A question on the ideal gas law and the functioning of a fridge

Question

First let me apologize: I am not a physicist and my question is perhaps not well-suited for this expert forum. I didn't find any other place to ask though.

We have the (simplified) formula $$ \frac{P * V}{T}=const $$ for an ideal gas. Here $P$ is the the pressure of the gas, $V$ is the volume and $T$ is the temperature. (Anticipating my question below, I guess my confusion lies in the misunderstanding of the concept of temperature.)

Suppose I have a closed system: Some amount of gas of a certain pressure $P$ and temperature $T$ inside a container of volume $V$. Suppose that I shink the container, i.e. decrease the volume $V$ to $V'$. Is it possible that only the pressure $P$ rises to $P'$ but the temperature $T=T'$ stays constant?

Phrased differently, I want to know whether there is another law in addition to the formula from above which states a dependence of $P$ and $T$. I conjecture the existence of such a dependence from my (possibly very bad) intuition: Let's take a fridge, for example. A compressor raises the pressure of some gas (assume for simplicity that the volume stays constant) but the temperature raises as well. On the other side (inside) of the fridge, the pressure decreases and the temperature decreases likewise.

If the answer to the question above is 'yes', how to explain the connection (i.e. the positive proportion between $P$ and $T$) in the example of the fridge?

Thank you for your help and sorry again, if the question is not appropriate.

Added later: Maybe my confusion is the following? The answer to the first question is 'yes', i.e. there may be no dependency between $P$ and $T$ beside the stated formula. The principle which plays a role in the fridge is the phase transition between gas and liquid (and vice versa) which raises the temperature (resp. lowers it). Put in other words: A fridge does not work if there is only gas in the cooling system - there has to be a phase transition. Is this correct?

Answer 1

A phase transition makes the operation of a refrigerator much more efficient.

The first thing to note is that your equation $\frac{PV}{T}$ is for an ideal gas and the working fluid in a fridge exhibits two phases - vapour and liquid.

A vapour is a technical term for a substance in the gaseous phase which can be converted to the liquid phase by pressure alone. To be a vapour the fluid must be below the critical temperature.

So you start with a vapour at low pressure and compress it using a compressor to such a degree that it is transformed into a liquid.
During this process the temperature of the vapour/liquid is increased because when the bonds are made between molecules energy is released (latent heat) which raises the temperature.
The reverse is when you have to supply heat to convert a liquid (water) into a vapour (steam).

The heat produced is expelled from the refrigerator into the surrounding air using a heat exchanger at the back of the refrigerator.

The high pressure liquid is then allowed to expand into a low pressure region becoming vapour and this process requires an input of energy which is heat from the inside of the refrigerator.

The low pressure vapour then goes back to the compressor and the process is repeated.

The key to the efficient operation of the refrigerator is the interaction between the molecules of the fluid which means that energy is taken in to separate molecules (going from the liquid to the vapour phase) and energy given out when the molecules form bonds between each other (going from the vapour to the liquid phase).

In theory compressing a vapour, allowing it to cool still as a vapour and then allowing the vapour to expand with the abstraction of heat from the surroundings will work but the process would not be very efficient.
Bonds would still be broken and made but the bonds would not require as much energy to make and break them and there would be fewer of them.
What you really want is lots of strong bonds (lots of energy involved) made between the molecules during the compression phase and lots of strong bonds broken during the expansion phase.

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Note that there are absorption refrigerators which are not driven by electricity but by a heat source.