Can I update an iMac 21.5" 2011 running High Sierra 10.13.6 to the latest macOS?

Peter Paul van de Wijs wrote:

I am using an old imac and it is running on High Sierra 10.13.6. Can i upgrade this to a more modern version of the OS?

No

MacOS 10.13 High Sierra at best

macOS Mojave

• MacBook (Early 2015 or newer)
• MacBook Pro (2012 or newer)
• MacBook Air (2012 or newer)
• Mac mini (2012 or newer)
• iMac (2012 or newer)
• iMac Pro (2017)
• Mac Pro (Late 2013)

Peter Paul van de Wijs wrote:

It seems a bit of a contradiction to replace my machine which is in a workable condition while donating it. If it works I should be able to continue using it right?

A 21.5" 2011 iMac in working condition might be useful for running applications that you already have, importing digital video from old MiniDV and Digital8 camcorders, or running old 32-bit games which are incompatible with Catalina and later.

You can still get a few applications that might help you get a bit more use out of that machine:

• The Firefox 115 ESR Web browser
• An older, unsupported version of the LibreOffice office suite, from the archives
• The GNU Image Manipulation Program (GIMP)
• Carbon Copy Coner and SuperDuper! – though not necessarily the current versions

But forget running modern Mac applications. Microsoft and Adobe abandoned High Sierra long ago. High Sierra isn't even enough to run the current versions of Firefox, Brave, Edge, Chrome, LibreOffice, or Affinity applications. Since even developers who support old versions of macOS have abandoned High Sierra, you really are fighting an uphill battle to keep using that machine, going forward, for applications that you do not already have.

If you still want to keep using it yourself, that's fine. If you donate it or sell it, you should make certain that the new owner knows that they are getting an outdated computer system. If they think they're getting a computer that will support modern applications, and be easy to find applications for, they'll be sadly disappointed.

Peter Paul van de Wijs wrote:

Thanks for the advice. It seems a bit of a contradiction to replace my machine which is in a workable condition while donating it. If it works I should be able to continue using it right?

Welcome

If it works - use it

Replace it - your choice

Donate it - also you choice

Three viable options offered

What you do with these suggestions are really your chose too

The 14 year old computer has had a long and productive life - no

Consider donating the machine if it is still in workable condition

Consider a new or newer machine

Explore the Apple Trade-In  Process where the Value of the exiting device could be used against the cost of a new Apple Computer, iPhone or iPad 

Notation - Some devices and depending on the computers age may have little to no value at Trade In time.

But at least the device will not arrive to the Land Fill Site and be recycled in a responsible way by Apple

Or Shop Refurbished Apple computers at an Apple Reduced Cost

Have used both previously and have been satisfied each time.

If you decide to sell, give away, or recycle the Mac, see:

What to do before you sell, give away, trade in, or recycle your Mac - Apple Support

If you are giving or selling the Mac to someone else, and it is one that came with a set of optical recovery discs, be sure to transfer those discs along with it. (Apple released 21.5" iMacs in Mid and Late 2011. The ones released in Mid 2011 shipped with a version of Snow Leopard, so I'm pretty sure they would have come with recovery discs.)

Donating the old iMac would be a noble gesture, but you really would not be doing anyone any favor because, yes, the computer is obsolete and its practical usefulness is becoming less with each passing year.

Peter Paul van de Wijs wrote:

And can i use it as second screen with a Macbook Air which runs Ventura.

No. See: Use your iMac as a display with target display mode - Apple Support

it will not work with target display for any newer macs