Principle of how refrigerator works: how is temperature dropping but pressure is not?

I'm taking "Physics of the everyday" on Brilliant. Here is the diagram of how refrigerators work:

Regarding the cooling step:

After exiting the compressor, this gas cools to room temperature in the external radiator, maintaining its high pressure.

How can the gas cool down but still maintain its high pressure? Isn't the lower the temperature, the slower the gas molecules move which means less of a force exert on the container which in turns mean lower pressure?

The ideal gas law (which is an approximation to real life, but good enough for a back of the envelope calculation) tells us that

$$PV \propto T$$

or, in an alternative formulation

$$P \propto \rho T$$

So if pressure $$P$$ stays the same but temperature $$T$$ decreases then density $$\rho$$ must increase. In other words, a given mass of gas takes up less space as it cools.

• Why does it take less space in this case? Doesn't the gas has the whole pipe (inside the radiator) for itself which mean its volume does not change? Jun 8, 2021 at 7:15
• @advidnewbie Remember that the gas is circulating continuously - we do not fill the radiator with hot gas, let it cool down, then let it leave the radiator. If we did, then yes, this would cause stresses on the radiator pipes because the cold gas would take up less space than the hot gas (and we would have invented the steam engine). But in actual operation the temperature of the continuously flowing gas in the radiator reaches a steady state where it is hotter at the inlet and colder at the outlet Jun 8, 2021 at 7:44
• I see, but that still doesn't explain why (at least not intuitively) the cold air should take up less space when it cools down Jun 9, 2021 at 1:38
• @advidnewbie You know that the gas molecules move more slowly as the gas cools. To keep the same pressure, the slower gas molecules must collide more frequently with the walls of their container. This means either reducing the size of the container (same density, smaller volume) or putting more gas molecules into the same space (higher density, same volume). Jun 9, 2021 at 5:23

If the temperature is lower that doesn't mean that pressure has to lower too. While the velocity of the particles might be less (imparting less momentum), there can be more particles to exert pressure. The lower temperature is compensated for by more particles.