Referring to Figure 5.17, we see that the maximum of each...

Referring to Figure $5.17$, we see that the maximum of each speed distribution plot is called the most probable speed $\left(u_{\mathrm{mp}}\right)$ because it is the speed possessed by the largest number of molecules. It is given by $u_{\mathrm{mp}}=\sqrt{2 R T / M}$. (a) Compare $u_{\mathrm{mp}}$ with $u_{\mathrm{rms}}$ for nitrogen at $25^{\circ} \mathrm{C}$. (b) The following diagram shows the Maxwell speed distribution curves for an ideal gas at two different temperatures $T_{1}$ and $T_{2}$. Calculate the value of $T_{2}$

Key Concepts

Relationship between Molecular Speeds and Temperature

In the kinetic theory of gases, the temperature is a measure of the average kinetic energy of the molecules. Both the most probable speed and the rms speed depend on temperature, and their specific formulas illustrate different aspects of the speed distribution. While the most probable speed identifies the peak of the distribution, the rms speed provides an average that is weighted more heavily by higher speeds.

Root-Mean-Square (rms) Speed

The root-mean-square speed is a measure of the average speed of the gas molecules, defined as the square root of the average of the squares of individual molecular speeds. It is directly related to the temperature of the gas through the kinetic energy of the molecules, providing a way to connect microscopic motion with the macroscopic temperature.

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution describes how the speeds of molecules in a gas are statistically distributed. It shows that while most molecules have speeds near a certain value, there is a spread of speeds due to random thermal motions. This distribution is central to the kinetic theory of gases and helps in understanding macroscopic properties like temperature and pressure in terms of molecular motion.

Most Probable Speed

The most probable speed is the speed at which the maximum number of molecules are found in a gas, corresponding to the peak of the Maxwell-Boltzmann distribution. It is derived by finding the speed that maximizes the probability density function and is given by u_mp = ?(2RT/?) in the context of the gas, indicating a direct relationship with temperature.

Transcript

00:01 For this question, we are going to be using the most probable speed and also the root mean square speed.

00:09 So we have two formulas that we're dealing with.

00:12 And we want to compare, first of all, the most probable speed of root mean square speed for nitrogen at 25 degrees celsius.

00:18 So we need to convert the temperature to calvin.

00:23 So i add 273 to get the temperature in calvin.

00:28 You convert the molar mass to kilograms for mole, so divide the molar mass of nitrogen by 1 ,000 to get that in kilograms per mole.

00:38 Then apply the formulas.

00:40 So the value for r is 8 .314 jule over mole times k.

00:47 And that's because we want the units of the speed we get to be meters per second.

00:53 So be sure that you're using the correct value for r.

00:57 And then be sure that temperature is in kelvin, divided by the mole and mass in kilograms per mole.

01:06 Then we need to take the square root of that entire thing.

01:12 So for the next part, we're just going to use the other formula.

01:16 So the only difference is this is it's going to be a square root of 3rt divided by the mole and mass.

01:22 So we multiply by 3 instead of 2.

01:25 So we're just going to plug in the numbers.

01:39 And then we're going to get the root matrix.

01:42 Mean square speed.

01:57 So now for the next part, we have two diagrams, or we have a diagram, and we have two curves...

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