# Gas undergoing isothermic compression

so Cv is measured in constant volume but is used in all processes? there must be an explanation to the constant volume notation, other than the effect on ideal gas that you described in post #20: "Yes, in a constant volume process, dQ = Cv dT and dU = CvdT for an ideal gas". so far i didn't find one.
$$\left(\frac{\partial U}{\partial V}\right)_TdV$$ change of volume against pressure is work, does this member account for the work? if yes, why is it missing in ideal gas?
If not please give me an example since i don't know how just a change in volume changes internal energy

Last edited:
Chestermiller
Mentor
so Cv is measured in constant volume but is used in all processes?
Yes.
there must be an explanation to the constant volume notation, other than the effect on ideal gas that you described in post #20: "Yes, in a constant volume process, dQ = Cv dT and dU = CvdT for an ideal gas". so far i didn't find one.
Actually, dU=CvdT applies generally to any single phase constant composition system at constant volume.
$$\left(\frac{\partial U}{\partial V}\right)_TdV$$ change of volume against pressure is work, does this member account for the work?
Not completely. The reversible work -PdV is one term that contributes to $$\left(\frac{\partial U}{\partial V}\right)_TdV$$. But that's not the only term. When you study the second law of thermo, you will find that there is also another term that contributes, related to the effect of volume on entropy. The complete expression is

$$\left(\frac{\partial U}{\partial V}\right)_TdV=\left(T\left(\frac{\partial P}{\partial T}\right)_V-P\right)dV$$

So, the full expression for dU in terms of changes in temperature and specific volume is:

##dU=C_vdT+\left(T\left(\frac{\partial P}{\partial T}\right)_V-P\right)dV##
if yes, why is it missing in ideal gas?
What is the term in parenthesis equal to for an ideal gas?
If not please give me an example since i don't know how just a change in volume changes internal energy
For an ideal gas, it doesn't. For a real gas, the term in parenthesis is not zero.

Chet

For ideal gas:
$$PV=nRT\rightarrow \left(\frac{\partial P}{\partial T}\right)_V=\frac{nR}{T}$$
$$\left(\frac{\partial P}{\partial T}\right)_V=T\frac{nR}{T}=P$$
$$\left(T\left(\frac{\partial P}{\partial T}\right)_V-P\right)=0$$

Chestermiller
Mentor
For ideal gas:
$$PV=nRT\rightarrow \left(\frac{\partial P}{\partial T}\right)_V=\frac{nR}{T}$$
$$\left(\frac{\partial P}{\partial T}\right)_V=T\frac{nR}{T}=P$$
$$\left(T\left(\frac{\partial P}{\partial T}\right)_V-P\right)=0$$
So you now kind of see how the internal energy and constant volume heat capacity of an ideal gas can be a function only of temperature? I know that the explanation is not totally satisfying, since you haven't learned about the 2nd law and entropy yet. Another way we know that the internal energy is not a function of volume is from experimental evidence, where, if we carry out an isothermal expansion or compression of a gas in the ideal gas region of low pressures, we find observationally that the heat added is equal to the work done, and thus, the change in internal energy is zero.

Chet

I made a mistake in the last post:
$$PV=nRT\rightarrow \left(\frac{\partial P}{\partial T}\right)_V=\frac{nR}{V}$$
$$\left(\frac{\partial P}{\partial T}\right)_V=T\frac{nR}{V}=P$$
Internal energy of non-ideal gas is also the potential energy between the molecules or atoms? is that the reason why U in non-ideal gas is a function of more than temperature?
And also we didn't talk at all about the dependence of the specific heat, Cv, on temperature:
So you now kind of see how the internal energy and constant volume heat capacity of an ideal gas can be a function only of temperature?

Chestermiller
Mentor
Internal energy of non-ideal gas is also the potential energy between the molecules or atoms?
Yes.
is that the reason why U in non-ideal gas is a function of more than temperature?
Yes.
And also we didn't talk at all about the dependence of the specific heat, Cv, on temperature:
Are you referring here to the fact you were taught that the specific heat of an ideal gas is a constant, independent of temperature?

Chet

Well i was taught that Cv varies with temperature, but please tell me what did you intend about that.
Does temperature, in general and also in liquids and solids, depend only on kinetic energy of the atoms/molecules? i understand from you that in biatomic gas it also reflects the vibratory energy, right? is temperature influenced also by the potential energy between atoms? if it were, i guess temperatures of metals and solids were high, no?

Chestermiller
Mentor
Well i was taught that Cv varies with temperature, but please tell me what did you intend about that.
Nothing in particular, really. Physicists often define the term "ideal gas" as one for which the heat capacity is constant. Engineers (I'm an engineer) define an ideal gas as the limiting behavior of real gases in the limit of low pressures; as such, in line with the engineering definition, the heat capacity is a function of temperature (which is available from experiments).

Does temperature, in general and also in liquids and solids, depend only on kinetic energy of the atoms/molecules? i understand from you that in biatomic gas it also reflects the vibratory energy, right?
Internal energy of a diatomic gas reflects kinetic and vibratory energy. As far as temperature is concerned, I'm a continuum guy, so the interpretation of temperature in terms of molecular arguments is not my area.
is temperature influenced also by the potential energy between atoms?
The internal energy is.

Chet

As you correctly indicated, if it were truly free expansion, the final temperature would be the same as the initial temperature.
In Wikipedia under Reversible Process i found:
"A classic example of irreversibility is allowing a certain volume of gas to be released into a vacuum. By releasing pressure on a sample and thus allowing it to occupy a large space, the system and surroundings will have completely left equilibrium, and heat dissipation will be large compared to the little work done."

Chestermiller
Mentor
In Wikipedia under Reversible Process i found:
"A classic example of irreversibility is allowing a certain volume of gas to be released into a vacuum. By releasing pressure on a sample and thus allowing it to occupy a large space, the system and surroundings will have completely left equilibrium, and heat dissipation will be large compared to the little work done."
If the process is adiabatic, the surroundings will not have left equilibrium if the expansion occurs within the system, say from one chamber to two chambers.

I have a feeling you have a question about all this. Can you please state your question.

Chet

Does a curved line in the P-V diagram for an adiabatic process have only a single temperature on each point of the line or can it have many?
In post #16 you showed that the temperature is -69 for work against pressure of 1[atm], and if we choose a different pressure, but with the same initial and final volumes, there will be different temperatures but the line will be the same since the P-V are the same, no? the lines follow ##PV^{\gamma}=Const##
Is it possible to have different temperatures for a single point? the temperature is determined by PV=nRT and it has one value at each point (for the same amount of material n)

Chestermiller
Mentor
Does a curved line in the P-V diagram for an adiabatic process have only a single temperature on each point of the line or can it have many?
In post #16 you showed that the temperature is -69 for work against pressure of 1[atm], and if we choose a different pressure, but with the same initial and final volumes, there will be different temperatures but the line will be the same since the P-V are the same, no? the lines follow ##PV^{\gamma}=Const##
Is it possible to have different temperatures for a single point? the temperature is determined by PV=nRT and it has one value at each point (for the same amount of material n)
This is all true for a reversible process, where the gas passes through a continuous sequence of thermodynamic equilibrium states. Under these circumstances, the pressure, temperature, and molar density within the gas are constant with spatial position, and are related by the ideal gas law.

But in an irreversible expansion, where the gas expands very rapidly, the local temperature, pressure, and molar density within the gas vary with spatial position from one parcel of gas to another at any given time during the expansion. Only in the final equilibrium state do the temperature, pressure, and molar density become uniform again. But, what about during the expansion process? What value of the pressure do you use to calculate the work done on the surroundings if the pressure is not the same at all locations within the gas? (And, even if it were, how would you calculate the pressure using the ideal gas law if the temperature is varying with spatial position within the gas in some unknown way?) Well, to get the work done on the surroundings, you use the pressure at the interface with the surroundings, which you can manually measure and/or impose on the gas. That's your 1 atm. in this problem.

When blogs were still a feature of Physics Forums, I had a blog that explains all this in detail. When the blogs were removed, I saved my blog in a file in case I ever wanted to use it. Here is write up that I saved:

FIRST LAW OF THERMODYNAMICS

Suppose that we have a closed system that at initial time ti is in an initial equilibrium state, with internal energy Ui, and at a later time tf, it is in a new equilibrium state with internal energy Uf. The transition from the initial equilibrium state to the final equilibrium state is brought about by imposing a time-dependent heat flow across the interface between the system and the surroundings, and a time-dependent rate of doing work at the interface between the system and the surroundings. Let $\dot{q}(t)$ represent the rate of heat addition across the interface between the system and the surroundings at time t, and let $\dot{w}(t)$ represent the rate at which the system does work on the surroundings at the interface at time t. According to the first law (basically conservation of energy),
$$\Delta U=U_f-U_i=\int_{t_i}^{t_f}{(\dot{q}(t)-\dot{w}(t))dt}=Q-W$$
where Q is the total amount of heat added and W is the total amount of work done by the system on the surroundings at the interface.

The time variation of $\dot{q}(t)$ and $\dot{w}(t)$ between the initial and final states uniquely characterizes the so-called process path. There are an infinite number of possible process paths that can take the system from the initial to the final equilibrium state. The only constraint is that Q-W must be the same for all of them.

If a process path is irreversible, then the temperature and pressure within the system are inhomogeneous (i.e., non-uniform, varying with spatial position), and one cannot define a unique pressure or temperature for the system (except at the initial and the final equilibrium state). However, the pressure and temperature at the interface can be measured and controlled using the surroundings to impose the temperature and pressure boundary conditions that we desire. Thus, TI(t) and PI(t) can be used to impose the process path that we desire. Alternately, and even more fundamentally, we can directly control, by well established methods, the rate of heat flow and the rate of doing work at the interface $\dot{q}(t)$ and $\dot{w}(t)$).

Both for reversible and irreversible process paths, the rate at which the system does work on the surroundings is given by:
$$\dot{w}(t)=P_I(t)\dot{V}(t)$$
where $\dot{V}(t)$ is the rate of change of system volume at time t. However, if the process path is reversible, the pressure P within the system is uniform, and

$P_I(t)=P(t)$ (reversible process path)

Therefore, $\dot{w}(t)=P(t)\dot{V}(t)$ (reversible process path)

Another feature of reversible process paths is that they are carried out very slowly, so that $\dot{q}(t)$ and $\dot{w}(t)$ are both very close to zero over then entire process path. However, the amount of time between the initial equilibrium state and the final equilibrium state (tf-ti) becomes exceedingly large. In this way, Q-W remains constant and finite.

SECOND LAW OF THERMODYNAMICS

In the previous section, we focused on the infinite number of process paths that are capable of taking a closed thermodynamic system from an initial equilibrium state to a final equilibrium state. Each of these process paths is uniquely determined by specifying the heat transfer rate $\dot{q}(t)$ and the rate of doing work $\dot{w}(t)$ as functions of time at the interface between the system and the surroundings. We noted that the cumulative amount of heat transfer and the cumulative amount of work done over an entire process path are given by the two integrals:
$$Q=\int_{t_i}^{t_f}{\dot{q}(t)dt}$$
$$W=\int_{t_i}^{t_f}{\dot{w}(t)dt}$$
In the present section, we will be introducing a third integral of this type (involving the heat transfer rate $\dot{q}(t)$) to provide a basis for establishing a precise mathematical statement of the Second Law of Thermodynamics.

The discovery of the Second Law came about in the 19th century, and involved contributions by many brilliant scientists. There have been many statements of the Second Law over the years, couched in complicated language and multi-word sentences, typically involving heat reservoirs, Carnot engines, and the like. These statements have been a source of unending confusion for students of thermodynamics for over a hundred years. What has been sorely needed is a precise mathematical definition of the Second Law that avoids all the complicated rhetoric. The sad part about all this is that such a precise definition has existed all along. The definition was formulated by Clausius back in the 1800's.

Clausius wondered what would happen if he evaluated the following integral over each of the possible process paths between the initial and final equilibrium states of a closed system:
$$I=\int_{t_i}^{t_f}{\frac{\dot{q}(t)}{T_I(t)}dt}$$
where TI(t) is the temperature at the interface with the surroundings at time t. He carried out extensive calculations on many systems undergoing a variety of both reversible and irreversible paths and discovered something astonishing. He found that, for any closed system, the values calculated for the integral over all the possible reversible and irreversible paths (between the initial and final equilibrium states) was not arbitrary; instead, there was a unique upper bound (maximum) to the value of the integral. Clausius also found that this result was consistent with all the "word definitions" of the Second Law.

Clearly, if there was an upper bound for this integral, this upper bound had to depend only on the two equilibrium states, and not on the path between them. It must therefore be regarded as a point function of state. Clausius named this point function Entropy.

But how could the value of this point function be determined without evaluating the integral over every possible process path between the initial and final equilibrium states to find the maximum? Clausius made another discovery. He determined that, out of the infinite number of possible process paths, there existed a well-defined subset, each member of which gave the same maximum value for the integral. This subset consisted of what we call today the reversible process paths. So, to determine the change in entropy between two equilibrium states, one must first dream up a reversible path between the states and then evaluate the integral. Any other process path will give a value for the integral lower than the entropy change.

So, mathematically, we can now state the Second Law as follows:

$$I=\int_{t_i}^{t_f}{\frac{\dot{q}(t)}{T_I(t)}dt}\leq\Delta S=\int_{t_i}^{t_f} {\frac{\dot{q}_{rev}(t)}{T(t)}dt}$$
where $\dot{q}_{rev}(t)$ is the heat transfer rate for any of the reversible paths between the initial and final equilibrium states, and T(t) is the system temperature at time t (which, for a reversible path, is equal to the temperature at the interface with the surroundings). This constitutes a precise mathematical statement of the Second Law of Thermodynamics.

Last edited:
This is all true for a reversible process, where the gas passes through a continuous sequence of thermodynamic equilibrium states. Under these circumstances, the pressure, temperature, and molar density within the gas are constant with spatial position, and are related by the ideal gas law.
Do you refer to ##PV^{\gamma}=Const##? so in the case of resistance of 1[atm] and irreversible processes the graph doesn't follow ##PV^{\gamma}=Const##? is this formula derived from the ideal gas law?
And in the case of 1[atm] resistance you cannot deduce anything except using the first law, as you did in post #16?

Chestermiller
Mentor
Do you refer to ##PV^{\gamma}=Const##?
Yes.
so in the case of resistance of 1[atm] and irreversible processes the graph doesn't follow ##PV^{\gamma}=Const##?
No, it doesn't. For an irreversible process, you can only plot the pressure at the piston face PI vs V, and it doesn't follow that equation.
is this formula derived from the ideal gas law?
It is derived exclusively for the case of an ideal gas undergoing an adiabatic reversible process (and nothing else).
And in the case of 1[atm] resistance you cannot deduce anything except using the first law, as you did in post #16?
You can still get the change in entropy (a 2nd law quantity) between the initial and final states, because it is a function of state only, and not of path.

Chet

So for any 2 states, (Pi, Vi) and (Pf, Vf) of an adiabatic process there is only one value of work done and it can be calculated using:
$$U=\frac{3}{2}nRT$$
And the first law? (ΔU=W)

Last edited:
In Wikipedia, under Specific Heat: ##dU=\delta Q-\delta W##
And if the work is by expanding the boundary: ##dU=\delta Q-PdV##
So for constant volume:
$$\left(\frac{\vartheta U}{\vartheta T}\right)=\left(\frac{\vartheta Q}{\vartheta T}\right)=C_v$$
So U is only a function of T. But you wrote:
$$dU=C_vdT+\left(\frac{\partial U}{\partial V}\right)_TdV$$
So is the article in Wikipedia about ideal gas or not? i know your formula is for real gasses. how to explain the contradiction

Chestermiller
Mentor
In Wikipedia, under Specific Heat: ##dU=\delta Q-\delta W##
And if the work is by expanding the boundary: ##dU=\delta Q-PdV##
So for constant volume:
$$\left(\frac{\vartheta U}{\vartheta T}\right)=\left(\frac{\vartheta Q}{\vartheta T}\right)=C_v$$
So U is only a function of T. But you wrote:
$$dU=C_vdT+\left(\frac{\partial U}{\partial V}\right)_TdV$$
So is the article in Wikipedia about ideal gas or not? i know your formula is for real gasses. how to explain the contradiction
For an ideal gas, $$\left(\frac{\partial U}{\partial V}\right)_T=0$$

Chet

Chestermiller
Mentor
Actually, dQ is not an exact differential unless the process path is reversible. But, for a reversible process path in which both the temperature and the volume are changing,
$$dQ=C_vdT+T\left(\frac{\partial P}{\partial T}\right)_VdV$$
You will learn how to derive this relationship once you study the 2nd Law.

Chet

Last edited:
Good but from the formula ##dU=\delta Q-PdV## there are no more members at all added to ##dU=C_vdT## and in your formula there is

Chestermiller
Mentor
Good but from the formula ##dU=\delta Q-PdV## there are no more members at all added to ##dU=C_vdT## and in your formula there is
Substitute the relationship in post #43 into your equation for dU. Then collect terms and substitute the ideal gas law into the coefficient of dV, and see what you get.

Chet

$$\left(\frac{\partial U}{\partial V}\right)_TdV=\left(T\left(\frac{\partial P}{\partial T}\right)_V-P\right)dV\rightarrow\left(\frac{\partial U}{\partial V}\right)_T=\left(T\left(\frac{\partial P}{\partial T}\right)_V-P\right)$$
Why is this equality? is it because we have dV at both ends so logically we understand that it has to be the energy change due to pressure change?
And why ##\left(\frac{\partial U}{\partial V}\right)_T## is at constant temperature? why not ##\left(\frac{\partial U}{\partial V}\right)_P##, at constant pressure?
Are those things taught in the second law?

Chestermiller
Mentor
$$\left(\frac{\partial U}{\partial V}\right)_TdV=\left(T\left(\frac{\partial P}{\partial T}\right)_V-P\right)dV\rightarrow\left(\frac{\partial U}{\partial V}\right)_T=\left(T\left(\frac{\partial P}{\partial T}\right)_V-P\right)$$
Why is this equality? is it because we have dV at both ends so logically we understand that it has to be the energy change due to pressure change?
No. This is the effect of the volume change on the internal energy at constant temperature, not the effect of the pressure change.

The first term in the parenthesis comes from dQ, not from dW. For a single phase ideal gas of constant composition undergoing a reversible change:

##dU=C_vdT##

##dQ=C_vdT+PdV##

##dW=PdV##

The fact that the internal energy of an ideal gas is a function only of temperature originated experimentally (by Joule), by doing tests on real gases at low pressures where they approach ideal gas behavior. It was found that, if the gas was expanded or compressed at constant temperature (i.e., the initial and final temperatures were equal, say by using a constant temperature reservoir), irrespective of whether the process was reversible or irreversible, the amount of heat added to the gas Q was virtually identical to the work done on the gas W. This meant that, when the gas volume was changed at constant temperature, its internal energy U did not change.
And why ##\left(\frac{\partial U}{\partial V}\right)_T## is at constant temperature? why not ##\left(\frac{\partial U}{\partial V}\right)_P##, at constant pressure?
The internal energy is a function of state. For a single phase substance of constant composition, the state of the substance is determined by any two independent physical parameters. Examples are P and T, P and V, and V and T (there are others). (In what we've been doing, U is the internal energy per mole, and V is the molar volume). All these combinations of pairs of physical parameters are equivalent. In the analysis we have been doing, we have been using the combination T and V (so U = U(T,V)). (The most convenient combinations to work with are T and V, and T and P). The key is that you have to choose the two that you will be working with. We can, of course, also do the analysis using dT and dP, and the partial of U with respect to T at constant P, and the partial of U with respect to P at constant T.
Are those things taught in the second law?
The equation
$$\left(\frac{\partial U}{\partial V}\right)_T=\left(T\left(\frac{\partial P}{\partial T}\right)_V-P\right)$$
is derived using the second law.

Chet

The internal energy is a function of state. For a single phase substance of constant composition, the state of the substance is determined by any two independent physical parameters. Examples are P and T, P and V, and V and T (there are others).
So Q is a function of path? that is why i can't analyze it as a function of 2 variables, (V,P), (T,P)... like you did for U?

Chestermiller
Mentor
So Q is a function of path? that is why i can't analyze it as a function of 2 variables, (V,P), (T,P)... like you did for U?
Q is a function of path, but that's not the only reason. Only for reversible paths can Q be evaluated in terms of variations of P and T. For irreverisble paths, P and T are not uniform within the system (they vary with spatial position), so there are no single unique values of these parameters to use in the calculation.

Chet

Thank you very much Chet