Kinetic Theory and Internal Energy

How do you derive the equation pV=1/3Nm<u²>

.

root mean squared

The average square of all molecules = <c^2>

Root mean squared = root of <c^2>

What is the use of root mean squared?

To find the typical speed of the molecules

Equation of the Internal energy of an ideal gas of an individual molecule

In ideal gases internal energy = kinetic energy only

1/2m<c^2> for an individual molecule

ideal gas equation pV=NkT

and pV=1/3Nm<c^2>

combining them you get 1/2m<c^2> =3/2kT

What does the equation 1/2m<c^2>=3/2kT show?

internal energy of an idea gas is proportional to its absolute temperature

so

the rise in absolute temperature increases the kinetic energy of each molecule, causing a rise in internal energy