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I was reading about the kinetic theory of gases and I came to know about this particular equation which piqued my interest.

I know about the ideal gas law which gives us $pV = nRT$.
But this one contained Boltzmann's Constant and it goes like this $PV = Nk_\mathrm{B}T$, where $k_\mathrm{B}$ is the Boltzmann constant and $T$ the absolute temperature.
Could I get some insights to how this form of the ideal gas law wasderived?
And how was the value of $k_\mathrm{B}$ and $R$ determined?

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$R$ is defined as $k_\mathrm{B}N_\mathrm{A}$ where $N_\mathrm{A}$ is Avogradro's number. Notice that these forms are equivalent once you realize that $N = N_\mathrm{A}n$.

I think $k_\mathrm{B}$ is measured experimentally. I learned the derivation via statistical mechanics.

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Both sides of the equation specify the total kinetic energy of the closed system. The left side in terms of the aggregate macro measurables of pressure and volume. The right side in terms of the aggregate average of micro molecular mass and squared velocity. The Boltzmann constant links average molecular kinetic energy to the absolute temperature scale. The zero of the absolute temperature scale is chosen by extrapolation to zero kinetic energy and cant be reached in practice.

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