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I encountered a question that asks for the reason that a perfectly fitting pot and its lid often stick after cooking when it cools down. The answer in the solution manual was that the pressure decreases when the temperature decreases. I understand that point, but my problem is the following: The pot before being heated during cooking is influenced only by the atmospheric value, and the food inside that pot is the atmospheric pressure also. So this is the initial state for the pressure inside the pot now. When heated the pressure inside the pot increases. When we let the pot cool down to the room tempreture, would the pressure return to its initial value which is the atmospheric pressure? Why does the solution manual consider a vacuum inside the pot?

As @Stephen C. Steel pointed out, the pressure increase in the pot (due to heating the air in the pot) relative to the outside pressure exerts a net force on the lid causing it to vent. However, it doesn’t matter if the lid is on before the cooking process starts, or put on just before turning off the heat. The important thing is to understand the mechanism involved for the lid sticking on cool down.

Let’s assume the cooking starts with the lid off. Generally you will have some steam (water vapor) developed in the pot. The lid is put on just before turning off the heat. Inside the pot we have a mixture of air and water vapor, likely near saturation, which is nearly equal to the outside pressure, 1 atm. But the pressure inside the pot is the sum of the partial pressures of the water vapor and dry air. As the mixture cools the water vapors condenses into a liquid. The pressure in the pot is now less due to the loss of the partial pressure of the water vapor. The lower pressure in the pot versus the pressure outside the pot causes a force holding down the lid, if it consists of a relatively good seal.

I should add the fact that in addition to the drop in pressure due to condensation, the drop in temperature of the dry air in essentially a fixed volume also results in a pressure drop. Considering air as an ideal gas, from the ideal gas law:

$$frac P_1V_1T_!=frac P_2V_2T_2$$

And for $V=$constant

$$frac P_1T_!=frac P_2T_2$$

Since $T_2<T_1$, then $P_2<P_1$

Hope this helps

You've made the wrong assumption about what a perfectly fitting lid does. Rather than sealing perfectly, it will vent gas when the interior is at a higher pressure (the lid will lift), but seal when the exterior pressure is higher (the lid is pressed down).