# How to find internal energy with constant temperature?

• EastWindBreaks
In summary, the heat formula Q=mc(T2-T1) is an approximation and only applies to closed systems at constant volume or pressure. The first definition of heat, as thermal energy transferred between systems, is correct while the second definition, as energy stored inside something, is incorrect. There is no single equation for calculating internal energy as it is a relative quantity and is expressed relative to a reference state of zero internal energy.
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?

## The Attempt at a Solution

EastWindBreaks said:
Q=mc(T2-T1) is just an approximation? correct?
Correct. More generally Q = m ∫c(T) dT.
EastWindBreaks said:
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."
EastWindBreaks said:
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.

EastWindBreaks
EastWindBreaks said:

## 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 and patrickmoloney
Got it, Thank you all!

## 1. What is internal energy?

Internal energy is the sum of all the microscopic forms of energy, such as the kinetic and potential energies of the molecules, within a system.

## 2. How is internal energy related to temperature?

Internal energy and temperature are directly proportional. This means that as the temperature of a system increases, the internal energy also increases.

## 3. Can internal energy be measured directly?

No, internal energy cannot be measured directly. It can only be calculated by measuring other thermodynamic quantities, such as temperature, pressure, and volume, and using them in the appropriate equations.

## 4. How is internal energy affected by a change in temperature at constant volume?

In a system with constant volume, an increase in temperature will result in an increase in internal energy because the molecules will have a higher average kinetic energy.

## 5. How can internal energy be calculated if the temperature is constant?

If the temperature is constant, the change in internal energy can be calculated using the equation ΔU = nCvΔT, where n is the number of moles of gas, Cv is the molar heat capacity at constant volume, and ΔT is the change in temperature.

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