Chapter 1-7 Review: Kinetic Molecular Theory and Intermolecular Forces (Video)

Course context: This week is a review; intermolecular forces are chapter 9 in Gilbert. Also reconnect with prior semester topics: gas concepts, including sociometry, multiple ratios, limiting reagents, etc. Emphasis on not discarding last semester content.
Kinetic Molecular Theory (KMT) basics discussed: temperature increases lead to faster molecular speeds; colder temps lead to slower speeds. Energy increases correlate with faster, more chaotic motion.
Conceptual takeaway from the graph (slide 36, part two gas slides): For nitrogen gas, as you increase temperature, the peak of the speed distribution shifts to higher speeds on the x-axis (molecular speed) while the y-axis shows the number of molecules at each speed. The distribution broadens with higher temperature because of greater chaos/entropy and greater variation in speeds due to more frequent molecular collisions.
Entropy note: Greater temperature leads to greater disorder in the system; distribution of speeds broadens as molecules move more chaotically and collide more often.
Practical reading of graphs: When interpreting graphs, examine axis labels first (x-axis = molecular speed, y-axis = number of molecules). Recognize that higher temperature broadens and shifts the distribution toward faster speeds.

Kinetic energy concept linkage: KE = \tfrac{1}{2} m u^2; in physics, KE relates to energy units (joules).
RMS velocity (root-mean-square speed) equation (for an ideal gas): u_{rms} = \sqrt{\frac{3RT}{M}}
- M is molar mass in kg/mol (not g/mol) for this form.
- R is the gas constant. Distinctions on constants: R can be given as
- R = 0.08206\ \mathrm{L\,atm\,mol^{-1}\,K^{-1}} (for PV=nRT in those units), or
- R = 8.314\ \mathrm{J\,mol^{-1}\,K^{-1}} (energy-based form).
Important unit note: M in the RMS formula is the molar mass in kilograms per mole (kg/mol). If your molar mass is given in g/mol, convert by dividing by 1000: M\big|{kg/mol} = \frac{M\big|{g/mol}}{1000}.
Relationship: RMS speed is inversely proportional to the square root of molar mass, i.e.
u_{rms} \propto \frac{1}{\sqrt{M}}.
Practical implication: Lighter molecules (smaller M) have larger RMS speeds at the same temperature than heavier molecules.

Objective: Determine how much faster a helium atom moves on average than a carbon dioxide molecule at the same temperature.
Key molar masses (typical values):
- Helium: M_{He} = 4.00\ \text{g mol}^{-1}
- Carbon dioxide: M{CO2} = 44.01\ \text{g mol}^{-1}
Using the RMS relation, the ratio of speeds is:
{u{rms, He} \over u{rms, CO2}} = \sqrt{ {M{CO2} \over M{He}} }\,.
- Substitution (in g/mol, since the ratio cancels units when using this form):
\sqrt{\frac{44.01}{4.00}} = \sqrt{11.0025} \approx 3.32.
Answer: Helium moves about 3.3 times faster on average than CO₂ at the same temperature (rounded).
Note on solving strategy: If you want to avoid converting to kg/mol