# How to derive Dalton's law of partial pressure for a non-ideal gas?

According to dalton's law,

$$p_i = p_°X_i$$

Where $$p_°$$ is total pressure of gas mixture and $$X_i$$ is the mole fraction of the $$i^{th}$$ component of mixture. This is very easy to derive for an ideal gas using the equation of state of an ideal gas.

However, I cannot derive it for non-ideal gases. Please help me to do so. Theoretical as well as mathematical proofs are both welcome.

Book is Physical chemistry by Peter Atkins. Relevant page is here :

• As far as I know, Dalton's law is empirical and only stricty true for ideal gases. For a mix of interacting gases, it doesn't really work (although it usually remains a good approximation). Commented Jan 29, 2023 at 9:10
• @Miyase I am studying these things in physical chemistry by Peter Atkins. He says that this is true for all non-reacting gaseous mixture and is not necessarily limited for ideal gases. I will add the link of book as well as the picture of that page in the question. Please wait. Commented Jan 29, 2023 at 9:31
• It's mostly true in practice, but not in theory as I said. When real gases interact, it can be through Van der Waals forces or because of a chemical reaction. The former are usually weak, so in the absence of chemical reaction, Dalton's law is a good approximation. But the later isn't, invalidating Dalton's law. So it's fine for "non-reacting" gases as your book says, but it's still an empirical law. Commented Jan 29, 2023 at 10:46
• After seeing your screenshot: it may be a problem of definition. Your book chooses to define partial pressure as xP, which makes partial pressures automatically add up to p. But this definition of partial pressure doesn't coincide with the individual pressures that gases have in a mixture. Another way to handle this (the one I'm familiar with) is to define partial pressure as the one the gas would have alone, and then partial pressures no longer add up to p (unless it's a mixture of ideal gases). Both approaches are correct, but aren't speaking of the same things. Commented Jan 29, 2023 at 11:35
• @Miyase The partial pressure approach you are familiar with i.e. the pressure of the gas if it would be alone , is only correct for ideal gas and not real gas. However , $p_i = X_i p_°$ approach is true for real gases also. This is what book says. But the book doesn't say these are approximations. This is what the last paragraph of book is all about. Commented Jan 29, 2023 at 12:39