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Does a PC's hardware temperature increase when it is in standby (sleep) mode or is it the same as if the PC is turned off?

If it does increase, can the hardware be damaged?

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The heat produced is directly proportional to the energy consumed, as all the energy eventually turns into heat. In sleep mode energy is mainly used for reserving contents of RAM. Depending on the setup, some USB devices may stay on, but external USB devices can be ruled out.

That's more than no energy at all (turned off), but...

Let's imagine a PC in the sleep mode consumes approximately 0,05 amperes from 230 V AC power, which sounds reasonable. That's less than 12 watts. You can easily touch a 12 W LED bulb, and even a small PC has more area than a bulb. The summer condition causes more heat than that, so you should avoid direct sunlight instead of sleep mode.

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Standby may not be the best word, as most computers are in a form of "Standby" when you tell them to shutdown as well - they are just in a deeper form of standby which only listens for network WOL packets, the power button, and possibly the keyboard, whereas the "Sleep" also keeps the ram active, and listens to the mouse. That said, the "Sleep" version will consume a small amount more power, and thus generate more heat - however, this heat is not anything that will significantly affect the lifespan of a component, as it's really "high temperatures" that wears out components like capacitors, and the rule of thumb is that for every 10°C of temperature increase, the component wears out twice as fast - i.e. the difference of going from 40°C to 41° won't matter nearly as much as going from 50°C to 51°C.

That said, temperature isn't the only thing that harms components - humidity does as well, to the point where some motherboard manufacturers have added a "feature" to turn on the fans & generate heat such that the humidity does not reach 100% (warmer air can absorb more moisture, thus less % humidity for same amount of water in the air), and even advertise this feature for when a computer goes to sleep / hibernate as it will monitor the environment and turn itself on when needed. How much this helps I don't know, but if heat ended up killing their products faster at any significant level due to this, it would not have been implemented.

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