Find the minimum attainable pressure of one mole of an ideal gas, if during its expansion its temperature and volume are related as T = `T_(0) + alpha V^(2)` where `T_(0)` and `alpha` are positive constants.

Explanation

Given that one mole of gas is used, thus from gas law, we have PV =RT or
P `(RT)/(V) = (R)/(V) ( T_(0) + alpha V^(2)) [ "as " T = T_(0) + alpha V^(2) ] `
Here pressure P will be minimum, when
`(dP)/(dV) = 0 " or " (dP)/(dV) = (RT_(0))/(V^(2)) + alpha R = 0 " or " V = sqrt((T_(0))/(alpha))`
Thus pressure of the gas is minimum, when its volume is `V = sqrt((T_(0))/(alpha))` and at this volume , its temperature is given as `T = T_(0) + alpha V^(2) = T_(0) + alpha ( sqrt((T_(0))/(alpha)) )^(2) = 2T_(0)`
Thus minimum value of pressure is
`P_(min) = (RT)/(V) = (R(2T_(0)))/(sqrt(T_(0)//alpha)) = 2R sqrt(T_(0) alpha)`