Making yogurt for the first time. Temp is set to 110 degrees. Mixture (1/2gal liquid) is in a 1 gallon glass jar. Will the yogurt eventually come up to temperature?
It will eventually. It may take some time due to the quantity for the heat to penetrate to the center.
Most cooks make yogurt in smaller containers that results in quicker processing and more uniform heat transfer. It’s a long slow trip for 110F heat to reach the centre of that gallon jar.
A benefit of your using that gallon jar and the resulting long cook time is it will reveal any deficiencies in your sanitation practices, something you might not have been planning on, but nice to know.
I suggest you pre-heat the milk to 43°C or so. The quantity you have a rather large, and you use one glass container, that is better than plastic, but still, glass is a rather bad heat conductor.
The temp will eventually be at what it should be, but that can take a while (1-2 hours??) and may meddle with the overall yogurt time.
If you have food thermometer, why don’t you try it out with water? This will give you a better idea.