follows,

Step by step solution

01

Step 1. Given.

Apply the Newton's second law of motion along -axis.

Rearrange the above expression for

From the free body diagram of rocket, the expression of net force along -axis is as follows, (there is only thrust and gravitational force along -axis)

Combine the above expression of net force along -axis and the expression of acceleration along -axis,

02

Step 2. Given.

Rearrange the derived expression of acceleration (in part a.) to find the mass of the rocket,

Further solve, to find the expression for mass,

Substitute for for and for in the above expression of mass, the mass of rocket remains is as follows,

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Most popular questions from this chapter

The analysis of this section applies also to liner polyatomic molecules, for which no rotation about the axis of symmetry is possible. An example is CO2, with ∈=0.000049eV. Estimate the rotational partition function for a CO2molecule at room temperature. (Note that the arrangement of the atoms isOCO, and the two oxygen atoms are identical.)

Estimate the probability that a hydrogen atom at room temperature is in one of its first excited states (relative to the probability of being in the ground state). Don't forget to take degeneracy into account. Then repeat the calculation for a hydrogen atom in the atmosphere of the starγ UMa, whose surface temperature is approximately 9500 K.

In the numerical example in the text, I calculated only the ratio of the probabilities of a hydrogen atom being in two different states. At such a low temperature the absolute probability of being in a first excited state is essentially the same as the relative probability compared to the ground state. Proving this rigorously, however, is a bit problematic, because a hydrogen atom has infinitely many states.

(a) Estimate the partition function for a hydrogen atom at 5800 K, by adding the Boltzmann factors for all the states shown explicitly in Figure 6.2. (For simplicity you may wish to take the ground state energy to be zero, and shift the other energies according!y.)

(b) Show that if all bound states are included in the sum, then the partition function of a hydrogen atom is infinite, at any nonzero temperature. (See Appendix A for the full energy level structure of a hydrogen atom.)

(c) When a hydrogen atom is in energy level n, the approximate radius of the electron wavefunction is a0n2, where ao is the Bohr radius, about 5 x 10-11 m. Going back to equation 6.3, argue that the PdV term is Tot negligible for the very high-n states, and therefore that the result of part (a), not that of part (b), gives the physically relevant partition function for this problem. Discuss.

Fill in the steps between equations 6.51 and 6.52, to determine the average speed of the molecules in an ideal gas.

Cold interstellar molecular clouds often contain the molecule cyanogen (CN), whose first rotational excited states have an energy of 4.7x 10-4 eV (above the ground state). There are actually three such excited states, all with the same energy. In 1941, studies of the absorption spectrum of starlight that passes | through these molecular clouds showed that for every ten CN molecules that are in the ground state, approximately three others are in the three first excited states (that is, an average of one in each of these states). To account for this data, astronomers suggested that the molecules might be in thermal equilibrium with some "reservoir" with a well-defined temperature. What is that temperature?
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