Question

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

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

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.

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.

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.

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.

Key concepts

Molecular Speed Distribution

The concept of molecular speed distribution is essential for understanding how gases behave at a molecular level. In a gas, molecules are in constant motion, each traveling at different speeds. These speeds are not uniform but rather distributed across a range of values. The Maxwell-Boltzmann distribution describes this range, showing the probability of molecules moving at particular speeds.

• The majority of gas molecules move at an average speed, known as the most probable speed.
• A smaller number of molecules travel at both much higher and much lower speeds.
• The distribution of these speeds can be represented in a graph, typically showing speed on the x-axis and the number of molecules on the y-axis.

In the gas world, understanding the molecular speed distribution allows us to visualize and predict how gases will behave under different conditions. For instance, when sketching the distribution for different gases and temperatures, as in our exercise, we look at the changes in the shape and width of these curves.

Temperature Effect on Gases

Temperature plays a crucial role in determining how fast molecules are moving in a gas. When the temperature increases, the kinetic energy of the gas molecules also increases. This change influences the molecular speed distribution in a notable way.

• As temperature rises, the average kinetic energy and the average speed of the molecules increase.
• The distribution curve becomes wider and flatter, indicating a broader range of molecular speeds.
• Higher temperatures cause molecules to move faster and disperse more widely across speed ranges.

In terms of our exercise, comparing krypton gas at -50°C to krypton gas at 0°C demonstrates this effect. The latter will have a wider and more spread out curve, showing the molecules moving at more varied speeds due to the higher temperature. This visual shift illustrates how temperature alterations can drastically change molecular motion.

Molar Mass and Molecular Speed

Molar mass significantly influences the molecular speed of gases. Heavier gases have molecules that generally move slower compared to lighter gases, assuming they're at the same temperature. Here's how molecular speed is affected by molar mass:

• Gases with a higher molar mass (like krypton) will have a molecular speed distribution that is taller and narrower. This means their molecules tend to have a smaller range of speeds.
• Conversely, gases with a lower molar mass (like argon) have a wider and flatter distribution curve, indicating that the molecules achieve a broader range of speeds.
• This is due to less massive molecules requiring less kinetic energy to attain higher speeds.

In our exercise, when krypton is compared to argon at the same temperature, argon's lighter molar mass makes its speed distribution curve appear wider and flatter. This contrast is a direct consequence of differing molar masses, impacting how gas molecules behave and distribute their speeds.