Problem 7 - Chemistry: The Central Science

Question

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

Step-by-Step Solution

Answer

To sketch the distribution of molecular speeds for \(Kr(g)\) at -50°C, \(Kr(g)\) at 0°C, and \(Ar(g)\) at 0°C on a single plot, consider the effects of temperature and molecular mass on the Maxwell-Boltzmann distribution curve. For \(Kr(g)\) at -50°C, draw a bell curve with a peak closer to the y-axis and a relatively narrower width. For \(Kr(g)\) at 0°C, sketch a bell curve with a peak further to the right and a broader width than the previous case. Finally, for \(Ar(g)\) at 0°C, draw a bell curve with a peak further to the right compared to the \(Kr(g)\) curve at 0°C and with the broadest width.

Step 1: To create a single plot for the three cases, first draw the x and y-axes. The x-axis represents the molecular speed, and the y-axis represents the number of molecules with a given speed. #create_title#Step 2: Sketch the distribution for Kr(g) at -50 degrees Celsius

Since the temperature is lower for this case, the average speed of the molecules will be lower for \(Kr(g)\) at -50 degrees Celsius than for the other two cases. Sketch a bell curve with peak closer to the y-axis and relatively narrower width. #create_title#Step 3: Sketch the distribution for Kr(g) at 0 degrees Celsius

Step 2: Now, we want to sketch the distribution for \(Kr(g)\) at 0 degrees Celsius. Since the temperature is higher for this case, the average speed of the molecules will be higher than for the previous case. This means that the peak of the bell curve will be shifted towards the right on the x-axis. The width of the distribution will also be broader, indicating that there are more molecules with higher speeds. Sketch a bell curve with a peak further to the right of the previous curve and a broader width. #create_title#Step 4: Sketch the distribution for Ar(g) at 0 degrees Celsius

Finally, we want to sketch the distribution for \(Ar(g)\) at 0 degrees Celsius. Since the temperature is the same as for Kr(g) at 0 degrees Celsius, the effect of temperature is the same for both gases. However, Argon has a lower molecular mass compared to Krypton, which means that the average speed of the molecules will be higher. Therefore, the peak of the bell curve will be further to the right on the x-axis compared to the \(Kr(g)\) curve at 0 degrees Celsius. Since the mass of Argon is lower than Krypton, its distribution will also be broader. Sketch a bell curve with a peak further to the right of the previous curve and with the broadest width. After completing these four steps, you will have a qualitative sketch of the distribution of molecular speeds for the three cases: \(Kr(g)\) at -50 degrees Celsius, \(Kr(g)\) at 0 degrees Celsius, and \(Ar(g)\) at 0 degrees Celsius. The shape, position, and width of the bell curves illustrate the impact of temperature and molecular mass on the distribution of molecular speeds.

Key Concepts

Maxwell-Boltzmann Distribution

In the realm of gases, molecules move at a variety of speeds. The Maxwell-Boltzmann distribution is a fundamental principle that helps us understand these speed variations among molecules in a gas. This concept is showcased using a graph, where the x-axis represents molecular speed and the y-axis shows the number of molecules at a given speed.
The resulting curve displays a bell-shaped distribution, also known as the Maxwell-Boltzmann distribution. This shape is not uniform and depends on factors like temperature and the type of gas.

• The peak of the curve shows the most probable speed of the gas molecules, which is where most molecules are found.
• The distribution's width illustrates the range of speeds within the gas.

The concept of the Maxwell-Boltzmann distribution is essential for understanding how different gases behave under varying conditions. It reflects how molecular speeds are distributed across populations at given conditions, highlighting areas with the most or least molecules.

Temperature Effect on Gas

Temperature plays a significant role in the behavior of gas molecules. When you increase the temperature of a gas, the average speed of its molecules increases. This is because higher temperatures provide more kinetic energy to the molecules, boosting their speed.
In a Maxwell-Boltzmann distribution graph, an increase in temperature shifts the bell curve to the right, indicating higher molecular speeds.

• Low temperatures result in fewer high-speed molecules and a peak closer to the y-axis.
• As temperature rises, more molecules gain enough energy to move faster, broadening the curve.

Temperature changes also affect the range of speeds. Higher temperatures mean greater variability in speeds, hence a wider curve. Temperature is a key factor in interpreting Maxwell-Boltzmann distributions since it directly affects how gases will distribute their molecular energies.

Kinetic Molecular Theory

The Kinetic Molecular Theory provides a framework for understanding the behavior of gases. It postulates that gas molecules are in constant, random motion, colliding with each other and the walls of their containers.
This theory builds on the idea that the temperature of a gas is proportional to the average kinetic energy of its molecules.

• When gas molecules move faster, they have higher kinetic energy, which reflects as increased temperature.
• Any change in temperature impacts molecular motion, affecting speed distributions.

By understanding the Kinetic Molecular Theory, we can better predict how changes in conditions such as temperature will affect gas behavior. This theory is crucial in explaining why gases react and move in particular ways, making it a cornerstone concept in chemistry and physics.