Dalton's Law of Partial Pressure

Introduction

According to the kinetic theory of gases, gas disperses to fill the space it occupies in a container since it has no intermolecular interactions. To put it another way, the particles in a gas mixture are spaced enough to behave independently of one another and not interact. There are no interactions with other substance particles, hence the pressure of an ideal gas is dictated by its interactions with the vessel rather than those with other substance particles. Without changing the pressure of another gas, a gas will grow to fill the vessel in which it is enclosed. Thus, it can be stated that a gas's pressure depends on its molecular weight, as well as its volume and temperature. The temperature (T) and volume (V) of the different gases in a mixture are the same since they are all contained in the same container. The total pressure of the gases contained in the container can be calculated by adding up the pressure which each gas generates in the system.

BlyumJ, Schematic Depicting Dalton's Law, CC BY-SA 4.0

What is partial pressure?

A particular gas pressure exerts within other gases mixture is referred to as pressure of that particular gas. For instance, if a vessel includes a mixture of 3 gases -oxygen, nitrogen, and 𝐶𝑂2-its partial pressure is equal to the pressure oxygen exerts on the container's walls, and its partial pressures are equal to those of nitrogen and carbon dioxide. The summation of the partial pressures of the gases $\mathrm{O_{2}\:,\:N_{2}\:,\:and\:CO_{2}}$ in the mixture exerts the total pressure on the container walls.

To put it another way, each gas that makes up a mixture of gases has a partial pressure. This represents the gas's hypothetical pressure at the same temperature if it filled the initial mixture.

A gas's thermodynamic activity is measured by its partial pressure. A gas's partial pressure might reveal a variety of its characteristics. For instance, a gas's partial pressure affects how reactive it is in a certain volume. The partial pressures of even gases determine how they dissolve and disseminate. This characteristic of gases aids in our comprehension and prediction of the chemical processes of gases when it comes to biology. The partial pressures of $\mathrm{O_{2}}$ and $\mathrm{CO_{2}}$ are significant factors in arterial blood gas testing.

The sign P with the gas's symbol in the subscript stands for the gas's partial pressure. As an illustration,$\mathrm{PO_{2}}$ denotes the partial pressure of $\mathrm{O_{2}}$

MaxD550, Dalton's law of partial pressures, CC BY-SA 4.0

What is Dalton's law of partial pressure?

English scientist, physicist, and meteorologist John Dalton developed and originally published the law of partial pressure in 1802. Dalton's partial pressure equation states that the overall pressure of a gas mixture is equal to the sum of the partial pressures of the constituent gases. Dalton's law is accurate for a perfect gas mixture. Because they are comparatively too far away, molecules in an ideal gas cannot communicate. The real gas mixture largely follows Dalton's law.

For instance, a mixture of ideal gases including nitrogen, hydrogen, and oxygen will exert the following overall pressure:

$$\mathrm{P_{total}\:=\:P_{N_{2}}\:+\:P_{H_{2}}\:+\:P_{O_{2}}}$$

In this case −

$\mathrm{P_{total}\:=\:ideal\:gas\:misture's\:total\:pressure}$

$\mathrm{P_{N_{2}}\:=\:Nitrogen\:partial\:pressure}$

$\mathrm{P_{H_{2}}\:=\:hydrogen\:partial\:presssure}$

$\mathrm{P_{O_{2}}\:=\:oxygen\:partial\:pressure}$

Let's have a look at an additional illustration. A nitrogen and oxygen gas mixture has been placed in a beaker.

A mixture of gases exerting total pressure will be 148 mm Hg + 590 mm Hg = 738 mm Hg if the partial pressures of 𝑂2 and 𝑁2 are 148mm Hg and 590 mm Hg, respectively.

Mole fraction and Partial pressure

We must first gain a fundamental understanding of mole fraction to comprehend the relationship between mole fraction and partial pressure. The mole fraction is the proportion of one gas component to the sum of all the other gas components in the mixture. By 𝑥𝑖, it is expressed.

Mole fraction formula −

$$\mathrm{x_{i}\:=\:\frac{n_{i}}{n_{total}}}$$

$\mathrm{x_{i}\:=\:the\:mole\:fraction}$

$\mathrm{n_{i}\:=\:number\:of\:moles\:of\:a\:specific\:gas\:that\:makes\:up\:the\:mixture}$

$\mathrm{n_{total}\:=\:sum\:of\:the\:moles\:of\:each\:component\:in\:the\:combination\:.\:Amount\:fraction\:is\:another\:name\:for\:mole\:fraction}$

Relationship between Mole Fraction and Partial Pressure

An ideal gas mixture's mole fraction of a certain gas component can be stated as -

$$\mathrm{x_{i}\:=\:\frac{n_{i}}{n}\:............(1)}$$

Where 𝑛𝑖 is the total number of moles in the ideal gas combination made up of each gas. n is the sum of all the elements' moles in the ideal gas mixture.

$\mathrm{mole\:fraction\:=\:x_{i}}$

Another way to express the mole fraction of a certain gas in an ideal gas mixture is as follows −

$$\mathrm{x_{i}\:=\:\frac{p_{i}}{p}..........(2)}$$

Where 𝑃𝑖 is the ideal gas mixture's optimum partial pressure of each gas separately. P is the optimal gas mixture overall pressure.

$\mathrm{mole\:fraction\:=\:x_{i}}$

Equations (1) and (2) allow us to write -

$$\mathrm{x_{i}\:=\:\frac{n_{i}}{n}\:=\:\frac{p_{i}}{p}}$$

Consequently, the ideal gas mixture's partial pressure of a particular gas can be stated as

$$\mathrm{P_{i}\:=\:x_{i}\:.\:p}$$

Given that the volumetric proportion of a gas constituent in a mixture of gases is like its mole fraction, we can write -

$$\mathrm{\frac{P_{X}}{P_{total}}\:=\:\frac{V_{X}}{V_{total}}\:\frac{n_{x}}{n_{total}}}$$

Where 𝑛𝑋= moles of component X of the gas

𝑛𝑡𝑜𝑡𝑎𝑙 = overall no. of moles of all the mixture's components.

𝑃𝑋 = partial gas pressure of X.

𝑃𝑡𝑜𝑡𝑎𝑙 is the full gas mixture pressure.

𝑉𝑋 = any given gas component's partial volume, X.

𝑉𝑡𝑜𝑡𝑎𝑙= the mixture's overall volume

Conclusion

In conclusion, we can say that the overall pressure of a gas mixture is equal to the total partial pressure that each gas exerts. Hence with the help of dalton's law, we can calculate the partial pressure of each gas. But the condition is that gases must be non- reactive.

The total pressure of the mixture −

$$\mathrm{P_{total}\:=\:P_{gas\:1}\:+\:P_{gas\:2}\:+\:P_{gas\:3}}$$

Dalton's law can also be expressed using the mole fraction of gas,

$$\mathrm{P_{gas\:1}\:=\:x_{1}P_{total}}$$

FAQs

1. What are the restrictions of the partial pressure law of Dalton?

At low pressure, the law is valid for actual gases, but it deviates dramatically at high pressure. The gas mixture is not reactive in any way. The interaction between the molecules of every individual gas is likewise considered to be the same as that between the particles in the mixture.

2. Which scenario precludes the use of Dalton's law?

Since $\mathrm{NH_{3}}$ and $\mathrm{HCl}$ are reacting gases, they are not covered by Dalton's law of partial pressure, which only applies to non-reacting gases.

3. Why doesn't the partial pressure rule of Dalton apply to reacting gases?

Dalton's law only holds for mixtures of non-reacting gases, meaning that the gases do not chemically react, and only in those situations will the sum of each gas's partial pressure equal the sum pressure of the mixture. Dalton's law does not apply to reaction mixtures made up of reacting gases because there will be no independent vapour pressure of the gases in those situations, so Dalton's will not be applied in those circumstances.

4. What did Dalton's atomic theory's key argument entail?

According to the first section of his theory, atoms-which cannot be divided-make up all stuff. The second portion of the theory states that the mass and characteristics of every atom in a certain element are the same. Compounds, according to the third section, are atoms that combine two or more distinct types.

5. 𝑲𝒑 and 𝑲𝒄—what are they?

A perfect gaseous mixture's equilibrium constants are 𝑲𝒑 and 𝑲𝒄. Whenever equilibrium concentrations are represented in terms of atmospheric pressure, the equilibrium constant employed is 𝑲𝒑, and when expressed in terms of molarity, it is 𝑲𝒄