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# PROBLEMSFORCHAPTER16

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PROBLEMSFORCHAPTER16

Problems for New Ch 16 [Old Ch 12] 1

PROBLEMS FOR CHAPTER 16

SECTION 16.2 (NUCLEAR PROPERTIES)

16.1 [old 12.1] ) Use Appendix D (XX) to find the numbers of protons and neutrons in

147204063206the following nuclei: H, He, Li, Ne, Ar, Cu, Pb.

16.2 [old 12.2] ) Use Appendix D to find the numbers of protons and neutrons in the

2492964209232following nuclei: H, He, Be, P, Zn, Bi, Th.

16.3 [old 12.2+] • Use Appendix D to find the chemical symbols and the natural percent abundances of the four isotopes with (a) Z = 2, A = 3; (b) Z = 6, A = 13; (c) Z =

26, A = 56; (d) Z = 82, A = 206.

16.4 [old 12.2++] • Use Appendix D to find the chemical symbols and the half-lives of

the four radioactive nuclei with (a) Z = 1, A = 3; (b) Z = 6, A = 14; (c) Z = 19, A = 40;

(d) Z = 94, A = 244.

16.5 [old 12.3] ) (a) Use Appendix D to make a list of all the stable isotopes, isobars,

702788and isotones of Ge. (b) Do the same for Al and (c) Sr.

16.6 [old 12.4] ) Make lists of all the stable isotopes, isobars, and isotones of each of

5636208the following: Fe, S, Pb.

1216.7 [old 12.5] ) Find the mass of the C atom in Appendix D. What fraction of its

mass is contained in its atomic electrons?

4He16.8 [old 12.5+] • What percent of the mass of the atom is contained in its

nucleus?

2516.9 [old 12.6] ) The mass of a lead atom's nucleus is 3.5 ? 10 kg and its half-density

3radius R is 6.5 fm. Estimate its density by assuming its volume to be 4R. Compare

16.10 [old 12.7] ) Atomic radii range from. a little less than 0.1 nm to 0.3 nm, while nuclear radii range from about 2 fm to about 7 fm. Taking the representative values R 0.2 nm and R 4.5 fm, find the ratio of the volume of an atom to that of a atomnuc

nucleus. What is the ratio of the corresponding average densities?

1/316.11 [old 12.8-] • Use the formula RRA, with , to find the R1.07fm00

approximate radii of the most abundant isotopes of carbon, iron and lead. (Mass numbers and abundances are in Appendix D.)

16.12 [old 12.8] ) The nuclear radius R is well approximated by the formula R =

1/3RA, where R = 1.07 fm. Find the approximate radii of the most abundant isotopes of 00

each of the following elements: Al, Cu, and U. (Mass numbers and abundances can be found in Appendix D.)

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Problems for New Ch 16 [Old Ch 12] 2

716.13 [old 12.9] ) The first excitation energy of the Li nucleus is 0.48 MeV. What is

7the wavelength of a photon emitted when a Li nucleus drops from its first excited state

to the ground state? Find the wavelength for the corresponding transition in the Li atom (excitation energy, 1.8 eV).

416.14 [old 12.10] ) The first excitation energy of the helium nucleus, He, is about 20

MeV. What is the wavelength of a photon that can just excite the helium nucleus from its ground state? Compare this with the wavelength of a photon that can just excite a helium atom (excitation energy 19.8 eV).

16.15 [old 12.11] )) (a) Treating the nucleus as a uniform sphere of radius (1.07

1/3356fm)A, calculate the density (in kg/m) of the Fe nucleus and compare it to that of

3solid iron (7800 kg/m). (b) Calculate the electric charge density of this nucleus (Z = 26)

3in coulombs/m. Compare this charge density to that which can be uniformly distributed in an insulating sphere of 1 m radius in air without sparking. The maximum

6electric field strength allowable at the surface of the sphere without sparking is 3 ? 10

V/m.

16.16 [old 12.12] )) The density of mass inside a nucleus is shown in Fig. 16.2. There is no simple theory that predicts the exact shape shown, but it is found that the shape can be approximated by the following mathematical form, known as the Fermi. function:

0 (16.45) ().rrRt()/1;e

173Here is a constant, = 3.25 ? 10 kg/m, R is the nuclear radius, and t is another 00

constant, t = 0.55 fm. Plot the function (16.45) for a nucleus of radius R = 5 fm. By all

means use computer software to plot this, if you have such. If you do it by hand, use your calculator to find (r) for r = 0, 5 fm, and 10 fm and then choose a suitable grid of intermediate points.

16.17 [old 12.13] )) The density (r) in a nucleus is well approximated by the formula

173(16.45) in Problem 16.16 [old 12.12], with = 3.25 ? 10kg/m, t = 0.55 fm, and R = 0

1/3208RA, where R = 1.07 fm. Plot (r) as a function of r from r = 0 to r = 15 fm for Pb. 00

16.18 [old12.14] )) The density in a nucleus can be approximated by the function (16.45) in Problem 16.16 [old 12.12]. (a) Prove that = /2 at r = R. (b) Show that 0

the maximum value of occurs at r = 0 and is very close to if R is much greater than 0

t (as it usually is). (c) Sketch as a function of r. (d) Prove that as r increases, the den-

sity falls from 90% to 10% of in a distance r = 4.4t. (Thus t characterizes the 0

thickness of the surface region.)

12C16.19 [Old 12.14+] •• Repeat Problem 16.17 [old 12.13] for the two nuclei and

244Pu, drawing both graphs on the same plot.

SECTION 16.3 (THE NUCLEAR FORCE)

16.20 [old 12.15] ) If there were no nuclear attraction, what would be the acceleration of either of two protons released at a separation of 4 fm? Compare your answer with g,

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Problems for New Ch 16 [Old Ch 12] 3

the acceleration of gravity. (Large as your answer is, it is still small compared to nuclear

accelerations. See Problem 16.24 [old 12.18].)

16.21 [old 12.16] ) Use Eq. (16.7) to estimate the minimum kinetic energy of a nucleon in a nucleus of diameter 8 fm.

16.22 [old 12.17-] •• Use Eq. (16.7), taking the size a to be twice the radius given by

12C(16.3), to find the minimum kinetic energy of a nucleon inside each of the nuclei and

244Pu.

16.23 [old 12.17] )) (a) The minimum kinetic energy K of a nucleon in a rigid min

cubical box of side a is given by (16.7). By setting a 2R, the nuclear diameter, and R =

1/3R A (with R = 1.07 fm), derive a formula for K as a function of the mass number A. 00min927238(b) Find K for Be, Al, and U. min

16.24 [old 12.18] )) A representative value for the kinetic energy of a nucleon in a nucleus is 20 MeV. To illustrate the great strength of the nuclear force, do the following classical calculation: Suppose that a nucleon in a nucleus oscillates in simple

harmonic motion with amplitude about 4 fm and peak kinetic energy 20 MeV. (a) Find

its maximum acceleration. (b) By how many orders of magnitude does this exceed the acceleration of gravity, g? (c) Find the force required to produce this acceleration. 16.25 [old 12.19] )) The charge independence of nuclear forces implies that in the

77absence of electrostatic forces, the energy levels of Li and Be would be the same. The

7main effect of the electrostatic forces is simply to raise all of the levels of Be compared

7to those of Li. Approximating both nuclei as uniform spheres of charge Q = Ze and the

7same radius R, estimate the difference in the binding energies for any level of and Be47the corresponding level of . (The electrostatic energy of a uniform charged sphere is Li3

23kQ/5R see Problem 16.49 [old 12.43]. The observed radius R is about 2.5 fm.)

Compare your rough estimate with the observed difference of about 1.7 MeV. (Note that the true charge distribution is not a uniform sphere with a well-defined radius R, but is

spread out somewhat beyond R. Therefore, the observed electrostatic energies would be expected to be somewhat smaller than your estimates.)

16.26 [old 12.20] )) Do the same calculations as in Problem 16.25 [old 12.19] but for

2323the nuclei Na and Mg. The observed difference is 4.8 MeV; use the formula R = R 01/3A to get the nuclear radius.

SECTION 16.4 (ELECTRONS VERSUS NEUTRONS AS NUCLEAR CONSTITUENTS)

16.27 [old 12.21] ) One argument against the proton-electron model of the nucleus

1concerns the total spins of nuclei. The proton, electron, and neutron all have spin , 2

and the total spin of any number of spin-half particles takes the familiar form

1ss(1);. If there is just one particle, then of course, s = . With two particles, the 2

spins can be parallel or antiparallel, giving s = 1 or 0. It can be shown that the general rule is this: For an odd number of particles, the total spin has some half-odd-integer

351value for s (s = , , , . . .). For an even number of particles, the total spin has s equal 222

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Problems for New Ch 16 [Old Ch 12] 4

14to some integer (s = 0, 1, 2, . . .). In light of this rule consider the nucleus N. (a)

14Assuming that N is made of 7 protons and 7 neutrons, predict the character of its total

14spin (integer or half-odd integer). (b) Repeat for the proton-electron model of N. (c)

14The observed total spin of N is integer; which model does this support?

216.28 [old 12.22] ) Repeat Problem 16.27 [old 12.21] for the deuteron H, whose

observed total spin is given by s = 1.

16.29 [old 12.23] )) Supposing that nuclei could contain electrons, estimate the

27potential energy of one such electron at the center of a nucleus of Al. Compare your

answer with the minimum kinetic energy (16.10) of an electron in a nucleus. [Hint: The

required potential energy is eV(0), where V(r) is given by Eq. (16.49) below. The radius

R is given by (16.3), and the total charge acting on the electron is Q = (Z + 1)e.]

16.30 [old 12.24] )) (a) Following the suggestions in Problem 16.29 [old 12.23], find

12the potential energy that an electron would have at the center of a nucleus. (b) C6

208Repeat for . (c) Compare your answers with the minimum kinetic energy, which is Pb82

of order 100 MeV.

16.31 [old 12.25] )) Because of the proton's magnetic moment, every energy level of the hydrogen atom, as calculated previously, actually consists of two levels, very close together. The proton's magnetic moment creates a magnetic field, which means that the energy of the H atom is slightly different depending on the relative orientations of the electron and proton moments an effect known as hyperfine splitting. You can esti-

mate the magnitude of the hyperfine splitting as follows: (a) The magnetic field at a

distance r from a magnetic moment ( is

((o (16.46) B32r

72where ( = 4 ? 10 N/A is the permeability of space. [For simplicity, we have given 0

the field at a point on the axis of (. At points off the axis the field is somewhat different, but is close enough to (16.46) for the purposes of this estimate.] Given that the magnetic moment of the proton is roughly equal to the nuclear magneton ( defined in N

(16.12), estimate the magnetic field at a distance a from a proton. (b) Taking your B

answer in part (a) as an estimate for the B field "seen" by an electron in the 1s state of a

6hydrogen atom, show that the atom's energy differs by roughly 10 eV for the cases

that the proton and electron spins are parallel or antiparallel. [The energy of a magnetic moment in a B field is given by (9.XX) (old10.10).] (c) This means that the 1s

state of hydrogen is really two very closely spaced levels. Compare your rough estimate with the actual separation of these levels given that the photon emitted when a hydrogen atom makes a transition between them has = 21 cm. (This 21-cm radiation

is used by astronomers to identify hydrogen atoms in interstellar space.) 16.32 [old 12.26] ))) It is often a useful approximation to treat a nucleus as a uniformly charged sphere of radius R and charge Q. In this problem you will find the

electrostatic potential V(r) inside such a sphere, centered at the origin. (a) Write down

the electric field E(r) at any point a distance r from the origin with r > R. (Remember

that this is the same as the field of a point charge Q at the origin.) (b) Use the definition

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Problems for New Ch 16 [Old Ch 12] 5

r2？？；？；VVrVrErdr()()() (16.47) 21?r1

to find the potential difference between points at r and r (both greater than R). It is 12

usual to define V(r) so that V(r) approaches 0 as r ~ . By choosing r = and r = r, 12

show that

kQ (16.48) VrrR()()？?r

2 (c) Now find E(r) for r R. (Remember that Gauss's law tells us this is kQ'/r, where Q'

is the total charge inside the radius r.) Check that your answers for parts (c) and (a)

agree when r = R. (d) Use Eq. (16.47) to find V(r) V(r) for any two points inside the 21

sphere. Now choose r = r and r = R, and use the value of V(R) from part (b) to prove 21

that

2：?kQr. (16.49) VrrR()3()？；：，?22RR??

In Problems 16.29 [old 12.23], 16.30[old 12.24], 16.33 [old 12.27], and 16.34 [old

12.28] this result is needed to find the potential energy of a charge inside a nucleus.

SECTION 16.5 (THE IPA POTENTIAL ENERGY FOR NUCLEONS) 16.33 [old 12.27] ) (a) Use the result (16.49) of Problem 16.32 [old 12.26] to estimate

12the electrostatic potential energy of a proton at the center of a C nucleus. (b) Repeat

208for a nucleus of Pb.

16.34 [old 12.28] )) The electrostatic potential energy U of a proton in a nucleus Coul

1)e can be approximated by assuming that it moves in a uniform sphere of charge (Z

and radius R. (a) Use the results (16.48) and (16.49) of Problem 16.32 [old 12.26] to

write down and sketch U(r). (b) Prove that U (0) = 1.5 U(R). (c) Setting R = CoulCoulCoul1/34238(1.07 fm)A, estimate U(0) for a proton in He and for a proton in U. Coul

SECTION 16.6 (THE PAULI PRINCIPLE AND THE SYMMETRY EFFECT)

121216.35 [old 12.29] ) Enlarge Fig. 16.11 to include the nuclei Be and O. By how much (in terms of the quantity E shown in Fig. 16.11) does the energy of each isobar with Z

12? N exceed the energy of C?

SECTION 16.7 (THE SEMIEMPIRICAL BINDING-ENERGY FORMULA)

1216.36 [old 12.30] ) Use the data of Appendix D to find the binding energy of C.

416.37 [old 12.31] ) (a) Find the mass (in u) of the He atom in Appendix D. (b) Find

4the mass of the He nucleus to seven figures (but ignore corrections due to the atomic

electrons' binding energy). (c) Do any of these seven figures change if you take into

account the electrons' binding energy (about 80 eV total)?

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Problems for New Ch 16 [Old Ch 12] 6

416.38 [old 12.32] ) Use the data of Appendix D to find the binding energies of He,

40120Ca, and Sn.

16.39 [old 12.33] ) Use the data of Appendix D to deduce the energy required to

1413remove one neutron from C to produce C and a neutron at rest, well separated from

13the C nucleus.

16.40 [old 12.34] )) (a) Use conservation of energy to write an equation relating the mass of a nucleus (Z, N) and its neutron separation energy S(Z, N) to the masses of the n13nucleus (Z, N 1) and the neutron. (b) Use the data in Appendix D to find S for C, n120200Sn, and Hg.

16.41 [old 12.35] )) (a) The proton separation energy S (energy to remove one proton) p198197for Hg is 7.1 MeV. Given that the total binding energy of Au is 1559.4 MeV, find

198directly from the mass of Hg given in Appendix D.

16.42 [old 12.36] )) (a) To estimate the importance of the surface term in the

27binding-energy formula, consider a model of Al that consists of 27 identical cubical

nucleons packed into a 3 ? 3 ? 3 cube. Each small cube has six faces. What fraction of all

125these faces are exposed on the nuclear surface? (b) Repeat for Te considered as a 5 ? 5

16.43 [old 12.37] )) The Coulomb energy of the charges in a nucleus is positive and hence reduces the binding energy B. If we approximate the nucleus as a uniform sphere of charge Q = Ze, then according to Eq. (16.50) of Problem 16.49 [old 12.43], the

Coulomb correction to B should be

23()kZeBU？；？；. CoulCoul5R

1/3Putting R = RA, show that this agrees with the form (16.26) of the Coulomb 0

correction given in Section 16.7. Use your answer here to calculate a in MeV and Coul

compare with the empirical value a = 0.711 MeV. (You should not expect precise Coul

agreement since the nucleus is certainly not a perfectly uniform sphere, but you should

get the right order of magnitude.)

16.44 [old 12.38] )) (a) Use the binding-energy formula (16.30) to predict the binding

40energy of Ca. (b) Compare your answer with the actual binding energy found from the masses listed in Appendix D.

16.45 [old 12.39] )) Use the semiempirical binding-energy formula to find B/A for

2060259Ne, Ni, No. Make a table showing all five terms for each nucleus and comment on their relative importance. Do your final answers follow the general trend of Fig. 16.14?

16.46 [old 12.40] )) Use the binding-energy formula (16.30) to find the total binding

56565656energy of NiCoFeMn, , and . Which terms are different in the four cases, and 28272625

why?

23523616.47 [old 12.41] )) If U captures a neutron to form U in its ground state, the

236235energy released is B(U) B(U). (a) Prove this statement. (b) Use the bind-

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Problems for New Ch 16 [Old Ch 12] 7

ing-energy formula (16.30) to estimate the energy released, and compare with the

236observed value of 6.5 MeV. (Note: We have assumed here that U is formed in its

ground state, and the 6.5 MeV is carried away, by a photon, for example. An important

236alternative is that U can be formed in an excited state, 6.5 MeV above the ground

state. This excitation energy can lead to oscillations that cause the nucleus to fission.)

16.48 [old 12.42] )) Compare the surface area of a sphere to that of a cube of the same volume. How much energy, in MeV, would be needed to distort the normally spherical 60Ni nucleus into a cubical shape if the volume and Coulomb energies changed by negligible amounts?

16.49 [old 12.43] ))) In this problem you will calculate the electrostatic energy of a uniform sphere of charge Q with radius R. The electric potential V(r) at any radius r < R

is given by (16.49) in Problem 16.32 [old 12.26], and the potential energy of a charge q

at radius r is qV(r). (a) Write down the total charge contained between radius r and r +

dr, and hence find the potential energy of that charge. (b) If you integrate your answer

to (a) from r = 0 to r = R, you will get twice the total potential energy of the whole sphere, since you will have counted the energy of any two charge elements twice. Show that the total Coulomb energy of the sphere is

23kQU (16.50) Coul5R

16.50 [old 12.44] ))) (a) Substitute the binding-energy formula (16.30) into the relation

B mZmNm？;；nucpn2c

to obtain the semiempirical mass formula. [To simplify matters, take A to be odd, so that

the pairing term in (16.30) is zero.] (b) Among any set of isobars, the nucleus with lowest mass is the most stable. To identify this nucleus, first write your mass formula in

terms of the variables A and Z (that is, replace N, wherever it appears, by A Z). Now,

with A fixed, differentiate m with respect to Z. The minimum mass is determined by the condition (m/(Z = 0. Show that this leads to a relation of the form

A1;Z (16.51) 2/321;A

where and are related to the coefficients (16.31) as follows:

2()mmcanpCoul and . 4a4asymsym

(c) For A = 37, 115, 185 find the values of Z that give the most stable nuclei and

compare with the observed values from Appendix D.

16.51 [old 12.45] ))) (a) Use the formula (16.51) of Problem 16.50 [old 12.44] to make a table showing the most stable values of Z and N for A = 25, 45, 65,. . . 245.(b)

Plot a curve of Z versus N for the most stable nuclei found in this way, and compare your picture with the observed curve in Fig. 16.1.

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Problems for New Ch 16 [Old Ch 12] 8

;SECTION 16.8 (THE SHELL MODEL)

16.52 [old 12.46] ) Which of the following nuclei has a closed subshell for Z? Which

10111727284248for N? B, B, O, Al, Si, Ca, Ca.

16.53 [old 12.47] ) Use the levels shown in Fig. 16.17 to predict j for the ground tot404148states of the nine nuclei Ca, Ca, ... Ca. Compare with the data in Appendix D.

16.54 [old 12.48] ) Use the levels shown in Fig. 16.17 to predict the total angular

2933375971momenta of the following nuclei: Si, S, Cl, Co, and Ga.

3916.55 [old 12.49] ) Use the shell model to predict j for the ground states of , Ktot1920

4041, and . CaSc20202120

3716.56 [old 12.50] ) The ground state of Li has j = and its first excited state has j tot2

1=. Explain these observations in terms of the shell model. 2

16.57 [old 12.51] ) (a) Use the levels shown in Fig. 16.17 to draw an occupancy

7diagram like Fig. 16.10(b) for the ground state of Be. (b) What is j? (c) What would tot

you predict for the first excited state? (Draw an occupancy diagram and give j.) tot

1716.58 [old 12.52] ) Answer the same questions as in Problem 16.57 [old 12.51] for F.

16.59 [old 12.53] ) In applying the shell model, we usually use the level ordering of Fig. 16.17, which is based on an assumed spherical potential well. For certain nonspherical nuclei this assumption is incorrect and the ordering of the levels is

19121different. Examples are F and Sb; check this by using the rule (16.41) and Fig. 16.17 to predict the angular momenta of these two nuclei and comparing with the observed values in Appendix D.

16.60 [old 12.54] )) (a) Use the energy levels of Fig. 16.17 to draw an occupancy

13diagram like Fig. 16.10(b) for C. (b) What is j for the ground state? (c) The first tot

excited state is formed by lifting one neutron from 1p to 1p. Draw an occupancy 3/21/2

diagram for this state and explain what you would predict for j of this state. (Does tot3your answer agree with the observed value, j = ?) (d) Now predict j for the ground tottot213and first excited states of N. (In this case the excitation involves a proton.) 16.61 [old 12.55] )) For each choice of n and l (with l ? 0) a nucleon has two different

1possible energy levels with j = . Prove that the sum of the degeneracies of these l?2

two levels is equal to the total degeneracy that the level nl would have had in the

absence of any S?L splitting.

16.62 [old 12.56] ))) The first magic number for both nuclei and atoms is 2. The second is 8 for nuclei and 10 for atoms. The difference occurs because what we call the

lp and 2s levels in nuclei are widely separated, whereas the corresponding levels in atoms (called 2s and 2p by atomic physicists) are very close together. To explain this difference, compare the shape of an atomic potential energy ( ? 1/r), to that of the

nuclear potential well. Noting that states with higher l have probability distributions

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Problems for New Ch 16 [Old Ch 12] 9

that are pushed out to larger radii, qualitatively explain the difference in the positions of these two levels.

16.63 [old 12.57] ))) The spin-orbit energy (16.34) is proportional to S?L. To evaluate

S?L note that since J = S + L,

222 JSL？;?;？;;?()()2SLSLSL

2222You can solve this for S?L and then put in the values for J, S, and L. [For instance, J

22111= where j = l ? .] Show that if j = l + then S?L = , and if j = l , l/2jj(1);222

2then S?L = . Use these answers to prove that the spin-orbit splitting ；;(1)/2l

increases with increasing l, as indicated in Fig. 16.17.

;SECTION 16.9 (MASS SPECTROMETERS)

16.64 [old 12.58] ) Where does the methionine molecule in Fig. 16.21 break to account for the two peaks at 74 u and 75 u in its mass spectrum?

16.65 [old 12.59] ) Repeat Problem 16.64 [old 12.58] for the peaks at 45 u and 104 u. 16.66 [old 12.60] )) (a) Find the radius R of the curved path for a once-ionized

nitrogen atom in a mass spectrometer with B = 0.05 T and V = 10 kV, using (16.44). (b) 0

Equation (16.44) was derived nonrelativistically. What would R be if calculated

relativistically? How many significant figures must you include to see the difference? 16.67 [old 12.61] )) Conditions in the ion source of a mass spectrometer can be adjusted to allow formation of multiply ionized atoms. (a) Rewrite Eq. (16.44) for the

412case of an n times ionized atom and solve for R. (b) Taking the masses of He, C, and

16O to be 4 u, 12 u, and 16 u, respectively, write down expressions for the radii, R, for

+3+4+He, C, and O. (c) Using the actual masses of the atoms, find the fractional

differences in these three radii (He versus C and C versus O). The tiny differences in these radii can be measured very accurately and allow precise comparisons of the three

12masses involved (one of which, C, defines the atomic mass scale).

16.68 [old 12.62] )) The chemical composition of a molecule can often be deduced from a very accurate measurement of its mass. As a simple example consider the following. Gas from a car's exhaust is analyzed in a mass spectrometer. Two of the resulting peaks are very close to 28 u. (a) Can you suggest what gases produced these two peaks? (b)

Careful measurements indicate that the masses of the neutral molecules concerned are 28.006 u and 27.995 u. What are the two molecules?

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