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F i = Δ p i Δ t = 2 m v i x 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 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 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 Δ t = 2 l / v x , and the expression for the force becomes

F i = 2 m v i x 2 l / v i x = m v i x 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 = i = 1 N F i = i = 1 N m v i x 2 l = m l i = 1 N v i x 2 .

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

F = N m l ( 1 N i = 1 N v i x 2 ) = N m 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

v 2 = v x 2 + v y 2 + v z 2 .

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

v 2 = 3 v i x 2 .

Substituting this into the expression for F gives

F = N m v 2 3 l .

The pressure is F / A , so we obtain

p = F A = N m v 2 3 A l = N m v 2 3 V ,

where we used V = A l for the volume. This gives the important result

p V = 1 3 N m v 2 .

Combining this equation with p V = N k B T gives

1 3 N m v 2 = N k B T .

We can get the average kinetic energy of a molecule, 1 2 m 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:

K = 1 2 m v 2 = 3 2 k B T .

The equation K = 3 2 k 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 int , we simply have E int = N K , or

E int = 3 2 N k B T .

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

E int = 3 2 n R T .

We can solve K = 1 2 m v 2 = 3 2 k 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 v 2 , is

v rms = v 2 = 3 k 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:

Questions & Answers

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
Opeyemi Reply
please how do dey get 5/9 in the conversion of Celsius and Fahrenheit
Gwam Reply
what is copper loss
timileyin Reply
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.
it is the work done in moving a charge to a point from infinity against electric field
Ashok Reply
what is the weight of the earth in space
peterpaul Reply
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...
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
Yeah got it... Earth and moon have specific value of g... But in case of sun ☀ it is just a huge sphere of gas...
Thats why it can't have a constant value of g ....
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
please why is the first law of thermodynamics greater than the second
Ifeoma Reply
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.
Mateshwar Reply
explain the lack of symmetry in the field of the parallel capacitor
Phoebe Reply
pls. explain the lack of symmetry in the field of the parallel capacitor
does your app come with video lessons?
Ahmed Reply
What is vector
Ajibola Reply
Vector is a quantity having a direction as well as magnitude
tell me about charging and discharging of capacitors
Ahemen Reply
a big and a small metal spheres are connected by a wire, which of this has the maximum electric potential on the surface.
Bundi Reply
3 capacitors 2nf,3nf,4nf are connected in parallel... what is the equivalent capacitance...and what is the potential difference across each capacitor if the EMF is 500v
Prince Reply
equivalent capacitance is 9nf nd pd across each capacitor is 500v
four effect of heat on substances
Prince Reply
why we can find a electric mirror image only in a infinite conducting....why not in finite conducting plate..?
Rima Reply
because you can't fit the boundary conditions.
what is the dimensions for VISCOUNSITY (U)
Practice Key Terms 8

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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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