U3.5 · Kinetic Molecular TheoryPractice · 分子运动论

At the same temperature, molecules of the two gases have equal average kinetic energy, ½mv², so speed is inversely proportional to the square root of molar mass (Graham’s law): rate($\ce{CH4}$) ÷ rate($\ce{SO2}$) = √(64.06 ÷ 16.04) = 2.00. $\ce{SO2}$ is about 4 times as heavy, so it moves about half as fast.

The dashed peak is farther left and taller: the molecules are slower overall, with speeds bunched more closely. Lowering the temperature reduces the average kinetic energy and does exactly that (the most probable speed drops from about 1117 m/s to about 790 m/s). Raising the temperature shifts the peak right and lowers it, and switching to lighter $\ce{H2}$ also shifts it right. The distribution shows the fraction of molecules at each speed, so adding more He at the same temperature doesn’t change its shape.

Pressure comes from molecules striking the walls. Cooling lowers the average kinetic energy, so the molecules move more slowly: they hit the walls less often per second and with less force each time, and the pressure drops. The container is sealed, so the number of molecules is unchanged, and the molecules don’t shrink.

Average kinetic energy is proportional to absolute temperature. 27 °C = 300.15 K; doubling gives 600.30 K, which is 600.30 − 273.15 = 327 °C. Doubling the Celsius value to 54 °C is wrong.

The first three are assumptions of kinetic molecular theory. At a given temperature, the particles have a range of speeds (the Maxwell–Boltzmann distribution); temperature fixes the average kinetic energy, not each particle’s speed.

A higher temperature means a higher average kinetic energy: the peak moves to a higher speed (422 → 731 m/s), and the distribution gets wider and lower. The area under each curve represents the total number of molecules, which is the same for both samples; Y’s peak is lower because its molecules are spread over a wider range of speeds. Heavier molecules at the same temperature would shift the peak to lower speeds, the opposite of Y.