[Firehchicken]

I have seen two formulas for 2nd order half lifes:

1/2*k*[initial concentration]

1/k*[initial concentration]

Which one is correct

[mjc123]

Neither is absolutely correct. It depends on the definition of k. How do you write the rate equation?

[Firehchicken]

Neither is absolutely correct. It depends on the definition of k. How do you write the rate equation?

They didn't give the rate equation, just the value of the rate constant.

[mjc123]

Well that is very naughty of them. Do they give a reaction equation? If so you must assume that the rate is defined in terms of that equation, and
Rate = (1/α)*d[X]/dt
where α is the coefficient of reagent X in the equation (negative for reactants, positive for products).
So if the equation is, say, 2A  B and
Rate = (-1/2)*d[A]/dt = k[A]²
then d[A]/dt = -2k[A]²
If, however, you write (as I would) d[A]/dt = -k'[A]²
then obviously k' = 2k
Now solve the differential equation and get the half-life, and you get
t_1/2 = 1/(k'[A]₀) = 1/(2k[A]₀)