For a reaction \(A\text{ + }B \longrightarrow C\text{ + }D\) if the concentration of A is doubled without altering the concentration of B, the rate gets doubled. If the concentration of B is increased by nine times without altering the concentration of A, the rate gets tripled. The order of the reaction is:

\(\frac{3}{2}\)

The correct answer is option 3. \(\frac{3}{2}\).

The given reaction is

\(A\text{ + }B \longrightarrow C\text{ + }D\)

 So,

\(r = k [A]^{\alpha} [B]^{\beta}\) -------(1)

\(2r = k [2A]^{\alpha} [B]^{\beta}\) -------(2)

\(3r = k [A]^{\alpha} [9B]^{\beta}\) -------(3)

Dividing equation (3) by (1),we get

\(3 = 9^{\beta}\)

or, \(3^1 = 3^{2 \beta}\)

or, \(2 \beta = 1\)

or, \(\beta = \frac{1}{2}\)

Dividing equation (2) by (1),we get

\(2 = 2^{\alpha }\)

or, \(2^1 = 2^{\alpha }\)

or, \(\alpha = 1\)

So, the order of the reaction will be

\(\alpha + \beta = 1 + \frac{1}{2} = \frac{3}{2}\)