What would happen to the difference and the ratio between most probable velocity and root mean square velocity if temperature is increased?

By the end of this section, you will be able to:

• Describe the distribution of molecular speeds in an ideal gas
• Find the average and most probable molecular speeds in an ideal gas

Particles in an ideal gas all travel at relatively high speeds, but they do not travel at the same speed. The rms speed is one kind of average, but many particles move faster and many move slower. The actual distribution of speeds has several interesting implications for other areas of physics, as we will see in later chapters.

The motion of molecules in a gas is random in magnitude and direction for individual molecules, but a gas of many molecules has a predictable distribution of molecular speeds. This predictable distribution of molecular speeds is known as the Maxwell-Boltzmann distribution, after its originators, who calculated it based on kinetic theory, and it has since been confirmed experimentally ((Figure)).

To understand this figure, we must define a distribution function of molecular speeds, since with a finite number of molecules, the probability that a molecule will have exactly a given speed is 0.

The Maxwell-Boltzmann distribution of molecular speeds in an ideal gas. The most likely speed

We define the distribution function

[Since N is dimensionless, the unit of f(v) is seconds per meter.] We can write this equation conveniently in differential form:

In this form, we can understand the equation as saying that the number of molecules with speeds between v and

We can now quote Maxwell’s result, although the proof is beyond our scope.

Maxwell-Boltzmann Distribution of Speeds

The distribution function for speeds of particles in an ideal gas at temperature T is

The factors before the

Calculating the Ratio of Numbers of Molecules Near Given Speeds In a sample of nitrogen

Strategy Since we’re looking at a small range, we can approximate the number of molecules near 100 m/s as

All we have to do is take the ratio of the two f values.

Solution

1. Identify the knowns and convert to SI units if necessary.
   
   
   

2. Substitute the values and solve.
   
   

(Figure) shows that the curve is shifted to higher speeds at higher temperatures, with a broader range of speeds.

The Maxwell-Boltzmann distribution is shifted to higher speeds and broadened at higher temperatures.

We can use a probability distribution to calculate average values by multiplying the distribution function by the quantity to be averaged and integrating the product over all possible speeds. (This is analogous to calculating averages of discrete distributions, where you multiply each value by the number of times it occurs, add the results, and divide by the number of values. The integral is analogous to the first two steps, and the normalization is analogous to dividing by the number of values.) Thus the average velocity is

Similarly,

as in Pressure, Temperature, and RMS Speed. The most probable speed, also called the peak speed

which is less than

The peak speed provides a sometimes more convenient way to write the Maxwell-Boltzmann distribution function:

In the factor

The first part of this equation, with the negative exponential, is the usual way to write it. We give the second part only to remark that

Problem-Solving Strategy: Speed Distribution

Step 1. Examine the situation to determine that it relates to the distribution of molecular speeds.

Step 2. Make a list of what quantities are given or can be inferred from the problem as stated (identify the known quantities).

Step 3. Identify exactly what needs to be determined in the problem (identify the unknown quantities). A written list is useful.

Step 4. Convert known values into proper SI units (K for temperature, Pa for pressure,

Step 5. Determine whether you need the distribution function for velocity or the one for energy, and whether you are using a formula for one of the characteristic speeds (average, most probably, or rms), finding a ratio of values of the distribution function, or approximating an integral.

Step 6. Solve the appropriate equation for the ideal gas law for the quantity to be determined (the unknown quantity). Note that if you are taking a ratio of values of the distribution function, the normalization factors divide out. Or if approximating an integral, use the method asked for in the problem.

Step 7. Substitute the known quantities, along with their units, into the appropriate equation and obtain numerical solutions complete with units.

We can now gain a qualitative understanding of a puzzle about the composition of Earth’s atmosphere. Hydrogen is by far the most common element in the universe, and helium is by far the second-most common. Moreover, helium is constantly produced on Earth by radioactive decay. Why are those elements so rare in our atmosphere? The answer is that gas molecules that reach speeds above Earth’s escape velocity, about 11 km/s, can escape from the atmosphere into space. Because of the lower mass of hydrogen and helium molecules, they move at higher speeds than other gas molecules, such as nitrogen and oxygen. Only a few exceed escape velocity, but far fewer heavier molecules do. Thus, over the billions of years that Earth has existed, far more hydrogen and helium molecules have escaped from the atmosphere than other molecules, and hardly any of either is now present.

We can also now take another look at evaporative cooling, which we discussed in the chapter on temperature and heat. Liquids, like gases, have a distribution of molecular energies. The highest-energy molecules are those that can escape from the intermolecular attractions of the liquid. Thus, when some liquid evaporates, the molecules left behind have a lower average energy, and the liquid has a lower temperature.

In the deep space between galaxies, the density of molecules (which are mostly single atoms) can be as low as

(a) Find the density in SI units of air at a pressure of 1.00 atm and a temperature of

a.

The air inside a hot-air balloon has a temperature of 370 K and a pressure of 101.3 kPa, the same as that of the air outside. Using the composition of air as

When an air bubble rises from the bottom to the top of a freshwater lake, its volume increases by

(a) Use the ideal gas equation to estimate the temperature at which 1.00 kg of steam (molar mass

One process for decaffeinating coffee uses carbon dioxide

a. supercritical fluid; b.

On a winter day when the air temperature is

On a warm day when the air temperature is

(a) People often think of humid air as “heavy.” Compare the densities of air with relative humidity and

The mean free path for helium at a certain temperature and pressure is

a.

The mean free path for methane at a temperature of 269 K and a pressure of

In the chapter on fluid mechanics, Bernoulli’s equation for the flow of incompressible fluids was explained in terms of changes affecting a small volume dV of fluid. Such volumes are a fundamental idea in the study of the flow of compressible fluids such as gases as well. For the equations of hydrodynamics to apply, the mean free path must be much less than the linear size of such a volume,

Find the total number of collisions between molecules in 1.00 s in 1.00 L of nitrogen gas at standard temperature and pressure (

(a) Estimate the specific heat capacity of sodium from the Law of Dulong and Petit. The molar mass of sodium is 23.0 g/mol. (b) What is the percent error of your estimate from the known value,

a.

A sealed, perfectly insulated container contains 0.630 mol of air at

Find the ratio

Unreasonable results. (a) Find the temperature of 0.360 kg of water, modeled as an ideal gas, at a pressure of

Unreasonable results. (a) Find the average speed of hydrogen sulfide,

a. 411 m/s; b. According to (Figure), the