I recently had my bike serviced, and found that I needed a new chain. I don't think they put on quite the right one. It works well for the most part but it skips in 10th gear. Does anyone know the size of a compatible chain, and where to get one?

A 1973 bike would have a 2 chainring x 5 cog drivetrain with a 1/2” x 3/32” derailleur chain. Any chain made for 5-6-7-8-speed bikes will work.

Since it only skips on one cog, was that cog inspected for wear?

REI if one is near, usually has a PC758 for 6,7,8 speed bikes.

Agreed, wrong chain is probably not the culprit. When you say 10th gear, do you mean easiest or hardest gear? If hardest, it’s likely your smallest freewheel cog is worn, or rear derailleur high limit needs a slight adjustment.

Yes, it is the smallest cog in the cassette, and it does look more worn than the others. I was thinking along those line myself. When I think about it, I do spend more time in 10th gear (the hardest) when on a flat when I have my pedal speed going where I like it. I was thinking that the cassette needs replacing.

This is usually an easy fix to the worn teeth on the smallest cog(s).

What is causing slippage isn't so intuitive.

The new chain isn't worn, so the rollers (the parts that actually contact the sprocket teeth) are slightly closer together than on your old chain.

The wear on the "driven" side edge of each tooth (where the roller contacts with driving force) is allowing the chain to travel in a slightly more-advanced position relative to the sprocket.
However, the wear to this same driven-side edge of the tooth, up near the tip of the tooth, is more like non-existent, leaving the driven-side corner near the tip of the tooth protruding rearward relative to where the roller contacts the tooth.
So, with the chain running in it's "advanced" position, the rollers will contact the unworn, rearward-protruding corners of the teeth as each roller is trying to mesh between adjacent teeth. This contact causes the rollers to fail to engage between the teeth momentarily, until the driving force momentarily eases off following gross slippage of the chain on the sprocket.
The slippage occurs at higher pedaling force because the chain roller and tooth both flex slightly under load, enough to pull the rollers forward to where they start contacting the protruding tip corners of the teeth with enough force to prevent engagement of the rollers between the teeth.

The cure is to grind a 1mm bevel at each protruding corner of the teeth of the affected sprocket, so the rollers of the new chain can engage between the teeth even as high tension force is forcing the sprocket to turn.

I use a Dremel tool for this with a 3/8" diameter stone, no need to remove it from the bike.
One could also remove the sprocket from the freewheel and then dress the corners on a bench grinder, which is a lot more work.

Do not remove too much metal from those corners or this may cause the chain to slip (even the old chain would still slip).

Below is an exaggerated cut made to a single tooth at roughly the 12-O'clock position.
Ideally, the cut would be more vertical, more like only 30 degrees away from the driven edge of the tooth.
Start with a lighter cut, then only cut more if the slippage persists.
As the chain wears, any tendency to slip will very slowly go away.

They aren't making these old freewheels like these any more, so it seems wasteful to replace such high-quality freewheels just because of such slight wear.

Note that severely-worn, "tombstoned" sprocket teeth can slip due to a completely different mechanism, and that such teeth cannot be restored using my method.
But a bike shop would have discarded your freewheel if the teeth looked like that.

And as far as the chain length, the shop would have likely matched the link count of your old chain. So unless the chain sags while on the smallest front and rear sprockets, your problem likely isn't due to chain length.

Can you post a photo of your freewheel?

While I understand the need/desire to have every component of a bicycle working perfectly, I would have to say that in the last 40+ years of cycling I have not spent much time in my absolute highest gear. How often do you pedal in the 10th gear?

A question worth asking!

Around here (Sierra Foothills), one frequently uses both lowest and highest gears, with want for more on both ends!

A super Course might have only a 14t small cog, which might find lot's of use on descending and even near-flat riding.

A skip in the drive can be dangerous at times as it occurs unexpectedly. I stubbed the toe of my brand-new Sidi shoes about ten years ago when my Ergolever over-shifted due to worn G-springs, ...I felt lucky to have stayed upright and not taken anybody out.

Go back to where you had this serviced and ask them to check the freewheel for wear. Most likely the skipping is because the freewheel cog is worn out and you need a new freewheel.

Strictly speaking what he needs is a new 14t cog, not the whole freewheel. For many brands, cogs aren't EASILY avalable, so it amounts to the same thing — replace the whole freewheel. Almost anything can be found if you're patient and know where to look, so I wouldn't rule out finding a new top-gear cog. I probably have some, I have many brands to cannibalize for parts, but I'm not a dealer, my junk pile is for working on my own bikes. But try Freewheel Spa maybe? That's BF's own pastorbobnlnh , highly regarded, the best!

By skipping in 10th gear, do you mean you have 5 cogs in the back and two chainrings one the front, (which would make 10th “big chainring-smallest cog”)? I assume you don’t mean you have an updated 10 cog cassette in the back.

And Pastor Bob will have the entire freewheel humming healthily.

Yes, two front chain rings and 5 rear cogs.

Historically I have spent much time in 10th gear. I've lived in mostly flat Florida since 2005, and I do like to get it going. The shop I'm working with had to order a replacement freewheel, which came in this past Thursday so I was pretty excited about that. I loaded up the bike and took it to the shop, which said it could swap it out in about 10 minutes. That didn't happen, because they have to now order a removal tool for the old one! They ordered that, and think it will be delivered in about 2 weeks. I guess that's to be expected. My bike bears little if any resemblance to the very pricey bikes in the shop, so I get that they don't have the right tool on hand. The only similarity is that it has two wheels.

Attached is an image of the freewheel

It's likely a mismatch between new chain and worn sprocket. If the wheel is original to the bike, it will have a freewheel, not a modern freehub/cassette, and probably only five sprockets. Individual sprockets will be hard to source and not worth the effort. Such freewheels are still available, and not expensive, and will have modern tooth profiles for better shifting performance than the OEM freewheel.

They have gotten a cassette that they claim will fit, if they can get the freewheel off. I'll let you know how it turns out in a few weeks.