How can I tell if a relay coil is rated for 240VAC
Your solution is unlikely to work, as the relay is going to buzz a lot, and probably start a fire. The coil voltage should be shown somewhere on the relay, though I don't see it on yours. In any case, \$640\Omega\$ is way too low for 240V, as it will be dissipating 90W, so instead of a relay, you'd have a lightbulb and a house fire. After looking at the data sheet (see TonyM's answer), and looking in the chart on page 2 for 'AZ-2110-1A-24D', we see that the coil voltage is 24V, and the winding resistance is ~\$660\Omega\$, which matches up nicely.

As for solving your problem, I'd recommend using an optocoupler. Something like this:







simulate this circuit &ndash; Schematic created using CircuitLab

Basically, it's a full bridge rectifier feeding the optocoupler (the led and transistor, the schematic editor didn't have a real optocoupler) with a current of ~5.5mA. The two 22k resistors will need to be 1W or larger to remain within temperature tolerances, and need to be rated for greater than 120V.