No, not much more happens in this chapter than in the last one.
Dmitri and Floyd shmooze.
Dmitri asks what the deal is with the epidemic, just like the stewardess does. Floyd stonewalls.
Okay, Shmoop gets it. Something's happening; Floyd isn't telling what it is; it's big; he's awesome.
Dmitri mentions the magic word "TMA-1". The security guy is startled to show that this is a big deal. Floyd doesn't react to show that he is even more awesome and tough than hard-headed security guards.
Dmitri tells Floyd to be careful.
Then off to another ship, this one on its way to the moon.
There's a bit about how the food is engineered so it doesn't float around the cabin in weightlessness. Seriously. Clarke loves that sort of thing.
Now Floyd looks at his Newspad, which is sort of like a low-tech Internet. Clarke was writing before the Internet, though, so he thinks the Newspad is really cool.
The stewardess is from Bali, an Indonesian island, and she performs a traditional dance for Floyd. Guys read Newspads; women dance. Clarke's future has nifty gadgets, but those 1960s gender roles seem like they will never go away.
Nothing else really happens; the moon gets bigger, then Floyd lands on it.