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Chapter 1: Q. 1.25 (page 17)

List all the degrees of freedom, or as many as you can, for a molecule of water vapor. (Think carefully about the various ways in which the molecule can vibrate.)

Short Answer

Expert verified

Total degree of freedom for water vapor is 3.

Step by step solution

01

Given information

water vapor molecule is given.

02

Explanation

The H-O bonds enclose 105 degrees, so the molecule is definitely nonlinear.
The rotational modes are around three axes, two of them in the molecular plane and one perpendicular to it.
We can see these rotational axis in the figure as below

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Most popular questions from this chapter

Imagine some helium in a cylinder with an initial volume of 1 liter and an initial pressure of 1 atm. Somehow the helium is made to expand to a final volume of 3 liters, in such a way that its pressure rises in direct proportion to its volume.

  1. Sketch a graph of pressure vs. volume for this process.
  2. Calculate the work done on the gas during this process, assuming that there are no 鈥渙ther鈥� types of work being done.
  3. Calculate the change in the helium鈥檚 energy content during this process.
  4. Calculate the amount of heat added to or removed from the helium during this process.
  5. Describe what you might do to cause the pressure to rise as the helium expands.

A 60-kghiker wishes to climb to the summit of Mt. Ogden, an ascent of5000vertical feet(1500m) .
aAssuming that she is 25%efficient at converting chemical energy from food into mechanical work, and that essentially all the mechanical work is used to climb vertically, roughly how many bowls of corn flakes (standard serving size 1ounce,100 kilocalories) should the hiker eat before setting out?
b As the hiker climbs the mountain, three-quarters of the energy from the corn flakes is converted to thermal energy. If there were no way to dissipate this energy, by how many degrees would her body temperature increase?
cIn fact, the extra energy does not warm the hiker's body significantly; instead, it goes (mostly) into evaporating water from her skin. How many liters of water should she drink during the hike to replace the lost fluids? (At25鈭�C, a reasonable temperature to assume, the latent heat of vaporization of water is 580cal/gmore than atrole="math" localid="1650290792399" 100鈭�C).

For a solid, we also define the linear thermal expansion coefficient, 伪, as the fractional increase in length per degree:

伪鈮�螖L/L螖T
(a) For steel, 伪 is 1.1 x 10-5 K-1. Estimate the total variation in length of a 1 km steel bridge between a cold winter night and a hot summer day.
(b) The dial thermometer in Figure 1.2 uses a coiled metal strip made of two different metals laminated together. Explain how this works.
(c) Prove that the volume thermal expansion coefficient of a solid is equal to the sum of its linear expansion coefficients in the three directions 尾=伪x + 伪y + 伪z. (So for an isotropic solid, which expands the same in all directions, 尾 =3 伪 .)

Even at low density, real gases don鈥檛 quite obey the ideal gas law. A systematic way to account for deviations from ideal behavior is the virial

expansion,

PV鈭�nRT(1+B(T)(V/n)+C(T)(V/n)2+鈰�)

where the functions B(T), C(T), and so on are called the virial coefficients. When the density of the gas is fairly low, so that the volume per mole is large, each term in the series is much smaller than the one before. In many situations, it鈥檚 sufficient to omit the third term and concentrate on the second, whose coefficient B(T)is called the second virial coefficient (the first coefficient is 1). Here are some measured values of the second virial coefficient for nitrogen (N2):

T(K)
B(cm3/mol)
100鈥�160
200鈥�35
300鈥�4.2
4009.0
50016.9
60021.3
  1. For each temperature in the table, compute the second term in the virial equation, B(T)/(V/n), for nitrogen at atmospheric pressure. Discuss the validity of the ideal gas law under these conditions.
  2. Think about the forces between molecules, and explain why we might expect B(T)to be negative at low temperatures but positive at high temperatures.
  3. Any proposed relation between P, V, andT, like the ideal gas law or the virial equation, is called an equation of state. Another famous equation of state, which is qualitatively accurate even for dense fluids, is the van der Waals equation,
    (P+an2V2)(V鈭�nb)=nRT
    where a and b are constants that depend on the type of gas. Calculate the second and third virial coefficients (Band C) for a gas obeying the van der Waals equation, in terms of aand b. (Hint: The binomial expansion says that (1+x)p鈮�1+px+12p(p鈭�1)x2, provided that |px|鈮�1. Apply this approximation to the quantity [1鈭�(nb/V)]鈭�1.)
  4. Plot a graph of the van der Waals prediction for B(T), choosing aand bso as to approximately match the data given above for nitrogen. Discuss the accuracy of the van der Waals equation over this range of conditions. (The van der Waals equation is discussed much further in Section 5.3.)

Geologists measure conductive heat flow out of the earth by drilling holes (a few hundred meters deep) and measuring the temperature as a function of depth. Suppose that in a certain location the temperature increases by20鈭�Cper kilometer of depth and the thermal conductivity of the rock is 2.5W/m鈰�K. What is the rate of heat conduction per square meter in this location? Assuming that this value is typical of other locations over all of the earth's surface, at approximately what rate is the earth losing heat via conduction? (The radius of the earth is 6400km.)
See all solutions

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