7. QUANTUM THEORY OF THE ATOM

Transcription

1 7. QUANTUM TEORY OF TE ATOM Solutions to Practice Problems Note on significant figures: If te final answer to a solution needs to be rounded off, it is given first wit one nonsignificant figure, and te last significant figure is underlined. Te final answer is ten rounded to te correct number of significant figures. In multiplestep problems, intermediate answers are given wit at least one nonsignificant figure owever, only te final answer as been rounded off. Starting wit Problem 7.9, te value.99 0 m/s will be used for te speed of ligt. 7.9 Solve c for : c.99 0 m/s /s m 7.30 Solve c for : c.99 0 m/s /s m (.33 cm) 7.3 Solve c for. Recognize tat 7 nm m, or m. c.99 0 m/s m /s 7.3 Solve c for. Recognize tat 656 nm m m. c.99 0 m/s /s m 7.33 Radio waves travel at te speed of ligt, so divide te distance by c: m s.99 0 m s 7.3 Electromagnetic signals travel at te speed of ligt, so divide te distance by c:

2 QUANTUM TEORY OF TE ATOM m s.99 0 m s (0.95 r) 7.35 To do te calculation, divide one meter by te number of wavelengts in one meter to find te wavelengt of tis transition. Ten, use te speed of ligt (wit nine digits for significant figures) to calculate te frequency: m,650, m c m/s /s m 7.36 To find te wavelengt, divide te speed of ligt (wit nine digits for significant figures) by te frequency: c m/s m 9,9,63, 770 /s 7.37 Solve for E, using E, and use four significant figures for : E ( s) ( /s) Solve for E, using E, and use four significant figures for : E ( s) ( /s) Recognize tat 535 nm m m. Ten, calculate and E. c.99 0 m/s m /s E ( s) ( /s) Recognize tat 5 nm m m. Ten, calculate and E. c.99 0 m/s m /s E ( s) ( /s)

3 CAPTER 7 7. First, calculate te wavelengt of tis transition from te frequency using te speed of ligt: c.99 0 m/s 3. 0 /s m (7 pm) Using Figure 7.5, note tat 7 nm is just on te edge of te red end of te spectrum and is barely visible to te eye. 7. First, calculate te wavelengt of tis transition from te frequency using te speed of ligt: c.99 0 m/s 5. 0 /s m (55 pm) Using Figure 7.5, note tat 55 nm is in te yellowgreen region of te spectrum and is visible to te eye. 7.3 Solve te equation E R /n for bot E 5 and E 3 equate to, and solve for. E 5 R 5 5 E R 5 R 9 6R 5 Te frequency of te emitted radiation is: 6R s /s 7. Solve te equation E R /n for bot E and E 3 equate to, and solve for. E R 6 E R 6 R 9 7 R Te frequency of te emitted radiation is: 7 R s /s 7.5 Solve te equation E R /n for bot E and E. E R E

4 QUANTUM TEORY OF TE ATOM 9 R R 3 R Te frequency of te emitted radiation is: 3 R s /s Te wavelengt can now be calculated. c.99 0 m/s /s m (near UV) 7.6 Solve te equation E R /n for bot E 5 and E solve for, and convert to. E 5 R 5 5 E 6 R 5 R 6 9 R 00 Te frequency of te emitted radiation is: 9 R s /s Te wavelengt can now be calculated. c.99 0 m/s /s m (near IR)

5 50 CAPTER Tis is te igest energy transition from te n 6 level, so te electron must undergo a transition to te n level. Solve te Balmer equation using Bor's approac: E 6 R 6 36 E R 36 R 35R 36 Te frequency of te emitted radiation is: 35R s /s Te wavelengt can now be calculated. c.99 0 m/s /s m (93. nm) 7. Tis is te lowest energy transition from te n 6 level, so te electron must undergo a transition to te n 5 level. Solve te Balmer equation using Bor's approac: E 6 R 6 36 E R 36 R 5 R 900 R s /s c.99 0 m/s /s m (760 nm)

6 QUANTUM TEORY OF TE ATOM Noting tat.7 nm m, convert te.7 nm to frequency. Ten convert te frequency to energy using E. c.99 0 m/s m /s E ( s) /s) Noting tat 5. nm m, convert te 5.nm wavelengt to frequency. Ten convert te frequency to energy using E. c.99 0 m/s m /s E ( s) /s) Te mass of a neutron kg. Its speed or velocity, v, of.5 km/s equals m/s. Substitute tese parameters into te de Broglie relation, and solve for : mv kg m / s kg.5 0 m/ s m or 95.3 pm 7.5 Te mass of a proton Its speed or velocity, v, of 6. km/s equals m/s. Substitute tese parameters into te de Broglie relation, and solve for : mv kg m / s kg 6. 0 m/ s m or 63. pm A wavelengt of 63. pm would be in te ray region of te spectrum.

7 5 CAPTER Te mass of an electron equals kg. Te wavelengt,, given as 0.0 pm, is equivalent to.00 0 m. Substitute tese parameters into te de Broglie relation, and solve for te frequency, : v m kg m / s kg.00 0 m m/s 7.5 Te mass of a neutron equals kg. Te wavelengt,, given as 0.0 pm, is equivalent to.00 0 m. Substitute tese parameters into te de Broglie relation, and solve for te frequency, : v m kg m / s kg.00 0 m m/s 7.55 Substitute te.5 0 kg mass of te baseball and te 30.0 m/s velocity, v, into te de Broglie relation, and solve for wavelengt (recall tat pm 0 m) kg m / s m mv.5 0 kg 30.0 m/ s.5 0 pm Because tis is muc smaller tan 00 pm, te wavelengt is muc smaller tan te diameter of one atom Te mass of O to tree significant figures is g kg. Substitute tis mass and te 5 m/s velocity, v, into te de Broglie relation, and solve for wavelengt (recall tat pm 0 m) kg m /s m mv kg 5m/s 3.9 pm Because tis is on te order of 00 pm, te wavelengt is on te order of an atomic diameter.

8 QUANTUM TEORY OF TE ATOM Te possible values of l range from zero to (n ), so l may be 0,,, or 3. Te possible values of m l range from l to + l, so m l may be 3,,, 0, +, +, or Te possible l values are 0,,, 3, and. Te possible m l values are, 3,,, 0,,, 3, or For te M sell, n 3 tere are tree subsells in tis sell (l 0,, and ). An f subsell as l 3 te number of orbitals in tis subsell is (3) + 7 (m l 3,,, 0,,, and 3) For te N sell, n tere are four subsells in tis sell (l 0,,, and 3). For a g subsell, te value of l te number of orbitals in tis subsell is () + 9 (m l, 3,,, 0,,, 3, and ). 7.6 a. 6d b. 5g c. f d. 6p 7.6 a. 3p b. d c. s d. 5f 7.63 a. Not permissible m s may be only + / or /. b. Not permissible l can only be as large as (n ). c. Not permissible m l may not eceed + in magnitude. d. Not permissible n may not be zero. e. Not permissible m s may only be + / or /. 7.6 a. Not permissible n starts at, not at zero. b. Not permissible l may only be as large as (n ). c. Permissible. d. Not permissible m l may not eceed l in magnitude. e. Permissible.

9 5 CAPTER 7 Solutions to General Problems 7.65 Use c to calculate frequency ten use E to calculate energy. c.99 0 m/s m /s E ( s) ( /s) Use c to calculate frequency ten use E to calculate energy. c.99 0 m/s m /s E ( s) (5.5 0 /s) Calculate te frequency corresponding to Ten convert tat to wavelengt. 9 E s /s c.99 0 m/s /s m 5 nm (blue) 7.6 Calculate te frequency corresponding to Ten convert tat to wavelengt. 9 E s /s c.99 0 m/s /s m 595 nm (yellow)

10 QUANTUM TEORY OF TE ATOM Solve for frequency using E. 9 E s /s 7.70 Solve for frequency using E. 9 E s /s 7.7 First calculate E p, te energy of te 35nm poton, noting tat it is equivalent to m. E p c ( s)( m m/s) Now, subtract te work function of Ca (Problem 7.69) from E p : Note tat, for tis situation, E /mv. Recall tat te mass of te electron is kg. Now calculate speed, v: v 9 E.7 0 m kg m/s 7.7 First calculate E p, te energy of te 5nm poton, noting tat it is equivalent to m. E p c ( s)( m m/s) Now, subtract te work function of Mg (Problem 7.70) from E p : Note tat, for tis situation, E /mv. Recall tat te mass of te electron is kg. Now calculate speed, v: v 9 E m kg m/s

11 56 CAPTER Tis is a transition from te n 5 level to te n level. Solve te Balmer equation using Bor's approac. E 5 R 5 5 E R 5 R R 00 R s /s c.99 0 m/s /s m (3 nm) 7.7 Tis is a transition from te n 3 level to te n level. Solve te Balmer equation using Bor's approac. E 3 R 3 9 E R 9 R 5 R 36 5 R s /s c.99 0 m/s /s m (656 nm) 7.75 Use 397 nm m, and convert to frequency and ten to energy. c.99 0 m/s m /s E ( s) ( ) (continued)

12 QUANTUM TEORY OF TE ATOM 57 Substitute tis energy into te Balmer formula recalling tat te Balmer series is an emission spectrum, so E is negative: E R R E i R n i R R n i n i E (of line) n i n i n i / ( n) 7.76 Convert m into frequency and ten to energy. c.99 0 m/s m /s E ( s) ( ) Substitute tis energy into te Balmer formula recalling tat te Lyman series is an emission spectrum, so E is negative, and n f : E R R E i R n i R R n i n i E (of line) n i (continued)

13 5 CAPTER 7 n i n i ( n) 7.77 Employ te Balmer formula using Z for te e + ion. E 3 ( ) R 3 ( ) 9 E ( ) ( ) R 9 R 5R 36 Te frequency of te radiation is 5 R s /s c.99 0 m/s /s m (6 nm near UV) 7.7 Employ te Balmer formula using Z 3 for te Li + ion. E ( 3 ) R ( 9 ) 6 E 3 ( 3 ) 3 ( 9 ) 9 R 9 6 R R 9 7 R s /s c.99 0 m/s /s m (0 nm UV)

14 QUANTUM TEORY OF TE ATOM First, use te wavelengt of 0.0 pm (.00 0 m) and te mass of kg to calculate te velocity, v. Ten, use te kinetic energy equation to calculate kinetic energy from velocity. v m kg m / s kg.00 0 m m/s E /mv / ( kg) ( m/s) E ev ev ev 7.0 First, use te wavelengt of 0.0 pm (.00 0 m) and te mass of kg to calculate te velocity, v. Ten, use te kinetic energy equation to calculate kinetic energy from velocity. v m kg m / s kg.00 0 m m/s E /mv / ( kg) ( m/s) E ev ev ev 7. a. Five b. Seven c. Tree d. One 7. a. Seven b. Nine c. One d. Tree 7.3 Te possible subsells for te n 6 sell are 6s, 6p, 6d, 6f, 6g, and Te possible subsells for te n 7 sell are 7s, 7p, 7d, 7f, 7g, 7, and 7i.

15 60 CAPTER 7 Solutions to CumulativeSkills Problems 7.5 First, use Avogadro's number to calculate te energy for one Cl molecule. 39 k mol 000 k mol molecules /molecule Ten, convert energy to frequency and finally to wavelengt. 9 E s /s c.99 0 m/s /s m (50 nm visible region) 7.6 First, use Avogadro's number to calculate te energy per molecule. 3 k mol 000 k mol molecules /molecule Ten, convert energy to frequency and finally to wavelengt. 9 E s /s c.99 0 m/s /s m (77 nm UV) 7.7 First, calculate te energy needed to eat te 0.50 L of water from 0.0 C to 00.0 C L 000 g L. (g C) (00.0 C 0.0C).36 0 Ten, calculate te frequency, te energy of one poton, and te number of potons. c.99 0 m/s 0.5m /s (continued)

16 QUANTUM TEORY OF TE ATOM 6 E of one poton ( s) ( /s).59 0 No. potons.36 0 poton potons 7. First, calculate te energy needed to eat te.00 L of water from 0.0 C to 30.0 C..00 L 000 g L. (g C) (30.0 C 0.0C). 0 Ten, calculate te frequency, te energy of one poton, and te number of potons. c.99 0 m/s m /s E of one poton ( s) ( /s) No. potons potons poton First, write te following equality for te energy to remove one electron, E removal : E removal E 5 nm E k of ejected poton Use E to calculate te energy of te poton. Ten, recall tat E k, te kinetic energy, is /mv. Use tis to calculate E k. E 5 nm c ( s)( m m/s) E k /mv / ( kg) (. 0 5 m/s) Subtract to find E removal, and convert it to k/mol: E removal ( ) /electron (continued)

17 6 CAPTER 7 E removal e e mol k k/mol 7.90 First, write te following equality for te energy to remove one electron, E removal : E removal E 05 nm E k of ejected electron Use E to calculate te energy of te poton. Ten, recall tat E k, te kinetic energy, is /mv. Use tis to calculate E k. E 05 nm c ( s)( m m/s) E k /mv / ( kg) ( m/s) Subtract to find E removal, and convert it to k/mol: E removal /electron E removal e e mol k k/mol 7.9 First, calculate te energy, E, in joules using te product of voltage and carge: E ( V) ( C) Now, use te kinetic energy equation, E k /mv, and solve for velocity: v Ek m kg m/s s mv 3 7 ( kg) ( m/s).9 0 m (9. pm)

18 QUANTUM TEORY OF TE ATOM First, calculate te energy, E, in joules using te product of voltage and carge: E (.00 0 V) ( C) Now, use te kinetic energy equation E k /mv, and solve for velocity: v Ek m kg m/s s.6 0 mv 3 7 ( kg) ( m/s).3 0 m (.3 pm)