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I’ve been taking a thermodynamics course for a while now, and one of the questions I get from time to time is “How can I actually calculate the difference in internal energy between two arbitrary states, A and B?”
How To Calculate Internal Energy
Thermodynamics H.B. Cullen gives a very good idea about this and, according to him, Joule, after his experiments: “In an adiabatic system, any two equilibrium states can be associated with a given mechanical process, and the work done in this mechanical process is the energy difference enter the inner $Delta U$”.
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I’ll start with a simple example of a “piston vessel” system with an “ideal gas” inside.
Suppose we need to find $Delta U$ between $(P_0, V_0)$ and $(P_0, 2V_0)$. But if we have an adiabatic system, we cannot jump between different adiabats, that is, we cannot find a mechanical process that takes $(P_0, V_0)$ to $(P_0, 2V_0)$. I also explained this problem with a diagram.
If I can’t find a mechanical process that connects these states, I can’t find the seemingly strange difference in internal energy between them, as we do in different thermodynamic problems. So if I were an experimenter, how would I find the internal energy difference between these two states?
If you can only expand or contract the piston, you cannot relate the two states to a mechanical process. But different mechanical ways of acting in the system can be conceived.
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, paragraphs 1-7: Through a very small hole in the adiabatic wall can be passed a thin shaft with a propeller blade at the inner end and a crank arm at the outer end. It would be very difficult to do this without letting the gas leak and ruining the system, but not impossible in principle (although in practice it is easier and equivalent to prime the system with the existing vane and handle from the start).
By turning the knob, you can do measurable mechanical work on the system that increases its internal energy. If you keep the external pressure at the piston tip constant at $P_0$, this increase in energy causes a gradual expansion. So you can go from $(P_0, V_0)$ to $(P_0, 2V_0)$.
EDIT: The problem is that we just need a process that connects two states, in which all energy transfers are directly monitored and measured. This is the goal of having adiabatic walls (heat flow is not directly observed or measured) and using only mechanical processes (measuring mechanical work is usually as easy as measuring displacement). When we know all the energy transfers involved in a process, we also know the energy difference between the initial state and the final state.
As noted in the comment below, the process can be unidirectional (as in our example), but it makes no difference as long as all energy transfers are under control.
Chapter 03 The First Law Of Thermodynamics (pp 59 81)
According to the Maxwell-Boltzmann distribution, the kinetic energy of molecules, in turn, is directly related to the temperature of the gas. So a change in internal energy (change in kinetic energy) inevitably means a change in temperature. This raises the question of how the change in internal energy ΔU is related to the change in temperature ΔT.
As mentioned above, the kinetic theory of gases shows a direct relationship between the average kinetic energy of the molecules in a gas and its temperature. Therefore, if you know the temperature of the gas (and the number of particles), a certain internal energy is directly related to it. It is independent of the pressure or volume of the gas. That’s why
The change in internal energy during a thermodynamic process is clearly defined in ideal gases if the initial temperature (hence the initial energy) and the final temperature (hence the final energy) are known. Whether it is isobaric, isochoric, isentropic or any other process is therefore irrelevant to the change in internal energy. The change in internal energy is determined only by the change in temperature!
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The fact that the internal energy depends only on the state of the gas (the temperature) and not on the type of process is called internal energy.
. In contrast, heat and work depend on a thermodynamic process (isochoric, isothermal, isobaric or isentropic) and are therefore considered.
So if you want to study the relationship between changes in internal energy and changes in temperature, you can do that in principle with any thermodynamic process. The results obtained in this process are also valid for any other thermodynamic process, that is, ideal gases in general. At this point, the isochoric process is ideal for studying the relationship, as such a process does not work in gas or on gas. Therefore, the heat input increases the internal energy by the same amount, so it is relatively easy to study. This will be discussed in more detail in the next section.
The following experiment is conducted to determine the relationship between the change in internal energy and the change in temperature. A volume of gas of mass m is confined in a constant container. By transferring heat, the temperature of the gas increases and with it the internal energy.
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Since the volume of the gas does not change during this isochoric process, energy cannot be transferred as work by expansion or compression (W = 0). According to the first law of thermodynamics, the heat Q transferred to the gas completely increases the internal energy ΔU:
Thus, it is not necessary to perform detailed studies of the kinetic energy of individual molecules to determine the relationship between the change in internal energy and the change in temperature. According to equation (ref), it is only necessary to study the relationship between the heat input Q of the isochoric process (which then corresponds directly to the change in internal energy ΔU) and the resulting temperature increase ΔT. The energy supplied by an electric heater is relatively easy to determine (“heat = electricity x time”).
Experiments show that the temperature change ΔT is proportional to the given heat Q, i.e. twice the temperature change requires twice the heat:
It is also observed that the more heat required, the greater the mass of the gas to be heated. If the mass is twice as large, twice as much heat Q is needed to heat the gas. Therefore, the heat Q and the mass m are proportional to each other:
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In general, the supplied heat Q is proportional to the mass m of the gas and the temperature change ΔT. O
The specific heat capacity shows how much energy is needed to change the temperature of 1 kg of matter by 1 K. Or in this case: how much does the internal energy of the mass change when the temperature changes by 1 K? In equation (ref), we found the desired relationship between the change in internal energy ΔU and the change in temperature ΔT for an ideal gas:
The proportionality between the change in internal energy and the change in temperature must be considered only as a constant.
This constant of proportionality is called the isochoric specific heat only because this quantity describes the heat given for an isochoric process (and only for an isochoric process!) – see Eq. (ref). But as for the change in internal energy, equation (ref) applies to any thermodynamic process of ideal gases – see “Preliminary Considerations for Determining the Change in Internal Energy”!
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Although only the change in internal energy ΔU is important in thermodynamics, for ideal gases the absolute internal energy U can also be determined. To do this, imagine a gas confined in a cylinder of constant volume cooled to absolute zero. In this state, all molecules are at rest, so the gas has no internal energy. Now heat Q is supplied to the gas at a constant volume until it reaches temperature T. All the heat required to heat the gas is in the form of internal energy U. Thus, the following internal energy of the gas at temperature T is U. :
For ideal gases it may also depend on temperature. In particular, when degrees of freedom are frozen with decreasing temperature due to quantum mechanical effects. Equations (ref) and (ref) are no longer valid without restrictions. However, if one were to guess
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