The length of the 4870x2 is 10.5". This is just under an inch longer than your motherboard is wide.
Measure with a tape measure and see if you have space. Take an extra 1/2 inch into account for the power plugs for the card. From the PCI-e backplate towards your hard drive, measure you shall.
Also, the 4870x2 is going to generate a lot more heat than a 4850. The Sonata is a great case, but with only the rear exhaust fan, is not a great cooling solution. I would suggest a HD57xx or HD58xx series card instead and would definitely limit any OC efforts. Good luck!!!