I'm not sure there is a true answer to this question because what you can or can't do as an intern is going to be dependent on who you are working with and what they're comfortable allowing you to do.
First, I don't think reading papers is a waste of time. It might not be what you want to be doing right now but it exposes you to ideas and current happenings in the chemical engineering world which is helpful. But I 100% understand where you're coming from. To answer your question, yes I think it's fine to talk to your professor again to see if there's other opportunities for you to do work.
I would avoid phrasing it as "can I finally do some actual work" and more along the lines of "I've been here a month and have enjoyed getting the opportunity to shadow my mentors and learn from reading papers. Is it possible to sit down and discuss my role and any future opportunities you think I might be able to help with?" or something along those lines. I wrote that quickly so obviously spend a bit more time thinking of what to say but make sure to present it in a professional way. Let them know you're thankful for the opportunity but that it would be helpful to set some expectations or a timeline for your role so you know what to expect over the coming weeks/months/year.
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