The identity element for the binary operation * defined on Q ~ {0} as a * b = ab/2 ∀ a, b ∈ Q ~ {0} is

The correct option is 2.

To find the identity element for the binary operation $*$ defined on the set of non-zero rational numbers $Q~\left\{0\right\}$, let us denote the identity element as $e$.

By definition, an element $e\in Q~\left\{0\right\}$ is the identity element for the binary operation $*$ if it satisfies the condition:
$a*e=e*a=a$
for all $a\in Q~\left\{0\right\}$.

The binary operation is given by:
$a*b=\frac{ab}{2}$

Using the definition of the operation, we set up the equation for the identity element:
$a*e=a$
Substitute the definition of $a*e$ into the equation:
$\frac{ae}{2}=a$

Since $a\in Q~\left\{0\right\}$, we know that $a\ne 0$. Therefore, we can divide both sides of the equation by $a$:
$\frac{e}{2}=1$

Multiplying both sides by $2$, we get:
$e=2$

Since $2\in Q~\left\{0\right\}$, the identity element for the given operation exists and is equal to $2$.