Question

The root mean square speed, \(v_{\text {rms }}\), of gas molecules was derived in Section 4.10. Using the Maxwell distribution of speeds, we can also calculate the mean speed and most probable (mp) speed of a collection of molecules. The equations used to calculate these two quantities are \(v_{\text {mean }}=(8 R T / \pi M)^{1 / 2}\) and \(v_{\mathrm{mp}}=(2 R T / M)^{1 / 2}\). These values have a fixed relation to one another. (a) Place these three quantities in order of increasing magnitude. (b) Show that the relative magnitudes are independent of the molar mass of the gas. (c) Using the smallest speed as the reference point, rank them by magnitude and determine the ratios of the larger values to the smallest.

Short answer

The speeds are ranked from smallest to greatest as: most probable, mean, and RMS. While the exact coefficients for the RMS speed are not given, it can generally be said that the ratio of mean speed to the most probable is \(\sqrt{\frac{4}{\pi}}\), and the RMS speed is greater still. The relative magnitudes are independent of the molar mass of the gas.

Step-by-step solution

Identify the equations for different speed measures

We have three expressions for the speed of gas molecules. The root mean square speed is given by the relation not provided in the exercise. The mean speed is given by, \(v_{\text{mean}} = \left(\frac{8RT}{\pi M}\right)^{1/2}\), and the most probable speed by \(v_{\text{mp}} = \left(\frac{2RT}{M}\right)^{1/2}\), where \(R\) is the gas constant, \(T\) is the temperature, and \(M\) is the molar mass of the gas.

Compare the constants in the speed equations

To compare the different speeds, we examine the coefficients in each equation: \(\frac{2RT}{M}\), \(\frac{8RT}{\pi M}\), and the coefficient in the RMS speed equation (not provided). Since \(R\) and \(T\) are common to all and \(M\) is the molar mass, their values do not change the order of magnitude. We now only need to compare the numerical values of the coefficients 2, \(\frac{8}{\pi}\), and the one for RMS speed.

Arrange the coefficients in ascending order

The number 2 for the most probable speed is the smallest coefficient. Since \(\pi\) is approximately 3.14159, \(\frac{8}{\pi}\) is greater than 2. The RMS speed typically has a coefficient higher than both mean and most probable speeds (the exact coefficient for RMS speed is needed to confirm its precise value, but we know it is higher than 2 and \(\frac{8}{\pi}\)). Therefore, the order from smallest to largest speed is: most probable, mean, and RMS speeds.

Demonstrate the independence of relative magnitudes

To show that relative magnitudes are independent of the molar mass, we note that each speed is proportional to \(\sqrt{\frac{T}{M}}\). Since \(M\) is in the denominator under a square root, any change in \(M\) affects all speeds equally. Thus, the relative order of magnitudes does not change with different molar masses.

Calculate the ratios of the larger speeds to the smallest

Using the most probable speed as reference (smallest speed), we find the ratio of the mean speed to the most probable speed by dividing their coefficients: \(\sqrt{\frac{8/\pi}{2}} = \sqrt{\frac{4}{\pi}} = \sqrt{\frac{4}{3.14159}}\). Similarly, the ratio of the RMS to the most probable speed (which we cannot calculate exactly without the RMS coefficient) will be higher, being the largest value. The calculated ratio provides a reference for mean speed relative to most probable speed.

Key concepts

Maxwell Distribution

Understanding how molecules in a gas move is critical for grasping concepts in kinetic theory and thermodynamics. Maxwell distribution describes the way gas molecule speeds are spread out at a given temperature. This mathematical model explains that in a sample of gas, there are molecules moving at various speeds. Some move slowly, some at a moderate pace, and others very fast, but they're most likely to be found at a particular speed range—the most probable speed.

What's unique about the Maxwell distribution is that it allows scientists to predict the likelihood of finding a particle at a certain speed within a gas. It's depicted with a curve that starts at zero, rises to a peak at the most probable speed, and then gradually tapers off. This distribution is fundamental not just in physics but also in chemistry where it explains rates of reactions, diffusion, and even the behavior of gas under different conditions.

Root Mean Square Speed

Now, let's dive into a specific term from the Maxwell distribution, the root mean square (rms) speed, which is a measure of the average velocity of gas molecules. It's derived from the individual speeds of all molecules, considering their direction and magnitude.

Mathematically, the rms speed, denoted as \(v_{\text{rms}}\), is calculated using the formula not explicitly given in the exercise. However, rms speed generally incorporates the mass of gas molecules and the temperature in its calculation. The concept is important because rms speed relates to the kinetic energy of gas particles - a fundamental aspect to understand in thermal physics. Note that rms speed is typically higher than both mean and most probable speeds because it squares the velocities before averaging, thus giving more weight to higher speeds.

Mean Speed of Gas Molecules

Apart from the root mean square speed, we also deal with the mean speed of gas molecules. This represents the average speed at which the gas molecules are moving. It doesn't square the velocities like the rms speed does but takes a more straightforward average. The mean speed, denoted as \(v_{\text{mean}}\), is given by the expression \(\left(\frac{8RT}{\pi M}\right)^{1/2}\) where \(R\) is the gas constant, \(T\) is the absolute temperature in Kelvins, and \(M\) is the molar mass.

The significance of the mean speed lies in its simplicity and frequent use in theoretical predictions. For instance, it's used when calculating the rate of effusion of gases through a small hole, known as Graham's law. Its calculation is less complex than that of the rms speed and provides a readily understandable measure of average molecular motion.

Most Probable Speed

The most probable speed, denoted as \(v_{\text{mp}}\), represents the speed at which you're most likely to find a gas molecule within a given sample, according to the Maxwell distribution. It's mathematically represented as \(\left(\frac{2RT}{M}\right)^{1/2}\) and is the speed corresponding to the peak of the Maxwell distribution curve.

Being the most common speed, it's a vital concept in understanding gas behavior at a microscopic level. It can give insights into phenomena like diffusion, where particles spread from an area of higher concentration to an area of lower concentration. When looking at the bigger picture, the most probable speed is critical for conceptualizing how gases behave and react when heated, compressed, or mixed with other substances.