Consider an oxygen (O₂) molecule in a bottle of air. Imagine that at a certain time, we locate the molecule along the x axis to within an uncertainty of 0.1 nm. (a) What is the minimum uncertainty in the molecule's x-velocity required by the Heisenberg Uncertainty Principle? (b) How does this compare (roughly) to the magnitude of the molecule's average x-velocity due to its thermal motion at room temperature? (Hints: Avogadro's number of oxygen molecules has a mass of about 32 g. At an absolute temperature T, a molecule has an average kinetic energy of kB T, where kB is Boltzmann's constant. Room temperature is about 295 K.) Answer

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Q9M.1
Consider an oxygen (O₂) molecule in a bottle of air. Imagine that at a certain time, we locate the molecule along the x axis to
within an uncertainty of 0.1 nm.
(a) What is the minimum uncertainty in the molecule's x-velocity required by the Heisenberg Uncertainty Principle?
(b) How does this compare (roughly) to the magnitude of the molecule's average x-velocity due to its thermal motion at
room temperature?
(Hints: Avogadro's number of oxygen molecules has a mass of about 32 g. At an absolute temperature T, a molecule has an
average kinetic energy of kB T, where kB is Boltzmann's constant. Room temperature is about 295 K.)
Answer
(a) about 10 m/s
(b) Av << average thermal speed.
Transcribed Image Text:Q9M.1 Consider an oxygen (O₂) molecule in a bottle of air. Imagine that at a certain time, we locate the molecule along the x axis to within an uncertainty of 0.1 nm. (a) What is the minimum uncertainty in the molecule's x-velocity required by the Heisenberg Uncertainty Principle? (b) How does this compare (roughly) to the magnitude of the molecule's average x-velocity due to its thermal motion at room temperature? (Hints: Avogadro's number of oxygen molecules has a mass of about 32 g. At an absolute temperature T, a molecule has an average kinetic energy of kB T, where kB is Boltzmann's constant. Room temperature is about 295 K.) Answer (a) about 10 m/s (b) Av << average thermal speed.
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