Speed Distribution of Ideal Gases: Videos & Practice Problems

Gas particles exhibit a range of speeds, represented by Maxwell-Boltzmann distributions. Key speeds include the most probable speed (V_MP), average speed (V_avg), and root mean square speed (V_RMS). The equations are: V_MP = √(2RT/M), V_avg = √(8RT/πM), and V_RMS = √(3RT/M). As temperature increases, all speeds rise, indicating greater energy and speed distribution among particles. Understanding these concepts is crucial for grasping gas behavior in thermodynamics.

Speed Distribution & Special Speeds of Ideal Gases

In the study of ideal gases, understanding different types of speeds is crucial. Among these, the most probable speed, average speed, and root mean square (RMS) speed are key concepts that describe the behavior of gas particles. These speeds can be visualized through Maxwell-Boltzmann distributions, which illustrate how gas particles do not all travel at the same speed. Instead, their speeds vary, and the distribution of these speeds can be plotted on a graph, showing the number of particles against their speeds at different temperatures.

The most probable speed (\(V_{MP}\)) is the speed at which the highest number of particles are found, corresponding to the peak of the distribution curve. The equation for calculating the most probable speed is given by:

\[ V_{MP} = \sqrt{\frac{2RT}{M}} \]

where \(R\) is the universal gas constant, \(T\) is the temperature in Kelvin, and \(M\) is the molar mass of the gas in kilograms per mole. For example, at a temperature of 300 K and a molar mass of 0.028 kg/mol, the most probable speed calculates to approximately 422 m/s.

The average speed (\(V_{avg}\)) is another important measure, representing the true average of the particle speeds. It is calculated using the formula:

\[ V_{avg} = \sqrt{\frac{8RT}{\pi M}} \]

Using the same values, the average speed at 300 K is found to be about 476.3 m/s. This value is typically higher than the most probable speed.

The RMS speed (\(V_{RMS}\)) is a specific type of average that gives more weight to higher speeds, calculated with the formula:

\[ V_{RMS} = \sqrt{\frac{3RT}{M}} \]

For the same conditions, the RMS speed is approximately 516 m/s, which is the highest of the three speeds discussed. This hierarchy of speeds—most probable speed being the lowest, followed by average speed, and then RMS speed—holds true across different temperatures.

As temperature increases, all three speeds rise, reflecting the increased kinetic energy of the gas particles. For instance, at 600 K, the most probable speed increases to about 596.9 m/s, indicating that higher temperatures lead to a broader distribution of speeds, with particles moving faster on average.

In summary, the most probable speed, average speed, and RMS speed are essential for understanding the kinetic behavior of gas particles, and their calculations provide insight into how temperature affects gas dynamics.

The escape velocity from the Earth is approximately 11.2 km/s. If the mass of helium atoms is 6.64 × 10^-27 kg, at what temperature would the average speed of helium atoms be equal to the escape velocity?

• A. 23,700 K
• B. 2.11 K
• C. 4.5 × 10^-42 K
• D. 3.9 × 10^-20 K

Answer: (A)

Probability Distribution Graph

In this problem, we analyze the probability distribution of oxygen gas at a specific temperature, which is represented by a curve on a graph. The farthest tick mark on the x-axis indicates a velocity of 1200 meters per second. To determine the temperature of the oxygen gas sample, we need to identify the most probable velocity, which occurs at the peak of the probability distribution curve.

The most probable velocity (\(V_{mp}\)) can be found by interpreting the graph. By working backward from the tick marks, we establish that the most probable velocity is 400 meters per second. This value is crucial as it will be used in the equation that relates the most probable velocity to temperature.

To find the temperature (\(T\)), we utilize the equation for the most probable velocity, which is given by:

\[ V_{mp} = \sqrt{\frac{2RT}{M}} \]

In this equation, \(R\) is the universal gas constant (8.314 J/(mol·K)), and \(M\) is the molar mass of the gas. For oxygen, the molar mass is provided as 0.032 kilograms per mole (after converting from grams per mole).

To isolate \(T\), we first square both sides of the equation:

\[ V_{mp}^2 = \frac{2RT}{M} \]

Next, we rearrange the equation to solve for \(T\):

\[ T = \frac{V_{mp}^2 \cdot M}{2R} \]

Substituting the known values into the equation, we have:

\[ T = \frac{(400 \, \text{m/s})^2 \cdot 0.032 \, \text{kg/mol}}{2 \cdot 8.314 \, \text{J/(mol·K)}} \]

Calculating this gives:

\[ T \approx 307.9 \, \text{K} \]

This result indicates that the temperature of the oxygen gas sample is approximately 307.9 Kelvin. Understanding the relationship between the most probable velocity and temperature is essential in thermodynamics and kinetic theory, as it illustrates how molecular motion correlates with temperature in gases.

Here’s what students ask on this topic:

What is the Maxwell-Boltzmann distribution in the context of ideal gases?

The Maxwell-Boltzmann distribution describes the distribution of speeds among particles in an ideal gas. It shows that gas particles have a range of speeds, with some moving very slowly and others very quickly. The distribution is temperature-dependent, meaning that as the temperature increases, the distribution curve flattens and shifts to higher speeds. This distribution helps in understanding the behavior of gas particles, including the most probable speed (V_MP), average speed (V_avg), and root mean square speed (V_RMS).

How do you calculate the most probable speed of gas particles?

The most probable speed (V_MP) is the speed at which the maximum number of gas particles are moving. It can be calculated using the equation:

$\sqrt{\frac{2}{RT}/M}$

where R is the universal gas constant, T is the temperature in Kelvin, and M is the molar mass of the gas. This speed corresponds to the peak of the Maxwell-Boltzmann distribution curve.

What is the difference between average speed and root mean square speed in ideal gases?

The average speed (V_avg) is the arithmetic mean of the speeds of all gas particles, calculated using:

$\sqrt{\frac{8}{RT}/\pi M}$

The root mean square speed (V_RMS) is a type of average that gives more weight to higher speeds, calculated using:

$\sqrt{\frac{3}{RT}/M}$

V_RMS is always higher than V_avg because it involves squaring the speeds before averaging, which skews the result towards higher values.

How does temperature affect the speed distribution of gas particles?

As temperature increases, the speed distribution of gas particles shifts to higher speeds and the distribution curve flattens. This means that at higher temperatures, gas particles have more kinetic energy and a wider range of speeds. The most probable speed (V_MP), average speed (V_avg), and root mean square speed (V_RMS) all increase with temperature, indicating that particles move faster on average.

What are the key equations for calculating the speeds of gas particles in an ideal gas?

The key equations for calculating the speeds of gas particles in an ideal gas are:

1. Most probable speed (V_MP):

$\sqrt{\frac{2}{RT}/M}$

2. Average speed (V_avg):

$\sqrt{\frac{8}{RT}/\pi M}$

3. Root mean square speed (V_RMS):

$\sqrt{\frac{3}{RT}/M}$

where R is the universal gas constant, T is the temperature in Kelvin, and M is the molar mass of the gas.