Editing Virial theorem (section)

=== Ideal-gas law ===
Consider a container filled with an ideal gas consisting of point masses. The only forces applied to the point masses are due to the container walls. In this case, the expression in the virial theorem equals
<math display="block">
 \Big \langle \sum_i \mathbf{F}_i \cdot \mathbf{r}_i \Big \rangle =
- P \oint \hat{\mathbf{n}} \cdot \mathbf{r} \,dA,
</math>
since, by definition, the pressure ''P'' is the average force per area exerted by the gas upon the walls, which is normal to the wall. There is a minus sign because 
<math>\hat{\mathbf{n}}</math> is the unit normal vector pointing outwards, and the force to be used is the one upon the particles by the wall.


Then the virial theorem states that
<math display="block">
 \langle T \rangle =
 \frac{P}{2} \oint \hat{\mathbf{n}} \cdot \mathbf{r} \,dA.
</math>
By the [[divergence theorem]], <math display="inline">\oint \hat{\mathbf{n}} \cdot \mathbf{r} \,dA = \int \nabla \cdot \mathbf{r} \,dV = 3 \int dV = 3V</math>. 

From [[equipartition]], the average total kinetic energy <math display="inline">\langle T \rangle = N \big\langle \frac 12 mv^2 \big\rangle = N \cdot \frac 32 kT</math>. Hence, <math>PV = NkT</math>, the [[ideal gas law]].<ref>{{Cite web |date=2018-03-22 |title=2.11: Virial Theorem |url=https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/02%3A_Review_of_Newtonian_Mechanics/2.11%3A_Virial_Theorem |access-date=2023-06-07 |website=Physics LibreTexts |language=en}}</ref>