Question

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

Short answer

To qualitatively sketch the distribution of molecular speeds for (a) Kr(g) at -50°C (223 K), (b) Kr(g) at 0°C (273 K), and (c) Ar(g) at 0°C (273 K), use the Maxwell-Boltzmann distribution and consider the effects of temperature and molar mass on the curves. The curve for (a) should show lower average speeds and be more narrow due to Kr's higher molar mass and lower temperature. The curve for (b) should have higher average speeds and be wider than curve (a) due to the higher temperature, but similar in narrowness as it has the same molar mass. The curve for (c) should be to the right of curve (b) and be slightly wider due to Ar's lower molar mass at the same temperature as (b). Plot the curves on a single graph, ensuring their order and shapes.

Step-by-step solution

The Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution, which is given by the formula: \(f(v) = 4\pi \left( \dfrac{m}{2\pi k T} \right)^{\frac{3}{2}} v^2 e^{-\frac{mv^2}{2kT}}\) where m is the molar mass of the gas, k is the Boltzmann constant, T is the temperature in Kelvin and v is the molecular speed. This distribution function essentially gives the probability of a molecule having a particular speed v in the gas at temperature T.

Convert temperatures to Kelvin

We need to use absolute temperatures (in Kelvin) to work with the Maxwell-Boltzmann distribution. Therefore, we need to convert the given temperatures to Kelvin: - -50°C = 223 K - 0°C = 273 K

Effect of temperature and molar mass on distribution

Higher temperature means that the molecules have more kinetic energy, which will result in a wider distribution of speeds and the curves will show higher average speeds. On the other hand, gases with higher molar mass (heavier gases) have less energetic molecules, resulting in a more narrow distribution of speeds. From the periodic table, we know that the molar mass of Krypton (Kr) is 83.8 g/mol and that of Argon (Ar) is 39.9 g/mol. Thus, Krypton is heavier than Argon.

Sketching the distribution of molecular speeds

Now we can qualitatively sketch the distribution of molecular speeds considering the effects of temperature and molar mass on each curve: - (a) Kr at -50°C = 223 K: The curve representing Krypton at lower temperature should show lower average speeds and be more narrow due to its higher molar mass. - (b) Kr at 0°C = 273 K: The curve for Krypton at a higher temperature should have higher average speeds. The curve will be wider than curve (a), but similar in narrowness as it has the same molar mass. - (c) Ar at 0°C = 273 K: Argon gas at the same temperature as Kr in case (b) will have a lower molar mass than Krypton, leading to a broader distribution of speeds. The curve should be to the right of curve (b) and be slightly wider. Now, plot the curves on a single graph, ensuring that curve (a) is the most narrow and lowest average speed, curve (b) is wider with higher average speeds, and curve (c) is the broadest curve with speeds similar to curve (b).

Key concepts

Molecular Speed Distribution

The Maxwell-Boltzmann distribution is a fundamental concept in understanding how the speeds of molecules in a gas are distributed. Imagine a bustling airport where you see people moving at various speeds. Similarly, molecules in a gas move at different speeds, from quite slow ones to super-fast ones. The distribution of these speeds can be visualized in a graph where the x-axis represents the speed and the y-axis represents the number of molecules moving at each speed.

In the graph, we typically see a bell-shaped curve. Most molecules move at a speed near the peak of the curve, called the most probable speed. Some molecules move slower, and some move faster, representing the tail ends of the curve. This spread of speeds comes about due to the chaotic nature of atomic movements, influenced by energy exchanges among molecules.

The mathematical representation of this is the Maxwell-Boltzmann distribution formula: \[ f(v) = 4\pi \left( \dfrac{m}{2\pi k T} \right)^{\frac{3}{2}} v^2 e^{-\frac{mv^2}{2kT}} \]where \(m\) is the molar mass, \(k\) is the Boltzmann constant, \(T\) is the absolute temperature, and \(v\) is the molecular speed. This equation helps to calculate the probability that a given molecule within the gas has a particular speed.

Effect of Temperature on Gas Particles

Temperature greatly affects the behavior and speed of gas particles. When you heat a gas, you're essentially providing more energy to the molecules. This added energy increases their velocity. If you think about a playground, a sunny day makes kids run faster whereas a cold day makes them sluggish. Similarly, gas molecules move quicker at higher temperatures.

In the context of our exercise, gas molecules at 0°C are warmer and thus, move faster compared to those at -50°C. This means the Maxwell-Boltzmann distribution curve for gases at higher temperatures will stretch wider and shift rightward, indicating a greater spread of speeds and a higher average speed.

• At higher temperatures, expect a broader and flatter curve, showcasing a wider range and more particles having higher speeds.
• At lower temperatures, curves are taller and narrower, indicating fewer higher-speed particles and a peak shifted to lower speeds.

Thus, as temperature rises, the graph evidences more chaos due to increased kinetic energy of the molecules.

Effect of Molar Mass on Gas Particles

Molar mass is another crucial factor that influences the speed distribution of gas particles. Think of different-sized cars going down a hill. Smaller cars (lighter molecules) generally accelerate more easily, while heavier trucks (heavier molecules) do not speed up as swiftly. In the gas world, lighter molecules like Argon (Ar) move faster on average than heavier molecules like Krypton (Kr).

This translates to the Maxwell-Boltzmann curves where gases with lighter molar masses have speed distributions that are broader. The peak is lower and further to the right, indicating faster molecular movement.

• Heavier gases have a narrow distribution, signifying more molecules moving at lower speeds.
• Lighter gases spread out more, showing a greater number of molecules at higher speeds.

This is why, in our exercise example, at the same temperature, Argon displays a wider speed distribution than Krypton. Such comprehension aids in visualizing the motion of gas particles in relation to their mass, ultimately affecting industrial and scientific applications where gas behavior must be precisely controlled.