To find the velocity at which a single Buckminsterfullerene (Buckyball) must be traveling to have a de Broglie wavelength larger than its diameter, we can use the de Broglie wavelength equation:

$\lambda = \frac{h}{mv}$

Where:

• $\lambda$ is the de Broglie wavelength,
• $h$ is Planck's constant ($6.626 \times 10^{-34}$ J·s),
• $m$ is the mass of the particle,
• $v$ is the velocity of the particle.

We are given:

• The diameter of the Buckyball ($d$) = 1 nm = $1 \times 10^{-9}$ m.
• The molar mass of the Buckyball = 0.7207 kg/mol.

To calculate the velocity, we will proceed as follows:

Step 1: Convert the molar mass to the mass of a single Buckyball.

The mass of one mole of Buckyballs is 0.7207 kg. To find the mass of a single Buckyball, we divide by Avogadro's number ($N_A = 6.022 \times 10^{23}$):

$m = \frac{0.7207}{6.022 \times 10^{23}} \, \text{kg}$

$m \approx 1.197 \times 10^{-24} \, \text{kg}$

Step 2: Use the de Broglie equation.

To have a wavelength larger than the diameter, we want $\lambda \geq d$. Setting $\lambda = d = 1 \times 10^{-9}$ m, we substitute into the de Broglie equation and solve for $v$:

$1 \times 10^{-9} = \frac{6.626 \times 10^{-34}}{(1.197 \times 10^{-24})v}$

Step 3: Solve for $v$.

Rearranging to solve for $v$:

$v = \frac{6.626 \times 10^{-34}}{(1.197 \times 10^{-24})(1 \times 10^{-9})}$

$v \approx 5.54 \, \text{m/s}$

Conclusion:

The velocity at which a single Buckyball must be traveling to have a de Broglie wavelength larger than its diameter is approximately 5.54 m/s.

Would you like more details or have any questions?

Related Questions:

1. How does the velocity change if the diameter of the particle is doubled?
2. What is the de Broglie wavelength for a particle moving at 10 m/s?
3. How does increasing the mass of a particle affect its de Broglie wavelength?
4. Can the de Broglie wavelength of macroscopic objects be observed?
5. How would the velocity change if the molar mass were increased to 1 kg/mol?

Tip:

De Broglie wavelengths are only significant for very small particles; for macroscopic objects, the wavelengths are typically too small to observe.