# What does the ratio of reservoir temperatures have to do with the efficiency of a heat engine?

I am having a conceptual difficulty with heat engine efficiency. I do not understand why having a larger difference in temperature between the hot and cold reservoirs have an effect on the thermal efficiency. In this question I am specifically referring to Carnot heat engines. To clarify, I do not understand the reason the exact same (carnot) heat engine is more efficient when placed between different reservoirs. What is the mathematical or physical reason behind this?

I found this question: Why is it that a Carnot heat engine will reject no heat to a zero temperature sink? which is very similar to what I am asking, but the answers there don't make sense to me and it seems that was the case for the OP too.

I hope the question is clear and cheers in advance!

## 1 Answer

Even though nobody seems to have come across this yet, I won't delete it just in case someone has a similar lack of understanding I had. This link here: http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/CarnotEngine.htm shows very clearly why the assumptions made of the Carnot engine must then mathematically lead to the ratio of temperature differences between the reservoirs (or indeed the ratio of $Q_{in}$ to $Q_{out}$) being part of the equation which defines the efficiency of the engine. On the page, just note that there is an error; instead of $$Q_c = nRT_cln\frac{V_a}{V_b}$$ it should be $$Q_c = nRT_cln\frac{V_b}{V_a}$$

I know this answer isn't very long or in depth, but the article I linked to is very clear, and so hopefully anyone with similar doubts can get use that to improve their understanding!