# 2.2 Pressure, temperature, and rms speed  (Page 3/18)

 Page 3 / 18
${F}_{i}=\frac{\text{Δ}{p}_{i}}{\text{Δ}t}=\frac{2m{v}_{ix}}{\text{Δ}t}.$

(In this equation alone, p represents momentum, not pressure.) There is no force between the wall and the molecule except while the molecule is touching the wall. During the short time of the collision, the force between the molecule and wall is relatively large, but that is not the force we are looking for. We are looking for the average force, so we take $\text{Δ}t$ to be the average time between collisions of the given molecule with this wall, which is the time in which we expect to find one collision. Let l represent the length of the box in the x -direction. Then $\text{Δ}t$ is the time the molecule would take to go across the box and back, a distance 2 l , at a speed of ${v}_{x}.$ Thus $\text{Δ}t=2l\text{/}{v}_{x},$ and the expression for the force becomes

${F}_{i}=\frac{2m{v}_{ix}}{2l\text{/}{v}_{ix}}=\frac{m{v}_{ix}^{2}}{l}.$

This force is due to one molecule. To find the total force on the wall, F , we need to add the contributions of all N molecules:

$F=\sum _{i=1}^{N}{F}_{i}=\sum _{i=1}^{N}\frac{m{v}_{ix}^{2}}{l}=\frac{m}{l}\sum _{i=1}^{N}{v}_{ix}^{2}.$

We now use the definition of the average, which we denote with a bar, to find the force:

$F=N\frac{m}{l}\left(\frac{1}{N}\sum _{i=1}^{N}{v}_{ix}^{2}\right)=N\frac{m\stackrel{\text{–}}{{v}_{x}^{2}}}{l}.$

We want the force in terms of the speed v , rather than the x -component of the velocity. Note that the total velocity squared is the sum of the squares of its components, so that

$\stackrel{\text{–}}{{v}^{2}}=\stackrel{\text{–}}{{v}_{x}^{2}}+\stackrel{\text{–}}{{v}_{y}^{2}}+\stackrel{\text{–}}{{v}_{z}^{2}}.$

With the assumption of isotropy, the three averages on the right side are equal, so

$\stackrel{\text{–}}{{v}^{2}}=3\stackrel{\text{–}}{{v}_{ix}^{2}}.$

Substituting this into the expression for F gives

$F=N\frac{m\stackrel{\text{–}}{{v}^{2}}}{3l}.$

The pressure is F / A , so we obtain

$p=\frac{F}{A}=N\frac{m\stackrel{\text{–}}{{v}^{2}}}{3Al}=\frac{Nm\stackrel{\text{–}}{{v}^{2}}}{3V},$

where we used $V=Al$ for the volume. This gives the important result

$pV=\frac{1}{3}\phantom{\rule{0.2em}{0ex}}Nm\stackrel{\text{–}}{{v}^{2}}.$

Combining this equation with $pV=N{k}_{\text{B}}T$ gives

$\frac{1}{3}\phantom{\rule{0.2em}{0ex}}Nm\stackrel{\text{–}}{{v}^{2}}=N{k}_{\text{B}}T.$

We can get the average kinetic energy of a molecule, $\frac{1}{2}\phantom{\rule{0.2em}{0ex}}m\stackrel{\text{–}}{{v}^{2}}$ , from the left-hand side of the equation by dividing out N and multiplying by 3/2.

## Average kinetic energy per molecule

The average kinetic energy of a molecule is directly proportional to its absolute temperature:

$\stackrel{\text{–}}{K}=\frac{1}{2}\phantom{\rule{0.2em}{0ex}}m\stackrel{\text{–}}{{v}^{2}}=\frac{3}{2}\phantom{\rule{0.2em}{0ex}}{k}_{\text{B}}T.$

The equation $\stackrel{\text{–}}{K}=\frac{3}{2}\phantom{\rule{0.2em}{0ex}}{k}_{\text{B}}T$ is the average kinetic energy per molecule. Note in particular that nothing in this equation depends on the molecular mass (or any other property) of the gas, the pressure, or anything but the temperature. If samples of helium and xenon gas, with very different molecular masses, are at the same temperature, the molecules have the same average kinetic energy.

The internal energy    of a thermodynamic system is the sum of the mechanical energies of all of the molecules in it. We can now give an equation for the internal energy of a monatomic ideal gas. In such a gas, the molecules’ only energy is their translational kinetic energy. Therefore, denoting the internal energy by ${E}_{\text{int}},$ we simply have ${E}_{\text{int}}=N\stackrel{\text{–}}{K},$ or

${E}_{\text{int}}=\frac{3}{2}\phantom{\rule{0.2em}{0ex}}N{k}_{\text{B}}T.$

Often we would like to use this equation in terms of moles:

${E}_{\text{int}}=\frac{3}{2}\phantom{\rule{0.2em}{0ex}}nRT.$

We can solve $\stackrel{\text{–}}{K}=\frac{1}{2}\phantom{\rule{0.2em}{0ex}}m\stackrel{\text{–}}{{v}^{2}}=\frac{3}{2}\phantom{\rule{0.2em}{0ex}}{k}_{\text{B}}T$ for a typical speed of a molecule in an ideal gas in terms of temperature to determine what is known as the root-mean-square ( rms ) speed of a molecule.

## Rms speed of a molecule

The root-mean-square (rms) speed    of a molecule, or the square root of the average of the square of the speed $\stackrel{\text{–}}{{v}^{2}}$ , is

${v}_{\text{rms}}=\sqrt{\stackrel{\text{–}}{{v}^{2}}}=\sqrt{\frac{3{k}_{\text{B}}T}{m}}.$

The rms speed is not the average or the most likely speed of molecules, as we will see in Distribution of Molecular Speeds , but it provides an easily calculated estimate of the molecules’ speed that is related to their kinetic energy. Again we can write this equation in terms of the gas constant R and the molar mass M in kg/mol:

A closely wound search coil has an area of 4cm^2,1000 turns and a resistance of 40ohm. It is connected to a ballistic galvanometer whose resistance is 24 ohm. When coil is rotated from a position parallel to uniform magnetic field to one perpendicular to field,the galvanometer indicates a charge
Using Kirchhoff's rules, when choosing your loops, can you choose a loop that doesn't have a voltage?
how was the check your understand 12.7 solved?
Who is ISSAAC NEWTON
he's the father of 3 newton law
Hawi
he is Chris Issaac's father :)
Ethem
how to name covalent bond
Who is ALEXANDER BELL
LOAK
what do you understand by the drift voltage
what do you understand by drift velocity
Brunelle
nothing
Gamal
well when you apply a small electric field to a conductor that causes to add a little velocity to charged particle than usual, which become their average speed, that is what we call a drift.
graviton
drift velocity
graviton
what is an electromotive force?
It is the amount of other forms of energy converted into electrical energy per unit charge that flow through it.
Brunelle
How electromotive force is differentiated from the terminal voltage?
Danilo
in the emf power is generated while in the terminal pd power is lost.
Brunelle
what is then chemical name of NaCl
sodium chloride
Azam
sodium chloride
Brunelle
Sodium Chloride.
Ezeanyim
How can we differentiate between static point and test charge?
Wat is coplanar in physics
two point charges +30c and +10c are separated by a distance of 80cm,compute the electric intensity and force on a +5×10^-6c charge place midway between the charges
0.0844kg
Humble
what is the difference between temperature and heat
Heat is the condition or quality of being hot While Temperature is ameasure of cold or heat, often measurable with a thermometer
Abdul
Temperature is the one of heat indicators of materials that can be measured with thermometers, and Heat is the quantity of calor content in material that can be measured with calorimetry.
Gamma
the average kinetic energy of molecules is called temperature. heat is the method or mode to transfer energy to molecules of an object but randomly, while work is the method to transfer energy to molecules in such manner that every molecules get moved in one direction.
2. A brass rod of length 50cm and diameter 3mm is joined to a steel rod of the same length and diameter. What is the change in length of the combined rod at 250°c( degree Celsius) if the original length are 40°c(degree Celsius) is there at thermal stress developed at the junction? The end of the rod are free to expand (coefficient of linear expansion of brass = 2.0×10^-5, steel=1.2×10^-5k^1)
A charge insulator can be discharged by passing it just above a flame. Explain.
of the three vectors in the equation F=qv×b which pairs are always at right angles?