Write the Van der Waals equation via the reduced parameters `pi, v` and `tau`, having taken the corresponding critical values for the units of pressure, volume, and temperature. Using the equation obtained, find how many times the gas temperature exceeds its critical temperature if the gas pressure is `12` times as high as critical pressure, and the volume of gas is equal to half the critical volume.

Explanation

`(p + (a)/(V_M^2))(V_m - b) = RT`
or `(p+(a)/(V_m^2))/(P_(Cr))xx (V_m - b)/(V_(M_Cr)) = (8)/(3) (T)/(T_(Cr))`
or `(pi + (a)/(P_(Cr) V_m^2)) xx (v - (b)/(V_(MCr))) = (8)/(3) tau`,
where `pi = (p)/(p_(Cr)), v = (V_m)/(V_(MCr)), tau = (T)/(T_(Cr))`
or `(pi + (27 b^2)/(V_M^2))(v - (1)/(3)) = (8)/(3) tau`, or `(pi + (3)/(v^2))(v - (1)/(3)) = (8)/(3) tau`
when `pi = 12` and `v = (1)/(2), tau = (3)/(8) xx 24 xx (1)/(6) = (3)/(2)`.