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It is a huge pet peeve of mine, when people donate musty-smelling books to the library where we live. I have tried many different methods for trying to remove the smell, and this is the best I've found.
Take the book and place a sheet of newspaper between the pages (need not be every single page, don't worry!), then seal in a tape box surrounded by more newspaper which you've crumpled into balls for three days. The newspapers absorbs the odor, and is better than using cat litter because the litter gets caught in the pages and spines of the books.
By AlaskanAurora from Dutch Harbor, AK
What I did was take the book and spread it out, blowing it with the blow dryer. I flipped the pages back and forth to get them all. Then, I sprayed the outer cover (it is vinyl) with some Febreeze fabric spray and then wiped it off. Then, I put it on a table in the sun. A few hours later - the musty is almost completely gone!
By Nacey 
To keep books from becoming musty, pull them out, away from the back of the shelf. Line up the spines along the front edge of the bookshelf. They'll look better, and the air will circulate behind them, so they'll smell fresher. Remember to dust behind them! And remember to READ them, too, so they'll bring you lots of pleasure.
If you have musty books, sprinkle talcum powder between pages, wrap the book in brown paper and store for several days. Remove and brush out powder.
By Randa from San Marcos TX
In damp summer weather, keep papers and books as dry as possible to help control mold growth. If you have an enclosed bookcase, keep a small electric light lit continuously in the bookcase or use a chemical dehumidifier, keeping the doors closed as tightly as possible.
Remove any dry,loose mold from paper with a clean, soft cloth. If mildewed paper is damp, dry it first--in an airy place, if possible.
To dry wallpaper, heat the room for several hours or even days to dry the plaster as well as the paper. Plaster should be dried slowly to prevent cracking.
If mildewed paper is washable, wipe it gently with a cloth wrung out of thick soapsuds, then with clear water. Take care not to wet the paper more than necessary. Do not scrub it. Finally pat with a soft, dry cloth. If stains remain, bleach with a solution of a household bleach, then sponge with a cloth wrung out of clear water. For small stains, a commercial ink eradicator may be useful.
Spread pages of books out fanwise to air. If the books are very damp, sprinkle cornstarch or talcum powder between the leaves to take up the moisture. Leave starch or powder on for several hours then brush off.
This article was written by Anne Field, Extension Specialist, Emeritus, with reference from the USDA bulletin, Mildew.
Source: MSU Extension
To get a musty odor out of old books, place the books outdoors in the sun and frequently fan out the pages so that the air can reach them. Brush off any mildew or mold. If the pages are damp, sprinkle cornstarch between them and brush it off after a few hours.
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Here are questions related to Removing Musty Smell from Books.
I have 3 boxes of valuable books. They have been stored for many years and when I took them out to check on them I saw that some of the books have mildew growing on them. What is the best way to remove the mildew?
More importantly how do I prevent these books from suffering another mildew attack. At the moment I put the books in an air tight zipper bag with silica beads. Is this a good way for long term storage?
BTW I live in a very humid country, combating humidity is an unwinable war.
I opened the strong box and all my important papers have a mildew smell. I have had the box for few years and just now has it developed an odor. I can throw the strong box away, but how do I get smell out of the papers?
By Barb B
I have 4 big boxes of smelly books that were stored in a garage for 19 months. I can't do any labour intensive work as I am sick, can I just set them out on the floor to hopefully air out?
By Lewissan 09/10/2012
The most common solution I have seen is to put newspaper between some of the pages, then wrap the book in newspaper and then put the books in a box full of more newspaper for few days. Some variations use charcoal as well. You read all the way down this page you can read lots of ideas about this subject:
How do I remove the musty smell from books?
By Booklover from Galesburg, IL
By Julia 05/10/2011
I use Snuggle dryer sheets when storing books or for moving, so maybe a week or so in a plastic bag with fresh dryer sheets might work. I also use fresh bars of Tone Soap when packing anything including books.
Thrifty Fun has been around so long that many of our pages have been reset several times. Archives are older versions of the page and the feedback that was provided then.
Would anyone know how to take the musty smell out of books?
Loretta from Windsor
Another good way to get rid of the smell is to put the offending book in a sealed plastic bag of dry cat litter (dollar stores usually have it).
Depending on how "bad" the book is, it usually comes out good as new without some horrible chemical smell. Try a couple of days for medium smell, maybe as long as 3 weeks for a really bad smell. Lots of other ways, but this one is cheap.
Hope this helps
bottom of bag
By AnaLisa Moyers, NMSU Library Tech(08/02/2004)
I really love to read and enjoy buying old books but a lot of them smell really musty! What's the best way to get rid of the musty smell? Rebecca from Iowa
Our neighbors at the Northeast Document Conservation Center say this: There is no guaranteed way to remove the musty smell from old books, but there is a strategy that may be successful. This musty smell is most often noted in books that have been moldy or mildewed in the past. The first step is to create an enclosed chamber. This is most easily done by using two garbage cans, one large (with a lid) and one small. The object to be "deodorized" should be placed in the smaller can, which is then placed inside the larger can. Some type of odor-absorbing material should then be placed in the bottom of the larger can. Odor-absorbing materials to try include baking soda, charcoal briquettes (without lighter fluid), or kitty litter. The lid should then be placed on the larger can, and the chamber should be left for some time. You will need to monitor periodically to see how long the materials need to be left inside the chamber. (08/10/2006)
By carla bledsoe
What's the best way to get rid of the musty smell from books? There seem to be many interesting approaches in removing musty smells from books.
How do you get rid of musty book smell?
By skag from Vancouver
How do I get a musty smell out of a used book I just bought?
By Flash Gordon from Petoskey, MI
By Ann Parker
How can I remove the moldy smell from an old book?
By ecline from Raleigh, NC