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Chapter 7: Problem 128

Alveoli are the tiny sacs of air in the lungs (see Problem 5.136 ) whose average diameter is \(5.0 \times\) \(10^{-5} \mathrm{~m} .\) Consider an oxygen molecule \(\left(5.3 \times 10^{-26} \mathrm{~kg}\right)\) trapped within a sac. Calculate the uncertainty in the velocity of the oxygen molecule. (Hint: The maximum uncertainty in the position of the molecule is given by the diameter of the sac.)

Short Answer

Expert verified
The uncertainty in the velocity of the oxygen molecule trapped within the alveolar sac in the lungs is equal to or larger than approximately \(0.033 m/s\).

Step by step solution

01

Understand the Heisenberg Uncertainty Principle

The Heisenberg's Uncertainty Principle states that you cannot simultaneously know the exact position and momentum (mass times velocity) of a particle. In terms of positioning and velocity, the principle can be represented as: \(\Delta x \cdot \Delta v \geq \frac{h}{4\pi}\) where \(\Delta x\) is the uncertainty in position, \(\Delta v\) is the uncertainty in velocity, and \(h\) is the Planck's constant (\(6.63 \times 10^{-34} Js\)).
02

Use the Given Data to Calculate the Uncertainty in Velocity

We know \(\Delta x\), the uncertainty in position, is the diameter of the alveoli sac \(5.0 \times 10^{-5} m\). Rearrange the Heisenberg Uncertainty Principle equation to solve for \(\Delta v\), we get: \(\Delta v \geq \frac{h}{4\pi \Delta x}\). Then substitute the given \(h = 6.62 \times 10^{-34} Js\) and \(\Delta x = 5.0 \times 10^{-5} m\) to calculate \(\Delta v\).
03

Calculate The Uncertainty in Velocity of The Oxygen Molecule

By substituting the values into the equation \(\Delta v \geq \frac{h}{4\pi \Delta x}\), we find \(\Delta v \geq 0.033 m/s\).

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Most popular questions from this chapter

In the beginning of the twentieth century, some scientists thought that a nucleus may contain both electrons and protons. Use the Heisenberg uncertainty principle to show that an electron cannot be confined within a nucleus. Repeat the calculation for a proton. Comment on your results. Assume the radius of a nucleus to be \(1.0 \times 10^{-15} \mathrm{~m} .\) The masses of an electron and a proton are \(9.109 \times 10^{-31} \mathrm{~kg}\) and \(1.673 \times 10^{-27} \mathrm{~kg},\) respectively. (Hint: Treat the diameter of the nucleus as the uncertainty in position.)

When two atoms collide, some of their kinetic energy may be converted into electronic energy in one or both atoms. If the average kinetic energy is about equal to the energy for some allowed electronic transition, an appreciable number of atoms can absorb enough energy through an inelastic collision to be raised to an excited electronic state. (a) Calculate the average kinetic energy per atom in a gas sample at \(298 \mathrm{~K}\). (b) Calculate the energy difference between the \(n=1\) and \(n=2\) levels in hydrogen. (c) At what temperature is it possible to excite a hydrogen atom from the \(n=1\) level to \(n=2\) level by collision? [The average kinetic energy of 1 mole of an ideal gas is \(\left.\left(\frac{3}{2}\right) R T .\right]\)

What is the de Broglie wavelength, in centimeters, of a 12.4-g hummingbird flying at \(1.20 \times 10^{2} \mathrm{mph} ?\) \((1 \mathrm{mile}=1.61 \mathrm{~km})\)

What is electron configuration? Describe the roles that the Pauli exclusion principle and Hund's rule play in writing the electron configuration of elements.

The electron configurations described in this chapter all refer to gaseous atoms in their ground states. An atom may absorb a quantum of energy and promote one of its electrons to a higher-energy orbital. When this happens, we say that the atom is in an excited state. The electron configurations of some excited atoms are given. Identify these atoms and write their ground-state configurations: (a) \(1 s^{1} 2 s^{1}\) (b) \(1 s^{2} 2 s^{2} 2 p^{2} 3 d^{1}\) (c) \(1 s^{2} 2 s^{2} 2 p^{6} 4 s^{1}\) (d) \([\mathrm{Ar}] 4 s^{1} 3 d^{10} 4 p^{4}\) (e) \([\mathrm{Ne}] 3 s^{2} 3 p^{4} 3 d^{1}\)
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