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I am reading the book Heat and Thermodynamics 7th Edition by Zemansky and Dittman.

There is a section of a chapter that I would like to understand (Ch. 5.3).

I'm going to go through the reasoning, paraphrasing some parts and quoting others.

Imagine a thermally insulated vessel with rigid walls, divided into two compartments by a partition. Suppose that there is a gas in one compartment and that the other contains a vacuum. If the partition is removed, the gas will undergo what is known as an adiabatic free expansion in which no work is done and no heat is transferred. From the first law, since both $Q$ and $W$ are zero, it follows that the internal energy remains unchanged during a free expansion.

The question of whether or not the temperature of a gas changes during a free expansion and, if it does, of the magnitude of the temperature change has engaged the attention of scientists for about a hundred years. Starting with Joule in 1843, many attempts have been made to measure either the quantity $(\partial T/\partial V)_U$, which is called the Joule coefficient, or related quantities that are all a measure, in one way or another, of the effect of an adiabatic free expansion, or as it is often called, Joule expansion.

Experiments that try to measure a temperature change during a free expansion are difficult in practice.

Instead of measuring a temperature change during free expansion for which the internal energy is constant, consider measuring a change of internal temperature for constant temperature.

The internal energy function of any gas is a function of any two of the coordinates $P,V$, and $T$. We can write

$$dU=\left ( \frac{\partial U}{\partial T} \right )_V dT +\left ( \frac{\partial U}{\partial V} \right )_T dV\tag{1}$$

and if no temperature change ($dT=0$) occurs in a free expansion ($dU=0$) then

$$\left (\frac{\partial U}{\partial V}\right )_T = 0\tag{2}$$

which means that $U$ does not depend on $V$

Similarly,

$$dU=\left ( \frac{\partial U}{\partial T} \right )_P dT +\left ( \frac{\partial U}{\partial P} \right )_T dP\tag{3}$$

and once more we have that if no temperature change ($dT=0$) occurs in a free expansion ($dU=0$) then

$$\left (\frac{\partial U}{\partial P}\right )_T = 0\tag{4}$$

ie, $U$ does not depend on $P$.

Then, it is apparent that, if no temperature change takes place in a free expansion of a gas, $U$ is independent of $V$ and $P$, and, therefore, $U$ is a function of $T$ only.

Thus, to determine if the internal energy is a function of temperature, one must perform an experiment where the temperature is constant and measure whether either $\left (\frac{\partial U}{\partial V}\right )_T$ or $\left (\frac{\partial U}{\partial P}\right )_T = 0$ is zero.

My question

The reasoning here seems to be that if we can determine that $U$ does not depend on $V$ or $P$ in general (actually, not sure about the use of the term "in general" here; I think actually we would be determining this for a particular level of $T$ and $V$ or $T$ and $P$), then this must mean that in the specific case of an adiabatic free expansion (ie, a Joule expansion) we have from (1) and (3) that

$$0=\left ( \frac{\partial U}{\partial T} \right )_V dT +0\cdot dV\tag{1a}$$

$$0=\left ( \frac{\partial U}{\partial T} \right )_P dT +0\cdot dP\tag{3a}$$

Now, assuming this is correct, I am not sure about the justification for $\left ( \frac{\partial U}{\partial T}\right )_V \neq 0$ or $\left ( \frac{\partial U}{\partial T}\right )_P \neq 0$, but if we assume this then it means that $dT=0$ in both equations.

Is this reasoning correct?

Note that I have a follow-up question concerning an actual experiment based on the theoretical analysis above.

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