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Ch 27 HW (Part 2)

The Hydrogen Spectrum

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Part B

What is the wavelength of the line corresponding to n=5 in the Balmer series?

Express your answer in nanometers to three significant figures.

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Part C

What is the smallest wavelength λmin in the Balmer's series?

Express your answer in nanometers to three significant figures.

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Part D

What is the largest wavelength λmax in the Balmer series?

Express your answer in nanometers to three significant figures.

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Part E

What is the smallest value of n for which the wavelength of a Balmer series line is smaller than 400nm, which is the lower limit for wavelengths in the visible spectrum?

Enter your answer as an integer.

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Encouraged by the success of Balmer's formula, other scientists extended the formula by simply changing the 22 term to 12 or 32, or more generally to m2, and verified the existence of the corresponding wavelengths in the hydrogen spectrum. The resulting formula contains two integer quantities, m and n, and it is given by

1λ=R(1m2−1n2),

where R is again the Rydberg constant. For m=2, you can easily verify that the formula gives the Balmer series. For m=1,3,4,5, the formula gives other sets of lines, or series, each one named after its discoverer. Note that for each value of m, n=m+1,m+2,m+3,....

Part F

If m=1, in what range are the wavelengths calculated from the generalized formula shown above?

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If , in what range are the wavelengths calculated from the generalized formula shown above?

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Part G

If m=3, in what range are wavelengths calculated from the generalized formula shown above?

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If , in what range are wavelengths calculated from the generalized formula shown above?

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The Hydrogen Spectrum When a low-pressure gas of hydrogen atoms is placed in a tube and a large voltage is applied to the end of the tube, the atoms will emit electromagnetic radiation and visible light can be observed. If this light passes through a diffraction grating, the resulting spectrum appears as a pattern of four isolated, sharp parallel lines, called spectral lines. Each spectral line corresponds to one specific wavelength that is present in the light emitted by the source. Such a discrete spectrum is referred to as a line spectrum. By the early 19th century, it was found that discrete spectra were produced by every chemical element in its gaseous state. Even though these spectra were found to share the common feature of appearing as a set of isolated lines, it was observed that each element produces its own unique pattern of lines. This indicated that the light emitted by each element contains a specific set of wavelengths that is characteristic of that element.	The first quantitative description of the hydrogen spectrum was given by Johann Balmer, a Swiss school teacher, in 1885. By trial and error, he found that the correct wavelength λ of each line observed in the hydrogen spectrum was given by 1λ=R(122−1n2), where R is a constant, later called the Rydberg constant, and n may have the integer values 3, 4, 5, .... If λ is in meters, the numerical value of the Rydberg constant (determined from measurements of wavelengths) is R = 1.097×107 m−1 . Balmer knew only the four lines in the visible spectrum of hydrogen. Thus, the original formula was written for a limited set of values of n. However, as more techniques to detect other regions of the spectrum were developed, it became clear that Balmer's formula was valid for all values of n. The entire series of spectral lines predicted by Balmer's formula is now referred to as the Balmer series. Part A What is the wavelength of the line corresponding to n=4 in the Balmer series? Express your answer in nanometers to three significant figures.
λ(n) =
nm
λ(n) =
nm
λmin =
nm
λmax =
nm
n =
microwave (1 to 10−4m)
infrared (10−3 to 7×10−7m)
visible (7×10−7 to 4×10−7m)
ultraviolet (4×10−7 to 10−8m)
X rays (10−8 to 10−13m)
microwave (1 to 10−4m)
infrared (10−3 to 7×10−7m)
visible (7×10−7 to 4×10−7m)
ultraviolet (4×10−7 to 10−8m)
X rays (10−8 to 10−13m)

Answer 1

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