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Chapter 10: Problem 8

On a single plot, qualitatively sketch the distribution of molecular speeds for (a) \(\mathrm{Kr}(g)\) at $-50^{\circ} \mathrm{C},(\mathbf{b}) \mathrm{Kr}(g)\( at \)0^{\circ} \mathrm{C}$ (c) \(\mathrm{Ar}(g)\) at \(0^{\circ} \mathrm{C}\). [Section \(\left.10.7\right]\)

Short Answer

Expert verified
To qualitatively sketch the distribution of molecular speeds for the given cases: 1. For Kr at -50°C, draw a relatively tall and narrow curve. 2. For Kr at 0°C, draw a wider and flatter curve than the curve for Kr at -50°C. 3. For Ar at 0°C, draw a curve that is wider and flatter compared to the curve for Kr at 0°C. Label each curve accordingly. The distribution curves demonstrate the effects of temperature and molar mass on molecular speeds based on Maxwell-Boltzmann distribution.

Step by step solution

01

Recall Maxwell-Boltzmann distribution's effect on temperature

When temperature increases, the distribution curve becomes wider and flatter because the gas molecules have more kinetic energy and can achieve a wider range of speeds. In other words, the average speed of the molecules increases with temperature.
02

Recall Maxwell-Boltzmann distribution's effect on molar mass

When the molar mass of a gas increases, the distribution curve becomes narrower and taller. This means that the heavier gas has a smaller range of molecular speeds. In other words, the molecules of a heavier gas, on average, move slower than those of a lighter gas at the same temperature.
03

Compare the cases in the problem

We have three cases to compare: 1. Krypton gas (Kr) at -50°C 2. Krypton gas (Kr) at 0°C 3. Argon gas (Ar) at 0°C When we compare cases 1 and 2 (both Kr at different temperatures), we can see that the temperature is higher for case 2. Hence, the distribution curve for case 2 will be wider and flatter than that for case 1. Now, comparing cases 2 and 3 (different gases at the same temperature), we can see that Argon has a lower molar mass than Krypton (Ar = 39.948 g/mol, Kr = 83.798 g/mol). Therefore, the distribution curve for case 3 will be wider and flatter than that for case 2.
04

Sketch the distributions qualitatively

Based on our analysis, we can now sketch the distributions qualitatively: 1. For Kr at -50°C, draw a curve that is relatively tall and narrow. 2. For Kr at 0°C, draw a curve that is wider and flatter than the curve for Kr at -50°C. 3. For Ar at 0°C, draw a curve that is wider and flatter compared to the curve for Kr at 0°C. Make sure to label each curve accordingly. This qualitative sketch should give an idea of how the distribution of molecular speeds varies with temperature and molar mass.

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