How to find internal energy with constant temperature?

[EastWindBreaks]

Homework Statement

since specific heat c changes with temperature, but its treated as a constant in the heat formula, so that means that heat formula Q=mc(T2-T1) is just an approximation? correct?
I see some texts define heat as Heat, q, is thermal energy transferred from a hotter system to a cooler system that are in contact.(from Khan Academy), and some texts define it as energy stored inside of something. from the heat formula, if there is no temperature change, then the heat is 0.so the second definition is incorrect?

do we have a single equation that calculates internal energy stored in a system with no temperature change, for example at a given instant of a bottle of water?

Homework Equations

The Attempt at a Solution

 

[kuruman]

Q=mc(T2-T1) is just an approximation? correct?

Correct. More generally Q = m ∫c(T) dT.

if there is no temperature change, then the heat is 0.so the second definition is incorrect?

Not really because you need to consider phase transitions. When you partially melt an ice cube and end up with some ice and some water, the energy content of the slush is higher than before. The energy that you transferred into the ice cube has gone into loosening the bonds between some of the water molecules and turn them from "ice" to "water."

do we have a single equation that calculates internal energy stored in a system with no temperature change, for example at a given instant of a bottle of water?

I don't believe so. There are many different kinds of phase transitions (Search the term if you wish) so there are many different mathematical descriptions of the energy stored in a system.

 

[Chestermiller]

Homework Statement

since specific heat c changes with temperature, but its treated as a constant in the heat formula, so that means that heat formula Q=mc(T2-T1) is just an approximation? correct?

These equations apply only to a closed system at constant volume ($C_v$) or at constant pressure ($C_p$). Otherwise, the heat transferred Q is not given by either. However, more general relationships which are always true (by definition) are given by:
$$\left(\frac{\partial U}{\partial T}\right)_V=C_v$$
$$\left(\frac{\partial H}{\partial T}\right)_P=C_p$$
In either case, if the heat capacity varies with temperature, then only the differential version is correct.

I see some texts define heat as Heat, q, is thermal energy transferred from a hotter system to a cooler system that are in contact.(from Khan Academy), and some texts define it as energy stored inside of something. from the heat formula, if there is no temperature change, then the heat is 0.so the second definition is incorrect?

According to how heat is described in thermodynamics, the first definition is correct and the second definition is definitely incorrect.

do we have a single equation that calculates internal energy stored in a system with no temperature change, for example at a given instant of a bottle of water?

Internal energy is a relative quantity (rather than absolute), so, in thermodynamics, it is always expressed relative to a specified reference state of zero internal energy. This is typically taken at a reference temperature and reference pressure.

 

[EastWindBreaks]

Got it, Thank you all!