17.17.What temperature is necessary to have twice as many
atoms in the ground state as in the first excited state, at
16.4 cm^1 , of C atoms? What temperature is necessary to
have equal populations in the ground state and the second ex-
cited state, at 43.5 cm^1? What temperature is necessary to
have equal populations in the first and second excited states?
The degeneracies of the ground, first, and second excited
states are 1, 3, and 5, respectively. (Note that these equilib-
rium ratios would not be possible at any temperature if the
degeneracies were equal.)
17.5 Thermodynamic Properties
17.18.Several times it has been mentioned that qis a con-
stant, but the expression for energy (as well as many other
thermodynamic functions) contains the derivative of q(or the
derivative of ln q). The derivatives of constants are zero. Why
aren’t thermodynamic state functions equal to zero, then?
17.19. (a)On the basis of their statistical thermodynamic de-
finition, which energy has the higher absolute value, Aor G?
(b)On the basis of their statistical thermodynamic definition,
can you tell which energy has the higher absolute value, Eor
G? Why or why not?
17.20.By following the steps outlined in the text, derive
equation 17.42 from equation 17.41.
17.21.For a chemical system with more than one component,
what is the restriction on the derivation of equation 17.46?
17.22.Derive equations 17.44 and 17.45.
17.23.Use statistical thermodynamic arguments to justify the
second-law spontaneity of the following situations. (a)A gas
expands as the volume of a system increases adiabatically.
(b)Ice is the unstable phase of H 2 O at 5°C.
17.24.Equations 17.44 and 17.45 for Aand Gdiffer only by
the 1 term in the definition of A. Where does this term come
from? (See exercise 17.22 above.)
17.25.Using L’Hôpital’s rule, determine the limit of Sas
T→0 and show that it equals kln g 0.
17.6 & 17.7 Monatomic Gases and
State Functions
17.26.Do a strict units analysis of equation 17.53 by break-
ing down all the units of all the quantities into their basic units
and show that they all cancel.
17.27.What change is there in the Sackur-Tetrode equation
if NNA?
17.28.In calculating thermodynamic properties for 1 mole of
a monatomic gas, we use the mass of a single atom, not the
mass of a mole of atoms. Explain why.
17.29.Verify equation 17.56, starting with equation 17.34.
17.30.Derive the Sackur-Tetrode equation, equation 17.61.
17.31.Calculate the thermal de Broglie wavelength of He at
25 K and 500 K. Are the different values to be expected?
17.32.Explain why the calculated value for the absolute en-
tropy of Kr at 120 K might not be very close to the experi-
mental value, even though the boiling point of Kr is 119.8 K.
17.33.Calculate Sfor (a)C atoms at 1000 K, (b)Fe atoms
at 3500 K, and (c)Hg atoms at 298 K. Compare your calcu-
lated values to 183.2, 239.6, and 174.9 J/(molK), respectively.
Assume 1 atm pressure. Can you explain the trend in agree-
ment between calculation and experiment?
17.34.Use equation 17.56 to determine the change in en-
ergy, E, when 1 mole of Ar is heated from 298 K to 348 K at
constant volume. Compare this result with the change in en-
ergy calculated using (mass)(specific heat)(change in temper-
ature). You will need to look up the specific heat of argon; see
the table of thermodynamic values in Appendix 2.
17.35.For an electron that has a velocity of 0.01c(where c
is the speed of light), at what temperature will its thermal de
Broglie wavelength equal its quantum-mechanical de Broglie
wavelength? (Note that the original de Broglie wavelength is
not directly dependent on temperature.)
17.36.Use the Sackur-Tetrode equation to derive the rela-
tionship SRln (V 2 /V 1 ) for an isothermal change and
SCVln (T 2 /T 1 ) for an isochoric change.17.37.Calculate the logarithm of N!, N1 to 100, explic-
itly and using Stirling’s approximation, and compare the
values. At what approximate value of Ndoes Stirling’s ap-
proximation agree with the true value to within 1%?
17.38.Consider a system of five energy levels, each of which
are doubly degenerate. The levels have energies of 0,
1 10 ^21 , 2.5 10 ^21 , 4 10 ^21 , and 6 10 ^21 J. Calculate
the partition function of this system at 50, 100, 200, 300, 500,
and 1000 K. Do you see a leveling off of the value of qas
the temperature increases? What is the interpretation of the
values of q?
17.39.Use a symbolic math program to take the symbolic
limit of equation 17.52 as napproaches infinity, and compare
the result to equation 17.54.
17.40.Program the Sackur-Tetrode equation, 17.61, into a
calculator or computer and calculate the molar entropy of all
noble gases at 298 and 1000 K.Exercises for Chapter 17 615Symbolic Math Exercises