The two isotopes of uranium, 238U and 235U, can be separated by effusion of thecorresponding UF6 gases.


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What is the ratio (in the form of a decimal) of the root-mean-square speed of 238UF6 to that of 235UF6 at constant temperature? Express your answer to three significant figures.

The root-mean-square (rms) speed of a gas is given by the formula:

\[
v_{\text{rms}} = \sqrt{\frac{3kT}{m}}
\]

where:
- \( v_{\text{rms}} \) is the root-mean-square speed,
- \( k \) is the Boltzmann constant,
- \( T \) is the temperature in kelvin, and
- \( m \) is the molar mass of the gas in kg/mol.

Since the question specifies that the temperature is constant, we can use Graham's Law of Effusion to compare the speeds of the two gases. Graham's Law states that the ratio of the speeds of two gases is inversely proportional to the square root of their molar masses:

\[
\frac{v_{\text{rms,238}}}{v_{\text{rms,235}}} = \sqrt{\frac{M_{235}}{M_{238}}}
\]

where:
- \( M_{238} \) is the molar mass of \( ^{238}\text{UF}_6 \),
- \( M_{235} \) is the molar mass of \( ^{235}\text{UF}_6 \).

### Step 1: Calculate the Molar Masses of \( ^{238}\text{UF}_6 \) and \( ^{235}\text{UF}_6 \)

1. **Molar Mass of \( ^{238}\text{UF}_6 \):**
   - Uranium-238 has an atomic mass of 238 g/mol.
   - Each fluorine (F) atom has an atomic mass of approximately 19 g/mol, and there are 6 fluorine atoms.
   - So, the molar mass of \( ^{238}\text{UF}_6 \) is:
     \[
     M_{238} = 238 + (6 \times 19) = 238 + 114 = 352 \text{ g/mol}
     \]

2. **Molar Mass of \( ^{235}\text{UF}_6 \):**
   - Uranium-235 has an atomic mass of 235 g/mol.
   - Therefore, the molar mass of \( ^{235}\text{UF}_6 \) is:
     \[
     M_{235} = 235 + (6 \times 19) = 235 + 114 = 349 \text{ g/mol}
     \]

### Step 2: Calculate the Ratio of Root-Mean-Square Speeds

Now we can plug the molar masses into the formula:

\[
\frac{v_{\text{rms,238}}}{v_{\text{rms,235}}} = \sqrt{\frac{349}{352}}
\]

Calculating the square root:

\[
\frac{v_{\text{rms,238}}}{v_{\text{rms,235}}} \approx \sqrt{0.9915} \approx 0.996
\]

### Answer

The ratio of the root-mean-square speed of \( ^{238}\text{UF}_6 \) to \( ^{235}\text{UF}_6 \) is approximately:

\[
\boxed{0.996}
\]

Let me know if you'd like further details, or if you have any questions.

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**Related Questions:**

1. What is the significance of the slight difference in speed between the two isotopes?
2. How would a change in temperature affect the rms speed of each gas?
3. What other methods besides effusion can separate isotopes?
4. Can Graham's Law be applied to solids or liquids, and why or why not?
5. How does the molecular mass impact the diffusion rate of gases in general?

**Tip:** When using ratios of speeds, remember that they are related to the inverse square root of masses, which is helpful in separating gases by effusion.

What is the ratio (in the form of a decimal) of the root-mean-square speed of 238UF6 to that of 235UF6 at constant temperature? Express your answer to three significant figures.

Learn how to calculate the root-mean-square speed ratio of uranium hexafluoride isotopes (238UF6 and 235UF6) at constant temperature using Graham's Law. This problem applies key gas law concepts, ideal for advanced high school students (Grades 11-12).

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