In a container, an certain number \(\alpha\) of ideal gases, are put inside the container such that rate of diffusion or effusion of every succeeding gas forms a geometric sequence with a common ratio \(\beta\).

Moreover, the mass of the preceding gas is directly proportional to the molar mass of every succeeding gas by the cube of the same common ratio \(\beta\).

If the total pressure inside the container is exactly the same as the gas constant \(R\), then what is its volume-to-temperature ratio in terms of the variables given?

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