Find and read the detailed manual for the RAID system you have. I am presuming that you plan to use the RAID management system built into the chipset on your mobo.
Most RAID1 systems I have seen do allow what you want to do. You have a 500 GB drive you do NOT plan to use in the RAID1 array, and it is your boot drive. You have received a 2 TB new HDD and you could add it to your system right now as a non-RAID unit, Partition and Format it, and start using it as a non-RAID data storage device. Later, when you receive the second 2 TB unit you can make the conversion.
The first step will be to install the second 2 TB unit. When you boot up it will not appear to be there, except in the lower right pane of Disk Management. That's OK. You should then boot into BIOS Setup and change the SATA port mode to RAID. This MAY be for ALL your SATA drives, even the one you don't plan to use that way, but don't worry. You save and exit and the machine will reboot. But this time during the boot process you will see a screen that tells you to push a certain key if you want to enter the RAID Setup and Management area. Do that.
Within the RAID management system you get to choose what type of RAID array you are trying to create, and which HDD units are assigned to the array. Any disk NOT assigned to an array continues to be a non-RAID disk, like your 500 GB original unit. For the RAID1 array you are creating, you will specify both 2 TB units as its members. It will ask about setting up the disks and Formatting them. When it asks you if it should completely wipe out all previous data, say NO. Use the menus to tell it to copy all the data from the drive that has some to the other drive as it sets up the array. You may need to be particularly careful to identify which unit already has the data to be copied.
As I said, most systems I've seen allow you to convert a single non-RAID disk into half of a RAID1 array and preserve its data on the new array. But before starting, make sure you read the manual and know how to do this on your system.
Once this is done, your Windows may NOT yet recognize the RAID1 array and show it as a single disk in My Computer. If it does not, install the RAID driver into your Windows so that it can use this new device type. You have to wait until the RAID array exists before you can do this step, however - I have seen Windows refuse to install a RAID driver if there is no RAID array present. If you're lucky, Windows may actually detect the existence of this new device and do the installation for you.