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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.)

2. Consider a classical particle moving in a one-dimensional potential well u(x), as shown The particle is in thermal equilibrium with a reservoir at temperature T, so the probabilities of its various states are determined by Boltzmann statistics.

{a) Show that the average position of the particle is given by

where each integral is over the entire xaxis.

A one-dimensional potential well. The higher the temperature, the farther the particle will stray from the equilibrium point.

(b) If the temperature is reasonably low (but still high enough for classical mechanics to apply), the particle will spend most of its time near the bottom of the potential well. In that case we can expand u(x)in a Taylor series about the equilibrium point x0: u(x)=ux0+x-x0dudxx0+12x-x02d2udx2x0

+13!x-x03d3udx3x0+⋯

Show that the linear term must be zero, and that truncating the series after the quadratic term results in the trivial prediction x=x0.

(c) If we keep the cubic term in the Taylor series as well, the integrals in the formula for xbecome difficult. To simplify them, assume that the cubic term is small, so its exponential can be expanded in a Taylor series (leaving the quadratic term in the exponent). Keeping only the smallest temperature-dependent term, show that in this limit x differs from zo by a term proportional to kT. Express the coefficient of this term in terms of the coefficients of the Taylor series foru(x)

(d) The interaction of noble gas atoms can be modeled using the Lennard Jones potential,

u(x)=u0x0x12-2x0x6

Sketch this function, and show that the minimum of the potential well is at x=x0, with depth u0. For argon, x0=3.9Aand u0=0.010eV. Expand the Lennard-Jones potential in a Taylor series about the equilibrium point, and use the result of part ( c) to predict the linear thermal expansion coefficient of a noble gas crystal in terms of u0. Evaluate the result numerically for argon, and compare to the measured value

α=0.0007K-1(at80K)

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.

For a diatomic gas near room temperature, the internal partition function is simply the rotational partition function computed in section 6.2, multiplied by the degeneracy Zeof the electronic ground state.

(a) Show that the entropy in this case is

S=NkInVZeZrotNvQ+72.

Calculate the entropy of a mole of oxygen at room temperature and atmospheric pressure, and compare to the measured value in the table at the back of this book.

(b) Calculate the chemical potential of oxygen in earth's atmosphere near sea level, at room temperature. Express the answer in electron-volts

For an O2molecule, the constant ∈is approximately 0.00018eV. Estimate the rotational partition function for an O2molecule at room temperature.
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