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For Gibbs free energy it says most reactions occur at constant temperature and pressure, but I don't understand how the temperature stays the same if the system is absorbing or losing heat (Delta H changes) ?
When a problem says that the reaction is occurring at constant temperature and pressure this means that the external temperature and pressure are not changing. The temperature or pressure of the system can still change but these changes would not affect the temperature or pressure of the often much larger surroundings enough to make a measurable difference.
I thought about it some more and let me just clarify what I was trying to say. So because exothermic reactions generate heat, they would likely increase the temperature of the system or reaction so essentially this would mean that the temperature is not constant. To approximately maintain constant temperature, there must be some way to cool the system, by maintaining the beaker with the reaction in a water bath with a constant temperature for example. Because water has a high specific heat, a large enough volume of water would likely be able to absorb the heat produced in the exothermic reaction without its temperature changing. There are likely better ways to maintain constant temperature but basically the surroundings have to be able to release or absorb heat from the system in order to maintain constant temperature. So yes heat can be released or absorbed by a system at constant pressure (Δ H) while maintaining constant temperature.
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