Consider a hypotheticla gas with molecules that can move along only a single axis. The following table givesfour situations. The velocities is meter per seocond of such a gas having four molecules. The plus and minus sign refer to the direction of the velcoity along the axis.
`{:(ul("Situation Velocities"),,,,),("a -2 +3 -4 +5",,,,),(-------------,,,,),("b +1 -3 +4 -6",,,,),(-------------,,,,),("c +2 +3 +4 +5",,,,),(-------------,,,,),("d +3 +3 -4 -5",,,,),(-------------,,,,):}`
In which situation root-square speed of the molecules is greatest

Options

A. `a`
B. `b`
C. `c`
D. `d`

Explanation

To determine which situation has the greatest root mean square (RMS) speed of the molecules, we will calculate the RMS speed for each situation provided in the question. The formula for the root mean square speed is given by:

\[ V_{\text{RMS}} = \sqrt{\frac{V_1^2 + V_2^2 + V_3^2 + V_4^2}{N}} \]

where \(V_1, V_2, V_3, V_4\) are the velocities of the molecules, and \(N\) is the total number of molecules (which is 4 in this case).

### Step-by-step Calculation:

1. **Situation A: Velocities = -2, +3, -4, +5**
\[ V_{\text{RMS}} = \sqrt{\frac{(-2)^2 + (3)^2 + (-4)^2 + (5)^2}{4}} \]
\[ = \sqrt{\frac{4 + 9 + 16 + 25}{4}} = \sqrt{\frac{54}{4}} = \sqrt{13.5} \approx 3.674 \]

2. **Situation B: Velocities = +1, -3, +4, -6**
\[ V_{\text{RMS}} = \sqrt{\frac{(1)^2 + (-3)^2 + (4)^2 + (-6)^2}{4}} \]
\[ = \sqrt{\frac{1 + 9 + 16 + 36}{4}} = \sqrt{\frac{62}{4}} = \sqrt{15.5} \approx 3.937 \]

3. **Situation C: Velocities = +2, +3, +4, +5**
\[ V_{\text{RMS}} = \sqrt{\frac{(2)^2 + (3)^2 + (4)^2 + (5)^2}{4}} \]
\[ = \sqrt{\frac{4 + 9 + 16 + 25}{4}} = \sqrt{\frac{54}{4}} = \sqrt{13.5} \approx 3.674 \]

4. **Situation D: Velocities = +3, +3, -4, -5**
\[ V_{\text{RMS}} = \sqrt{\frac{(3)^2 + (3)^2 + (-4)^2 + (-5)^2}{4}} \]
\[ = \sqrt{\frac{9 + 9 + 16 + 25}{4}} = \sqrt{\frac{59}{4}} = \sqrt{14.75} \approx 3.843 \]

### Summary of RMS Speeds:
- Situation A: \( \approx 3.674 \)
- Situation B: \( \approx 3.937 \)
- Situation C: \( \approx 3.674 \)
- Situation D: \( \approx 3.843 \)

### Conclusion:
The situation with the greatest root mean square speed is **Situation B** with an RMS speed of approximately \(3.937\).