Have lowest root-mean-square speed — States of Matter Chemistry Question

STEPS:

1. Identify the Core Concept: The root-mean-square (rms) speed ($u_{rms}$) of gas molecules is a measure of the average speed of particles in a gas. According to the Kinetic Molecular Theory, it is defined by the formula $u_{rms} = \sqrt{\frac{3RT}{M}}$.
2. Analyze the Variables: From the formula, it is clear that molecular speed depends on two factors:
* Temperature ($T$): Speed is directly proportional to the square root of the absolute temperature. Higher temperatures result in higher speeds.
* Molar Mass ($M$): Speed is inversely proportional to the square root of the molar mass. Heavier molecules move more slowly than lighter ones at the same temperature.
3. Eliminate Non^-Factors: Note that pressure ($P$) does not appear in the $u_{rms}$ formula. While pressure affects the frequency of collisions, it does not change the average speed of the individual molecules at a given temperature.
4. Compare Temperatures: The options provide two temperatures: $200^\circ\text{C}$ (A, B, C, D) and $300^\circ\text{C}$ (E). To find the lowest speed, we should first look at the lower temperature, $200^\circ\text{C}$.
5. Compare Molar Masses at $200^\circ\text{C}$: Using the provided periodic table, calculate the approximate molar masses for the gases at this temperature:
* $H_2$: $\approx 2\text{ g/mol}$
* $O_2$: $\approx 32\text{ g/mol}$
* $SO_2$: $\approx 64\text{ g/mol}$ (Sulfur $\approx 32$ + two Oxygens $\approx 32$)
* $NH_3$: $\approx 17\text{ g/mol}$
6. Conclusion: At the same temperature ($200^\circ\text{C}$), $SO_2$ is the heaviest molecule. Because it has the largest molar mass and is at the lowest temperature provided for that mass, it will have the lowest root-mean-square speed (Option C).

WHY_OTHERS_WRONG:

• $H_2(g)$ (A): This is the lightest molecule ($\approx 2\text{ g/mol}$). At any given temperature, it will have the highest speed of all the options.
• $O_2(g)$ (B) and $NH_3(g)$ (D): At $200^\circ\text{C}$, both $O_2$ ($\approx 32\text{ g/mol}$) and $NH_3$ ($\approx 17\text{ g/mol}$) are lighter than $SO_2$ ($\approx 64\text{ g/mol}$), meaning they will move faster than $SO_2$ molecules.
• $NH_3(g)$ at $300^\circ\text{C}$ (E): This option is incorrect because it features both a lighter mass than $SO_2$ and a higher temperature, both of which serve to increase the molecular speed rather than decrease it.