Maximum difference between two successive real number will occur at extremes. This is because numbers are represented upto mantissa bits and as the exponent grows larger, the difference gets multiplied by a larger value. (The minimum difference happens for the least positive exponent value).

Biasing will be done by adding 31 as given in question. So, actual value of exponent will be represented value - 31. Also, we can not have exponent field as all 1's as given in question (usually taken for representing infinity, NAN etc). So, largest value that can be stored is 111110 = 62.

Largest number will be $1.111111111 \times 2^{62-31} = \left(2 - 2^{-9}\right) \times 2^{31}$

Second largest number will be $1.111111110 \times 2^{62-31} = \left(2 - 2^{-8}\right) 2^{31}$

So, difference between these two numbers $= \left(2 - 2^{-9}\right) \times 2^{31} - \left(2 - 2^{-8}\right) \times 2^{31}\\= 2^{31} \left[\left(2 - 2^{-9}\right) - \left(2 - 2^{-8}\right) \right] \\= 2^{31} \left[ 2^{-8} - 2^{-9} \right] \\= 2^{31} \times 2^{-9} = 2^{22}$