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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

In this section we computed the single-particle translational partition function,Ztr, by summing over all definite-energy wave functions. An alternative approach, however, is to sum over all position and momentum vectors, as we did in Section 2.5. Because position and momentum are continuous variables, the sums are really integrals, and we need to slip a factor of 1h3to get a unitless number that actually counts the independent wavefunctions. Thus we might guess the formula

role="math" localid="1647147005946" Ztr=1h3d3rd3pe-EtrkT

where the single integral sign actually represents six integrals, three over the position components and three over the momentum components. The region of integration includes all momentum vectors, but only those position vectors that lie within a box of volume V. By evaluating the integrals explicitly, show that this expression yields the same result for the translational partition function as that obtained in the text.

Derive equation 6.92 and 6.93 for the entropy and chemical potential of an ideal gas.

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 partition function for the hypothetical system represented in Figure 6.3. Then estimate the probability of this system being in its ground state.

Although an ordinary H2 molecule consists of two identical atoms, this is not the case for the molecule HD, with one atom of deuterium (i.e., heavy hydrogen, 2H). Because of its small moment of inertia, the HD molecule has a relatively large value of ϵ:0.0057eV At approximately what temperature would you expect the rotational heat capacity of a gas of HD molecules to "freeze out," that is, to fall significantly below the constant value predicted by the equipartition theorem?

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