A molecule of roughly spherical shape has a mass of $1.80 \times 10 ^ { - 25 } \mathrm {~kg}$ and a diameter of $0.6 \mathrm {~nm}$. If the uncertainty in the measured position of the molecule is equal to the molecular diameter, what is the minimum uncertainty in the speed of the molecule? ( $h = 6.626 \times 10 ^ { - 34 } \mathrm {~J} \cdot \mathrm { s }$ )
A) $1 \mathrm {~m} / \mathrm { s }$
B) $0.01 \mathrm {~m} / \mathrm { s }$
C) $100 \mathrm {~m} / \mathrm { s }$
D) $10 \mathrm {~m} / \mathrm { s }$
E) $0.1 \mathrm {~m} / \mathrm { s }$

Question

Quizplus · Accepted Answer

The uncertainty in the speed of the molecule is given by the Heisenberg uncertainty principle:$$\Delta x \Delta p \geq \frac{h}{4\pi}$$where $\Delta x$ is the uncertainty in the position of the molecule, $\Delta p$ is the uncertainty in the momentum of the molecule, and $h$ is the Planck constant.We are given that $\Delta x = 0.6 \mathrm {~nm}$ and that the mass of the molecule is $1.80 \times 10 ^ { - 25 } \mathrm {~kg}$. The momentum of the molecule is given by:$$p = mv$$where $m$ is the mass of the molecule and $v$ is its speed.Substituting this into the Heisenberg uncertainty principle, we get:$$\Delta x \Delta v \geq \frac{h}{4\pi m}$$Solving for $\Delta v$, we get:$$\Delta v \geq \frac{h}{4\pi m \Delta x}$$Substituting in the given values, we get:$$\Delta v \geq \frac{6.626 \times 10 ^ { - 34 } \mathrm {~J} \cdot \mathrm { s }}{4\pi (1.80 \times 10 ^ { - 25 } \mathrm {~kg})(0.6 \times 10 ^ { - 9 } \mathrm {~m})}$$$$\Delta v \geq 1 \mathrm {~m} / \mathrm { s }$$Therefore, the minimum uncertainty in the speed of the molecule is 1 m/s.

A Molecule of Roughly Spherical Shape Has a Mass Of 1.80