# What is Enthalpy?

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?

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.