# Does isothermal expansion not beat the efficiency of Carnot Heat Engine?

## Ques:

If I just take the 1st step of a Carnot Heat Engine (CHE) than it turns out to be more efficient than the CHE itself.

## Explanation:

The 1st step of a CHE is the pure isothermal expansion of (ideal) gas where all the heat($$Q$$) taken from the source reservoir is converted into work($$W$$). That is essentially what happens in iosothermal expansion:

$$Q$$=$$W$$
(i.e all the Heat given to the system is converted to work done by the system)

So, the efficiency for the (quasi-static) isothermal expansion is 100%.
So, if i am not missing anything than we should just be continuously isothermally expanding gases in our cars and getting the best possible efficiency forever. (I know that an ideal isothermal expansion is not possible but so is the CHE!)

P.S

1. I know that the CHE is the most efficient Heat engine between two reservoirs at temperatures T1 and T2.
2. Also i think there is some restrictions by definition that a heat engine operates in cycle. That is to say that it returns to its original state after a cycle. (Is that a defining characteristic of Heat engines by the way? I kind of got it as an intuition).
3. So, the isothermal process by no means satisfies the above 2 points but that is not what i am going for. I was just thinking of some way to beat the efficiency of a CHE and the above method was pretty simple to come by.

You have to distinguish between the efficiency of a process and the efficiency of a cycle. A reversible isothermal expansion process can be 100% efficient. That does not violate the second law. It’s the efficiency of a cycle that can’t be 100%.

So, yes, if you had a one cylinder car performing a reversible isothermal expansion you could theoretically continue converting heat to work as long as the cylinder (and car containing it) keep expanding forever (they become infinitely large).

But even more importantly, since $$PV=$$ constant, as the volume increases the pressure, and thus the force exerted by gas, decreases. That limits the amount of work that can be done by the gas in a single expansion.

So if you want an engine of finite size that can do more work than can be done in a single expansion, you need to perform a cycle. Heat is rejected during isothermal compression making the engine efficiency less than 100%.

Hope this helps.