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I'm looking at Elements of Physical Chemistry by Atkins and de Paula. In section 1A.3, they state Dalton's law as

The pressure exerted by a mixture of perfect gases is the sum of the pressures that each gas would exert if it were alone in the container at the same temperature:

$$ p = p_A + p_B + ... \tag{1}$$

In this expression, $p_J$ is the pressure that the gas $J$ would exert if it were alone in the container at the same temperature. Dalton's law is strictly valid only for mixtures of perfect gases [...].

They then go on to define the partial pressure as

$$ p_J = x_J p \tag{2}$$

where $x_J$ is the mole fraction of $J$ and $p$ is the total pressure of the mixture.

So my question is this: Eq. 1 holds for all gases given the definition in Eq. 2. So when they say Dalton's law only holds for perfect gases, do they mean that because $p_J$ in Dalton's law is not the same as Eq. 2, Dalton's law doesn't hold for all gases? Using the same notation for the two different meanings for $p_J$ seems to be widespread and potentially quite confusing.

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    $\begingroup$ It's the same meaning, just insert Eq 2 into Eq 1: $p = \sum p_i = \sum x_i p = p \sum x_i = p$. $\endgroup$
    – Buck Thorn
    Dec 3, 2022 at 19:52
  • $\begingroup$ They are definitely not the same meaning. $p$ for a real gas may be considerably lower than $nRT/V$. If $p_J$ is the pressure the gas would exert if it were alone in the container, then it is not the same as $px_J$ $\endgroup$
    – scmartin
    Dec 3, 2022 at 19:56
  • $\begingroup$ Hmmm, I think that Eq 1 can be satisfied simultaneously with the condition that "$p_J$ is the pressure that the gas J would exert if it were alone in the container" only if Eq 2 is true. Otherwise the equality in Eq 1 would not hold true. But you can write $\sum p_i = \sum p x_i$ without requiring that Eq 2 hold for each i. $\endgroup$
    – Buck Thorn
    Dec 3, 2022 at 20:06
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    $\begingroup$ The updated related question, quoting the same book, gives a hint that Atkins and de Paula mean by "perfect gas" what is now called an ideal gas, with the current term "perfect gas" having narrower meaning as an ideal gas with constant heat capacity. $\endgroup$
    – Poutnik
    Dec 3, 2022 at 20:19
  • $\begingroup$ Perfect gas usually means a gas above its critical T that cannot be liquified; it is usually not a synonym for an ideal gas that anyway is a mathematical figment. the question seems to be [and a good one] "are the deviations from the ideal gas law the same in a pure gas as they are in a mixture of gases?" A test might be to measure the vapor pressure of a precise amount of completely evaporated liquid in a vacuum and in an atmosphere of helium and then in air. $\endgroup$
    – jimchmst
    Dec 3, 2022 at 22:12

2 Answers 2

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After discussion with other commenters and answers, here is a summary of the issues with the text quoted in the question.

First, there are two non-equivalent definitions of the partial pressure $p_i$ being used.

Definition 1

The first is given only in the text immediately preceding and following Eq. 1 in the quoted text in the question. In mathematical form, this definition is

$$ p_i = \frac{n_i RT}{V} \: .$$

Using this definition for the partial pressure, we can show that

\begin{align} p_\mathrm{tot}^{(id)} &= \frac{n_\mathrm{tot}RT}{V} \\ &= \frac{\sum_i{n_i}RT}{V} \\ &= \sum_i p_i \: . \tag{A1} \end{align}

This is Dalton's law, which is only true for ideal (perfect in Atkins and de Paula's parlance) gases.

Definition 2

The second definition of $p_i$ is based on the actual pressure of the real gas and the mole fraction of species $i$, and it is given by Eq. 2 in the question

$$p_i = x_i p \: .$$

From this definition, and using the fact that $\sum_i x_i = 1$, we can express the total pressure $p$ as

\begin{align} p &= p\sum_i x_i \\ &= \sum_i x_i p \\ &= \sum_i p_i \: . \tag{A2} \end{align}

This expression holds for all gases, ideal or not.

Summary

Note that the final lines of Eqs. A1 and A2 look similar, but A1 is the ideal gas pressure for $n_\mathrm{tot}$ molecules, whereas A2 is based on the true pressure of the system and does not depend on any assumptions about the form of the equation of state. The confusion that led to my question is the fact that $p_i$ was used with these two different meanings in very close proximity and no special attention paid to differentiating them.

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There is no inconsistency as far the pasted text goes. Before studying any derivation or law, we should know their assumptions. Once we accept those conditions and the assumptions, the derived equations become valid. With the following assumptions, Equation (1) and (2) are perfectly consistent as long as the following hold.

So what are assumptions for Dalton's law? (0) The gases in the mixture will not react, even if they are ideal gases. (1) There is no interaction among gas molecules (2) Molecular volume is negligible. This does not hold true at higher pressures. (3) Temperature is not too high or too low (say, we are not at the Sun's surface or Pluto)

If we start talking about real gases for equation (1) and ideal gases for equation (2), it is comparing apples and oranges.

BTW there are better textbooks on Physical Chemistry and it is a good habit to read the same topic from multiple books! It sometimes gives interesting insights.

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  • $\begingroup$ I agree, this isn't my favorite P Chem text. I suppose the problem I have is that the presentation in the text as quoted above seems to imply that Eq. 1 is Dalton's law, but possibly more accurately, Dalton's law should be given mathematically as $p = n_A RT/V + n_B RT/V + ... $ to make clear that the partial pressures given are assuming ideal gas behavior, whereas the definition they give immediately below is a general definition for all gases. $\endgroup$
    – scmartin
    Dec 3, 2022 at 20:47
  • $\begingroup$ @scmartin, I agree. I am not exactly sure (or aware of exceptions) when equation 2 is not valid either for real gases. $\endgroup$
    – AChem
    Dec 3, 2022 at 21:23
  • $\begingroup$ I don't believe it is possible to have exceptions to equation 2. $\sum_i x_i = 1$ so $p = p \sum_i x_i = \sum_i p x_i = \sum p_i$. This is why I think the implication in the Atkins & de Paula text is a bit unclear and even possibly misleading. $\endgroup$
    – scmartin
    Dec 3, 2022 at 21:38
  • $\begingroup$ But you are implicitly using Dalton's law in summations. I mean nothing is absolute in gases laws. There is no universal equation of state. All laws have a limited validity range. This is what I am trying to convey, say you have a mixture of gases, you keep increasing pressure, then suddenly one of the gases starts to liquefy...then equation 2 may break down. I do not know about the universal truth of (2). $\endgroup$
    – AChem
    Dec 3, 2022 at 21:47
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    $\begingroup$ No, the summation $ \sum_i x_i =1$ doesn't have anything to do with Dalton's law, it is part of the definition of mole fraction. So if we define $p_i$ to be $ x_i p$, then $p = \sum_i p_i$ is always true. $p$ and $x_i$ are well-defined thermodynamic properties for any system. That is why I thought the Atkins explanation was so confusing. They seem to use $p_i$ to mean both $n_i RT/V$ and $x_i p$, but these are not equivalent. In the first instance, the use it in the Dalton's law sense of ideal gas behavior, but in the second instance they use it in the more general sense of $x_i p$. $\endgroup$
    – scmartin
    Dec 3, 2022 at 22:53

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