1.8 dc to power on 120v AC

Almost there, a current limiting resistor will be required on the adrino output, also remember that relays that will handle mains voltage come in set coil voltages, nominally 6, 12, 24, 120, & 240. Because the relay requires a certain wattage to operate, the lower the coil voltage, the lower the coil resistance, the higher the coil current, the transistor biasing will need to be looked at. Because of this, these circuits normally use 12 or 24 Vdc to operate. http://everycircuit.com/circuit/5515365507661824

How u have done it ? can u explain ?

I was not sure if the arduino could power a mosfet but I notice a lot of craziness going with the transistor if it's on for to long. I have not tested on real life yet. Still looking for input to help me understand how this could all work

Measure the current draw of the relay, if you can, ramp the voltage up because the pull in current is higher than the holding current, look up the specification for the transistor, particularly the gain. Say the relay takes 30 mA to pull in, and the gain of transistor is 100. Choose a resistor that will allow 300 uA at 1.8 Vdc, approximately 6.2k (I'd give it a bit more drive, use a 5.6k). I don't think the Adrino will have any problems with this, if it does use a higher gain Darlington transistor. A Fet needs a higher gate voltage to work successfully, you may need to drive it with a transistor. Don't forget the anti fly-wheel diode on the relay.