Pressure in a deodorant can is greater than the surrounding pressure but still the can is in equilibrium with the surrounding . Why does this happen. If we apply $F = PA$ then $P_{in}A= P_{atm}A$ and since both the areas are equal then both pressures must also be equal.

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A deodorant can contains a liquid hydrocarbon, typically a propane/butane mixture, and the pressure inside the can is due to the vapour pressure of this hydrocarbon. The pressure can be set to any desired value by varying the composition of the propellant - more propane makes a higher pressure while more butane makes a lower pressure. For a deodorant the pressures are typically in the range 1 - 3 atmospheres.

As long as there is liquid propellant left in the can the pressure will stay roughly constant, because as you spray material out the can some of the liquid evaporates to keep the pressure constant.

In your question you say:

the can is in equilibrium with the surrounding

but this is not the case. Liquid propellant is not in equilibrium at room temperature and a pressure of one atmosphere. That's why as soon as you lower the pressure in the can (by spraying from it) the propellant starts evaporating. The can is able to keep the propellant in a non-equilibrium state because it is rigid.


Think of an inflated balloon. Pressure inside the balloon is greater than atmospheric pressure. Entire system is in mechanical equilibrium because of tension provided by balloon surface. If you keep inflating the balloon, pressure inside the balloon keeps increasing, and so does tension in balloon surface, until it can take no more and bursts, thus equalising pressure. If the deodorant can does not burst, it is because the can wall is strong enough to tolerate the stresses generated by difference in pressure between inside and outside of can. So $P_{in}A=P_{atm}A$ is incorrect, because we have not taken into account forces due to can walls.


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