Atoms

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SCI 270: On Nuclear Technology Practice Problems

3. Atoms - the tiny wonders

When heated to very high temperature or passing a electric discharge through, a hydrogen gas glows, emitting a purplish light. When analyzed by a prism, the light consists of a few lines with wavelengths listed in the Table.

Wavelength, l	Wavenumber, 1/l	Frequency, c/l	Photon energy, h v	n in Balmer series.
nm	10⁶ m^-1	10¹⁴ Hz	10^-19 J	Integer
656.3
486.1
434.0
410.1
396.9
389.0

Convert the wavelengths l to wavenumbers and frequencies using the given formulas in multiples and units as indicated. Also evaluate the n in the Balmer series.

From the values above, draw a spectrum indicating where the lines will be observed based on a linear frequency scale.
| | | | | |4____________________5____________________6____________________7___________/e14 Hz
The Rydberg relationship is usually given in the form as follows:

1
---
l = - R ( 1
---
n² - 1
--- )
4

The Rydberg constant R is
________________________ m^-1

However, the Rydberg relationship can be written for frequency (v) as follows:

v = - R ( 1
---
n² - 1
--- )
4

The Rydberg constant R is
________________________ Hz
The Bohr model and the quantum mechanical approach result in an expression for the Rydberg constant R:
1. 2 p2 m Z2 e4
  R = --------------- = 10967700 m-1
  c h3
where m, Z, c, h and e are the mass of the electron, atomic number, speed of light, Planck's constant, and charge of an electron respectively.

Apply this number to calculate the wave numbers of the 5 lines (n_f = 2, 3, 4, 5, and 6; n_i = 1) of the Lyman series.

The formula used:
_____________________________

Wave numbers of the five lines

_____________
_____________
_____________
_____________
_____________

In what regions are these lines within the electromagnetic radiation spectrum, visible, UV, IR, Microwave, or X-ray?

These lines are in the region of

_______________________________

The Rydberg constant R is	________________________ m^-1

The Rydberg constant R is	________________________ Hz

The formula used:	_____________________________
Wave numbers of the five lines
_____________	_____________	_____________	_____________	_____________

These lines are in the region of
_______________________________

The characteristic X-rays of some elements are listed here,

Atomic number	Element	Wavelength	Frequency	Frequency^(1/2)	Photon energy, h v
		nm	10¹⁸ Hz	10⁹ Hz^(1/2)	10^-15 J
23	V	0.2503	1.199
24	Cr	0.2289	1.310
25	Mn	0.2102	1.427
26	Fe	0.1936	1.550
27	Co	0.1789	1.677
28	Ni	0.1657	1.810
29	Cu	0.1541	1.947
30	Zn	0.1435	2.090
42	Mo	0.0746	4.020
47	Ag	0.0559	5.363
79	Au	0.0180	16.650

Complete the Table by taking the square root of the frequencies, and plot them against the atomic number Z on a graph to see if these data obey the Moseley's law.

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Evaluate the photon energies and plot them against Z² to see if they fit a straight line.

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Apply the Moseley's law to estimate the wavelength of characteristic X-rays from hydrogen, (H, Z = 1), aluminium (Al, Z = 13), tin (Sn, Z = 50), and uranium (U, Z = 92).

The wavelengths are given below

H Al Sn U

_____________
_____________
_____________
_____________

Answers to Assignment 3 is available, but do your part first. Your skills are your best friends they are yours forever.

Answers

SCI 270: On Nuclear Technology Practice Problems
Answers are given for your reference only. Please do your part of the learning, because no one else will be able to do that for you. Your skills of problem solving are tested and useful in the future. Answers are useless. Acquiring skills and abilities are the goals of learning, marks indications of your performances on these tasks.

Wavelength l nm	Wavenumber 1/l /10⁶ m^-1	Frequency c/l /10¹⁴ Hz	Photon energy h v /10^-19 J	n in Balmer series.
656.3	1.524	4.571	3.029	3
486.1	2.057	6.172	4.090	4
434.0	2.304	6.912	4.580	5
410.1	2.438	7.315	4.847	6
396.9	2.520	7.556	5.007	7
389.0	2.570	7.712	5.110	8

Convert the wavelengths l to wavenumbers and frequencies using the given formulas and units. Also evaluate the n in the Balmer series (see page 62 of the lecture notes).

I hope the repetitive calculation will let you learn the formulas and theory well.

In order to avoid the repitition of common factors and units, I have used /10⁶ m^-1 to mean millions per meter.

Unfortunately, some students considered the the factors after the "/" as part of the formula without knowing why and calculate the numbers.

From the values above, draw a spectrum based on a linear scale using the frequencies, and evaluate the Rydberg constant R both in wavenumbers (m^-1) and in Hz.

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|4____________________5____________________6____________________7___________/e14 Hz

The line spectrum in in e14 Hz are given above. If you have plotted the frequency as a function of 1/n², we will not mark you wrong.

The Rydberg constant R is	__3.29212e15 Hz__

There are many methods to calculate R. If you use a plot of frequency vs. 1/n² to evaluate it, that is fine. The simplest way is to convert the known value in m^-1 to Hz by multiplying it by the speed of light or use any two lines to evaluate it.

The Bohr model and the quantum mechanical approach result in an expression for the Rydberg constant R:
where m, Z, c, h and e are the mass of the electron, atomic number, speed of light, Planck's constant, and charge of an electron respectively.

Apply this number to calculate the wave numbers of the 5 lines (n_f = 2, 3, 4, 5, and 6; n_i = 1) of the Lyman series.

The formula used:
wavenumber = -R(1/n² - 1)

Wave numbers of the five lines

8.226e6 9.750e6 10.28e6 10.53e6 10.66e6 m^-1

You can also plot a spectrum in the UV region for the Lyman series. Having generated these numbers, you have learned how photons are emitted. The energy differences from various energy levels are given out as photons.

In what regions are these lines within the electromagnetic radiation spectrum, visible, UV, IR, Microwave, or X-ray?

These lines are in the region of

__ Ultraviolet (UV) __

The formula used:	wavenumber = -R(1/n² - 1)
Wave numbers of the five lines
8.226e6	9.750e6	10.28e6	10.53e6	10.66e6 m^-1

The characteristic X-rays of some elements are listed here,

Atomic number	Photon energy h v /10^-15 J	Element	Wavelength nm	Frequency /10¹⁸ s	Frequency^(1/2) /10⁹
23	0.794	V	0.2503	1.199	1.095
24	0.868	Cr	0.2289	1.310	1.145
25	0.946	Mn	0.2102	1.427	1.195
26	1.027	Fe	0.1936	1.550	1.245
27	1.111	Co	0.1789	1.677	1.295
28	1.199	Ni	0.1657	1.810	1.345
29	1.290	Cu	0.1541	1.947	1.395
30	1.385	Zn	0.1435	2.090	1.446
42	2.664	Mo	0.0746	4.020	2.005
47	3.554	Ag	0.0559	5.363	2.316
79	11.03	Au	0.0180	16.650	4.080

Complete the Table by taking the square root of the frequencies, and plot them against the atomic number Z on a graph to see if these data obey the Moseley's law.

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Evaluate the photon energies and plot them against Z² to see if they fit a straight line.

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|   You should get a linear plot
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Apply the Moseley's law to estimate the wavelength of characteristic X-rays from hydrogen, (H, Z = 1), aluminium (Al, Z = 13), tin (Sn, Z = 50), and uranium (U, Z = 92).

The wavelengths are given below
H	Al	Sn	U
_____________	_____________	_____________	_____________

Element, Z	SQRT(f)	Frequency	Wavelength	Wavenumber /m^-1
H, 1	41924543	2.2968e15	131 nm	7.63e6
Al, 13	6.23e8	3.8815e17	0.773 nm	1.29e9
Sn, 50	2.396e9	5.742e18	0.0522 nm	1.92e10
U, 92	4.409e9	1.944e19	1.54e-11 m 0.0154 nm	6.49e10

ing on the two points you may choose, the value you estimated may be slightly different. If you used a least-squares line to get the slope, you are doing it properly. Since we do not require math skills here, we choose to use a simple approach in this hint.

Note the wavenumbers for the Lyman series range from 8 to 11 million and the estimate for H here is 7.6 million. Depending on the slope one uses, the agreement can be very good.

You should get a linear plot

Contributors and Attributions

Chung (Peter) Chieh (Professor Emeritus, Chemistry @ University of Waterloo)