That's like asking if you purchase a $0.42 stamp, does the 42 cents go to the federal government. Both have the same answer. Technically speaking the answer is no. The central bank is deemed to be independant of the US Gov't just as the post office is deemed to be independant of the US Gov't (private shippers complained that they can't compete in fair enterprise if the government owns the post office so the gov't "spun off" the post office to make it a separate entity). If you read about the Federal Reserve (go to their website or Wikipedia) you will see it's comprised of "member banks" which are most of the biggest banks in the US. This way the balance sheet is separate. That's how criminal Hank Paulson gives his buddies $700B and the balance sheet changes by $700B while Uncle Ben doles out trillions but that does not change the US Gov't balance sheet.
If you really want to study the Federal Reserve read "Web of Debt," by Ellen Brown. It's easily the best book I've ever read. Unfortunately, you'll need a double dose of Prozac before you read it because you're sure to get awfully depressed. Aside from that, every American Citizen should be required to read this book.