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An electron confined to a one-dimensional box has energy levels given by the equationEn=n2h2/8mL2where n is a quantum number with possible values of 1,2,3,…;m is the mass of the particle; and L is the length of the box. a. Calculate the energies of the n=1,n=2, and n=3 levels for an electron in a box with a length of 170 pm .b. Calculate the wavelength of light required to make a transition from n=1→n=2 and from n=2→n=3.c.In what region of the electromagnetic spectrum do these wavelengths lie?radiomicrowaveinfraredvisible lightultravioletX-raygamma ray

Question

An electron confined to a one-dimensional box has energy levels given by the equation

En=n2h2/8mL2

where n is a quantum number with possible values of 1,2,3,…;m is the mass of the particle; and L is the length of the box.

 

a. Calculate the energies of the n=1,n=2, and n=3 levels for an electron in a box with a length of 170 pm .

b. Calculate the wavelength of light required to make a transition from n=1→n=2 and from n=2→n=3.

c.
In what region of the electromagnetic spectrum do these wavelengths lie?
radio
microwave
infrared
visible light
ultraviolet
X-ray
gamma ray

 

 

check_circleAnswer
Step 1

a. An electron confined to a one-dimensional box has energy levels given by the equation

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Step 2

The energy of each level is calculated as,

h-6.626x103 m2kg/s
m9.11x103kg
L-170pm 170x10^2m
1x (6.626x103 m2kg/s)
8mL2 8x9.11x103'kgx(170x101m) 2.085x101
E1
22 x(6.626x103m2kg/s)
8mL2 8x9.11x10 kgx(170x10 8.339x1013
E2
12
32 x(6.626x103 m 'kg/s)
8x9.11x103kgx(170x10 m1.876x10
E
8mL
-12
help_outline

Image Transcriptionclose

h-6.626x103 m2kg/s m9.11x103kg L-170pm 170x10^2m 1x (6.626x103 m2kg/s) 8mL2 8x9.11x103'kgx(170x101m) 2.085x101 E1 22 x(6.626x103m2kg/s) 8mL2 8x9.11x10 kgx(170x10 8.339x1013 E2 12 32 x(6.626x103 m 'kg/s) 8x9.11x103kgx(170x10 m1.876x10 E 8mL -12

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Step 3

b. The wavelength required for tran...

n 1to n 2
1
R
2
1 1
109677/cm
3
109677/cm
4
1A°-10cm
-1.216x10m =1216A°
help_outline

Image Transcriptionclose

n 1to n 2 1 R 2 1 1 109677/cm 3 109677/cm 4 1A°-10cm -1.216x10m =1216A°

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