Two containers of identical size contain 1 mole each of the same ideal monatomic gas. Container 1 is at a higher temperature than container 2. Are the following statements true sometimes, always or never (justify your answer) The most probable speed of gas atoms in container 1 is faster than the root mean square speed of gas atoms in container 2.

To determine if the statement is true, false, or sometimes true, let’s analyze the relationship between the **most probable speed** and **root mean square speed** for an ideal monatomic gas.

### Key Formulas
For an ideal gas, the **most probable speed** \( v_{\text{mp}} \) and the **root mean square speed** \( v_{\text{rms}} \) are given by:

1. **Most probable speed (v_mp)**:
   \[
   v_{\text{mp}} = \sqrt{\frac{2 k_B T}{m}}
   \]
   where:
   - \( k_B \) is the Boltzmann constant,
   - \( T \) is the temperature of the gas,
   - \( m \) is the mass of a single gas atom.

2. **Root mean square speed (v_rms)**:
   \[
   v_{\text{rms}} = \sqrt{\frac{3 k_B T}{m}}
   \]

### Analysis
- Since \( v_{\text{mp}} \) depends on \( T \) and so does \( v_{\text{rms}} \), increasing the temperature increases both values.
- In container 1, which is at a higher temperature than container 2, both \( v_{\text{mp}} \) and \( v_{\text{rms}} \) will be higher than in container 2 because both speeds are proportional to the square root of temperature.

### Comparing \( v_{\text{mp}} \) and \( v_{\text{rms}} \)
We can see that for any given temperature:
\[
v_{\text{rms}} = \sqrt{\frac{3}{2}} \cdot v_{\text{mp}}
\]
which means that for an ideal gas, \( v_{\text{rms}} \) is always greater than \( v_{\text{mp}} \).

### Conclusion
Since container 1 has a higher temperature, the most probable speed \( v_{\text{mp,1}} \) in container 1 will be greater than the most probable speed \( v_{\text{mp,2}} \) in container 2. However, due to the relationship \( v_{\text{rms}} > v_{\text{mp}} \) at any temperature, the root mean square speed \( v_{\text{rms,2}} \) in container 2 will still be greater than the most probable speed \( v_{\text{mp,1}} \) in container 1, regardless of temperature.

Therefore, the statement is **never true**.

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Would you like further details or have any questions?

### Related Questions
1. How do temperature changes affect the root mean square speed of gas molecules?
2. How does the mass of gas molecules affect the most probable speed?
3. What is the significance of the Maxwell-Boltzmann distribution in kinetic theory?
4. Can the root mean square speed ever be equal to the most probable speed?
5. How does the average speed compare to the most probable and root mean square speeds?

**Tip:** For monatomic gases, temperature directly affects molecular speeds, as they depend solely on temperature and molecular mass.

Explore the relationship between the most probable speed and root mean square speed of gas atoms in containers at different temperatures. This problem demonstrates key principles of kinetic theory and thermodynamics, including calculations of molecular speeds for ideal gases. Ideal for undergraduate physics students.

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