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Following the original approach of Maxwell-Boltzmann(MB), we derive a4-velocity distribution function for the relativistic ideal gas. Thisdistribution function perfectly reduces to original MB distribution in thenon-relativistic limit. We express the relativistic equation of state(EOS),$\rho-\rho_0=(\gamma-1)^{-1}p$,\ in the two equations: $\rho=\rho_0f(\lambda)$,\ and $p=\rho_0 g(\lambda)$, where $\lambda$\ is a parameterrelated to the kinetic energy, hence the temperature, of the gas. In the bothextreme limits, they give correct EOS:\ $\rho=3p$\ in the ultra-relativistic,and\ $\rho-\rho_0=3/2p$ in the non-relativistic regime. Using these equationsthe adiabatic index $\gamma$ (=$\frac{c_p}{c_v}$) and the sound speed $a_s$ arecalculated as a function of $\lambda$. They also satisfy the inequalities: $4/3\le \gamma \le 5/3$ and $a_s \le \frac{1}{\sqrt{3}}$ perfectly.

4-Velocity distribution function using Maxwell-Boltzmann's original approach and a new form of the relativistic equation of state