So , why are you saying there'll be a current of 1.1 A, given Vgs=5?
The datasheet spec's the max on-resistance to be 0.35 ohms when the Vgs is 4.5V and the mosfet is cool. Its max allowed power is 0.41W and its max allowed current is 0.86A.
I guessed that the max on-resistance is 0.338 ohms when Vgs is 5.0V so the max power dissipation is 0.41W.
There is a load of 5 Ohms , so given I want to achieve a current of 500 mA , Vds will be reduced to 2.5 V if I'm right...
With a max on-resistance of 0.338 ohms and a current of 500mA then the max heating in the Mosfet is only 0.0845W which is much less than the max allowed dissipation of 0.41W. Vds will be a max of 0.338 ohms x 500mA= 0.169V, not anywhere near 2.5V.