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Edited on Sat Sep-25-10 04:10 AM by PJPhreak
I went to Wiki to look this up,The Wiki read like a Ginormas Algebra Equation...
Let q = (qx, qy, qz) and p = (px, py, pz) denote the position vector and momentum vector of a particle of an ideal gas, respectively. Let F denote the net force on that particle. Then the time average momentum of the particle is: \begin{align} \langle \mathbf{q} \cdot \mathbf{F} \rangle &= \Bigl\langle q_{x} \frac{dp_{x}}{dt} \Bigr\rangle + \Bigl\langle q_{y} \frac{dp_{y}}{dt} \Bigr\rangle + \Bigl\langle q_{z} \frac{dp_{z}}{dt} \Bigr\rangle\\ &=-\Bigl\langle q_{x} \frac{\partial H}{\partial q_x} \Bigr\rangle - \Bigl\langle q_{y} \frac{\partial H}{\partial q_y} \Bigr\rangle - \Bigl\langle q_{z} \frac{\partial H}{\partial q_z} \Bigr\rangle = -3k_{B} T, \end{align} where the first equality is Newton's second law, and the second line uses Hamilton's equations and the equipartition theorem. Summing over a system of N particles yields
3Nk_{B} T = - \biggl\langle \sum_{k=1}^{N} \mathbf{q}_{k} \cdot \mathbf{F}_{k} \biggr\rangle.
By Newton's third law and the ideal gas assumption, the net force on the system is the force applied by the walls of their container, and this force is given by the pressure p of the gas. Hence
-\biggl\langle\sum_{k=1}^{N} \mathbf{q}_{k} \cdot \mathbf{F}_{k}\biggr\rangle = p \oint_{\mathrm{surface}} \mathbf{q} \cdot d\mathbf{S},
where dS is the infinitesimal area element along the walls of the container. Since the divergence of the position vector q is
\boldsymbol\nabla \cdot \mathbf{q} = \frac{\partial q_{x}}{\partial q_{x}} + \frac{\partial q_{y}}{\partial q_{y}} + \frac{\partial q_{z}}{\partial q_{z}} = 3,
the divergence theorem implies that
p \oint_{\mathrm{surface}} \mathbf{q} \cdot d\mathbf{S} = p \int_{\mathrm{volume}} \left( \boldsymbol\nabla \cdot \mathbf{q} \right) dV = 3pV,
where dV is an infinitesimal volume within the container and V is the total volume of the container.
Putting these equalities together yields
3Nk_{B} T = -\biggl\langle \sum_{k=1}^{N} \mathbf{q}_{k} \cdot \mathbf{F}_{k} \biggr\rangle = 3pV,
which immediately implies the ideal gas law for N particles:
pV = Nk_{B} T = nRT,\,
where n = N/NA is the number of moles of gas and R = NAkB is the gas constant.
The readers are referred to the comprehensive article Configuration integral (statistical mechanics) where an alternative statistical mechanics derivation of the ideal-gas law, using the relationship between the Helmholtz free energy and the partition function, but without using the equipartition theorem, is provided.
Please,for those of us who don't carry a Phd in Math,explain this in plain english!
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