Dalton's law clarification

Question

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.

Answer 1

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.

Answer 2

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.