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At school, they teach the following two formulas: $$\Delta H = -mc\Delta T$$ and $$q= mc\Delta T$$ What i am not sure is that my teacher says that q and $\Delta H$ are the same thing but to think of q as a magnitude and hence it is always positive. However what I am confused about is how can q always be positive? What if the change in temperature is negative? Also, what is the difference between q and $\Delta H$. My current understanding is that they are the same thing and they measure the change in heat energy. Is this right?

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Enthalpy is thought as the total potential energy of a system. The actual thermodynamic equation for enthalpy is: $$\Delta H = \Delta U + \Delta {(pV)}$$ where $\Delta U$ is the change in internal energy. Internal energy can be thought as the total amount of heat energy there is in a system. Hence $\Delta U$ can be thought as the total energy that is required to make a system, while $\Delta {(pV)}$ can be thought as how much energy is required to make space for the system.

Q on the other hand is actually a different thing to enthalpy. Q refers to the heat transfer of a system. Note that this means that q can be positive or negative (if heat energy moves from the system to the surroundings) and hence it is not always positive. The correct formula for heat transfer is: $$q = \Delta U -w $$ So actually, q doesn't always have to equal to enthalpy change. However at a school level it is taught that they equal the same thing. This is because of we assume that pressure is constant (which is a common assumption made as usually most experiments have a constant pressure), enthalpy becomes: $$\Delta H = \Delta U + p\Delta V$$ Also since $w = p\Delta V$, enthalpy becomes: $$\Delta H = \Delta U - w = q$$ Hence, enthalpy and q equal the same values only when pressure is constant.

The equation that your teacher has given you is actually an equation for heat transfer, not for enthalpy change. However because enthalpy equals heat transfer in most cases, it is commonly said that enthalpy change also equals to that equation. However the one mistake is that there shouldn't be a minus in the equation for enthalpy change. The equation for heat transfer and enthalpy should both be the same.

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