Buckminsterfullerene (Buckyballs) have a diameter of~Inm and a molar mass of~0.7207kg/mol. Approximately what velocity must a single Buckyball be travelling at to have a de Broglie wave length larger than its diameter?

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.

Buckminsterfullerene (Buckyballs) have a diameter of ~1nm and a molar mass of ~0.7207kg/mol. Approximately what velocity must a single Buckyball be travelling at to have a de Broglie wavelength larger than its diameter?

Find the velocity of a Buckminsterfullerene (Buckyball) to have a de Broglie wavelength larger than its 1 nm diameter. This problem involves quantum mechanics and uses the de Broglie wavelength equation, Planck's constant, and molar mass conversion. Suitable for undergraduate physics students.

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