# 2.4 Distribution of molecular speeds  (Page 2/9)

 Page 2 / 9

[link] shows that the curve is shifted to higher speeds at higher temperatures, with a broader range of speeds.

With only a relatively small number of molecules, the distribution of speeds fluctuates around the Maxwell-Boltzmann distribution. However, you can view this simulation to see the essential features that more massive molecules move slower and have a narrower distribution. Use the set-up “2 Gases, Random Speeds”. Note the display at the bottom comparing histograms of the speed distributions with the theoretical curves.

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

$\overline{v}={\int }_{0}^{\infty }vf\left(v\right)dv=\sqrt{\frac{8}{\pi }\phantom{\rule{0.2em}{0ex}}\frac{{k}_{\text{B}}T}{m}}=\sqrt{\frac{8}{\pi }\phantom{\rule{0.2em}{0ex}}\frac{RT}{M}.}$

Similarly,

${v}_{\text{rms}}=\sqrt{\stackrel{\text{–}}{{v}^{2}}}=\sqrt{{\int }_{0}^{\infty }{v}^{2}f\left(v\right)dv}=\sqrt{\frac{3{k}_{\text{B}}T}{m}}=\sqrt{\frac{3RT}{M}}$

as in Pressure, Temperature, and RMS Speed . The most probable speed    , also called the peak speed     ${v}_{p},$ is the speed at the peak of the velocity distribution. (In statistics it would be called the mode.) It is less than the rms speed ${v}_{\text{rms}}.$ The most probable speed can be calculated by the more familiar method of setting the derivative of the distribution function, with respect to v , equal to 0. The result is

${v}_{p}=\sqrt{\frac{2{k}_{\text{B}}T}{m}}=\sqrt{\frac{2RT}{M}},$

which is less than ${v}_{\text{rms}}.$ In fact, the rms speed is greater than both the most probable speed and the average speed.

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

$f\left(v\right)=\frac{4{v}^{2}}{\sqrt{\pi }{v}_{p}^{3}}{e}^{\text{−}{v}^{2}\text{/}{v}_{p}^{2}}$

In the factor ${e}^{\text{−}m{v}^{2}\text{/}2{k}_{\text{B}}T}$ , it is easy to recognize the translational kinetic energy. Thus, that expression is equal to ${e}^{\text{−}K\text{/}{k}_{\text{B}}T}.$ The distribution f ( v ) can be transformed into a kinetic energy distribution by requiring that $f\left(K\right)dK=f\left(v\right)dv.$ Boltzmann showed that the resulting formula is much more generally applicable if we replace the kinetic energy of translation with the total mechanical energy E . Boltzmann’s result is

$f\left(E\right)=\frac{2}{\sqrt{\pi }}{\left({k}_{\text{B}}T\right)}^{-3\text{/}2}\sqrt{E}{e}^{\text{−}E\text{/}{k}_{\text{B}}T}=\frac{2}{\sqrt{\pi }{\left({k}_{\text{B}}T\right)}^{3\text{/}2}}\phantom{\rule{0.2em}{0ex}}\frac{\sqrt{E}}{{e}^{E\text{/}{k}_{\text{B}}T}}.$

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 ${e}^{E\text{/}{k}_{\text{B}}T}$ in the denominator is ubiquitous in quantum as well as classical statistical mechanics.

## 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, ${\text{m}}^{3}$ for volume, molecules for N , and moles for n ). In many cases, though, using R and the molar mass will be more convenient than using ${k}_{\text{B}}$ and the molecular mass.

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.

What is differential form of Gauss's law?
help me out on this question the permittivity of diamond is 1.46*10^-10.( a)what is the dielectric of diamond (b) what its susceptibility
a body is projected vertically upward of 30kmp/h how long will it take to reach a point 0.5km bellow e point of projection
i have to say. who cares. lol. why know that t all
Jeff
is this just a chat app about the openstax book?
kya ye b.sc ka hai agar haa to konsa part
what is charge quantization
it means that the total charge of a body will always be the integral multiples of basic unit charge ( e ) q = ne n : no of electrons or protons e : basic unit charge 1e = 1.602×10^-19
Riya
is the time quantized ? how ?
Mehmet
What do you meanby the statement,"Is the time quantized"
Mayowa
Can you give an explanation.
Mayowa
there are some comment on the time -quantized..
Mehmet
time is integer of the planck time, discrete..
Mehmet
planck time is travel in planck lenght of light..
Mehmet
it's says that charges does not occur in continuous form rather they are integral multiple of the elementary charge of an electron.
Tamoghna
it is just like bohr's theory. Which was angular momentum of electron is intral multiple of h/2π
determine absolute zero
The properties of a system during a reversible constant pressure non-flow process at P= 1.6bar, changes from constant volume of 0.3m³/kg at 20°C to a volume of 0.55m³/kg at 260°C. its constant pressure process is 3.205KJ/kg°C Determine: 1. Heat added, Work done, Change in Internal Energy and Change in Enthalpy
U can easily calculate work done by 2.303log(v2/v1)
Abhishek
Amount of heat added through q=ncv^delta t
Abhishek
Change in internal energy through q=Q-w
Abhishek
please how do dey get 5/9 in the conversion of Celsius and Fahrenheit
what is copper loss
this is the energy dissipated(usually in the form of heat energy) in conductors such as wires and coils due to the flow of current against the resistance of the material used in winding the coil.
Henry
it is the work done in moving a charge to a point from infinity against electric field
what is the weight of the earth in space
As w=mg where m is mass and g is gravitational force... Now if we consider the earth is in gravitational pull of sun we have to use the value of "g" of sun, so we can find the weight of eaeth in sun with reference to sun...
Prince
g is not gravitacional forcé, is acceleration of gravity of earth and is assumed constante. the "sun g" can not be constant and you should use Newton gravity forcé. by the way its not the "weight" the physical quantity that matters, is the mass
Jorge
Yeah got it... Earth and moon have specific value of g... But in case of sun ☀ it is just a huge sphere of gas...
Prince
Thats why it can't have a constant value of g ....
Prince
not true. you must know Newton gravity Law . even a cloud of gas it has mass thats al matters. and the distsnce from the center of mass of the cloud and the center of the mass of the earth
Jorge
please why is the first law of thermodynamics greater than the second
every law is important, but first law is conservation of energy, this state is the basic in physics, in this case first law is more important than other laws..
Mehmet
First Law describes o energy is changed from one form to another but not destroyed, but that second Law talk about entropy of a system increasing gradually
Mayowa
first law describes not destroyer energy to changed the form, but second law describes the fluid drection that is entropy. in this case first law is more basic accorging to me...
Mehmet
define electric image.obtain expression for electric intensity at any point on earthed conducting infinite plane due to a point charge Q placed at a distance D from it.
explain the lack of symmetry in the field of the parallel capacitor
pls. explain the lack of symmetry in the field of the parallel capacitor
Phoebe