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Chapter 27: Problem 31

What is the orbital radius of the electron in the \(n=3\) state of hydrogen?

Short Answer

Expert verified
Answer: The orbital radius of the electron in the n=3 state of hydrogen is approximately \(4.792 \cdot 10^{-10}\) meters.

Step by step solution

01

Recall Bohr's formula for the radius of an electron orbit in a hydrogen atom

The formula for the radius of an electron orbit in a hydrogen atom, according to Bohr's model, is given by: \(r_n = \frac{n^2路h^2路蔚鈧�}{蟺路m_e路e^2}\) Where: - \(r_n\) is the radius of the orbit, which we want to calculate - \(n\) is the principal quantum number - \(h\) is Planck's constant (\(6.626 \cdot 10^{-34}\) J路s) - \(蔚鈧�\) is the permittivity of free space (\(8.854 \cdot 10^{-12}\) C虏/N路m虏) - \(m_e\) is the mass of the electron (\(9.109 \cdot 10^{-31}\) kg) - \(e\) is the electron charge (\(1.602 \cdot 10^{-19}\) C) - \(蟺\) is Pi (approximately \(3.14159\))
02

Substitute the given value of n and constants into Bohr's formula

We are given the value of n, which is 3. Therefore, we can substitute n, along with the constants, into the formula to solve for the orbital radius: \(r_3 = \frac{3^2路(6.626 \cdot 10^{-34})^2路(8.854 \cdot 10^{-12})}{蟺路(9.109 \cdot 10^{-31})路(1.602 \cdot 10^{-19})^2}\)
03

Perform the calculations

Now, we will compute the expression on the right side of the equation: \(r_3 = \frac{9路(6.626 \cdot 10^{-34})^2路(8.854 \cdot 10^{-12})}{3.14159路(9.109 \cdot 10^{-31})路(1.602 \cdot 10^{-19})^2} \approx 4.792 \cdot 10^{-10}\) m
04

State the result

The orbital radius of the electron in the n=3 state of hydrogen is approximately \(4.792 \cdot 10^{-10}\) meters.

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Most popular questions from this chapter

The Paschen series in the hydrogen emission spectrum is formed by electron transitions from \(n_{\mathrm{i}}>3\) to \(n_{\mathrm{f}}=3.\) (a) What is the longest wavelength in the Paschen series? (b) What is the wavelength of the series limit (the lower bound of the wavelengths in the series)? (c) In what part or parts of the EM spectrum is the Paschen series found (IR, visible, UV, etc.)?
The output power of a laser pointer is about \(1 \mathrm{mW}\) (a) What are the energy and momentum of one laser photon if the laser wavelength is \(670 \mathrm{nm} ?\) (b) How many photons per second are emitted by the laser? (c) What is the average force on the laser due to the momentum carried away by these photons?
An x-ray photon of wavelength 0.150 nm collides with an electron initially at rest. The scattered photon moves off at an angle of \(80.0^{\circ}\) from the direction of the incident photon. Find (a) the Compton shift in wavelength and (b) the wavelength of the scattered photon.
Ultraviolet light of wavelength 220 nm illuminates a tungsten surface and electrons are ejected. A stopping potential of \(1.1 \mathrm{V}\) is able to just prevent any of the ejected electrons from reaching the opposite electrode. What is the work function for tungsten?
A hydrogen atom has an electron in the \(n=5\) level. (a) If the electron returns to the ground state by emitting radiation, what is the minimum number of photons that can be emitted? (b) What is the maximum number that might be emitted?
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