Move SharePoint 2010 site from one content database to another

steps :

1. Create the new content database ( Using Central Admin or PowerShell ) on the same farm and web application.
2. move the site to the newly created content database in step 1 by using the following PowerShell  :
   

   Move-SPSite http://servername/sites/sitename -DestinationDatabase NewContentDatabaseName

   please note during this process the databases will be locked and users will not be able to modify the site ( add modify documents and lists ) you need to run this command with Admin rights.
   

3. Reset IIS on each front end.
4. Release Old content DB storage by starting the timer job “Gradual Site Delete“.  you can use central administration –> monitoring –> Job definitions -> run now

Ref :
Move-SPSite