# Partial pressure

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Since partial pressure and partial volume are concepts related to gas mixtures, we first determine what a mixture of ideal gases is. So, a mixture of gases is a combination of several different gases that do not enter into a chemical reaction under given conditions. Under other conditions (for example, increasing pressure), the same gases may react chemically. Mixtures are characterized by such a physical quantity as the weight concentration of the \$ g_i \$ i -th gas, which is a component of the mixture, with:

where N is the total number of different gases in the mixture,

and molar concentration of \$ x_i i-th \$ gas in the mixture, with:

where \$ < nu> _i \$ is the number of moles of \$ i-th \$ gas in the mixture.

## What is partial pressure?

Partial pressure is a characteristic of the state of the components of a mixture of ideal gases.

Partial pressure \$ (p_) \$ \$ i-th \$ gas in the mixture is called the pressure that this gas would create if, in addition to it, all other gases were absent, but the volume and temperature remained unchanged.

where \$ V- \$ volume of the mixture, \$ T \$ - temperature of the mixture. It should be noted here that due to the equality of the average kinetic energies of the molecules of the mixtures, we can speak of the equality of temperatures of all components of the mixtures in the state of thermodynamic equilibrium.

The pressure of a mixture of ideal gases p is determined by Dalton's law:

Therefore, partial pressure can be expressed as:

## What is partial volume

Another important parameter of the state of a gas mixture is the partial volume.

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The partial volume of \$ V_i \$ \$ i-th \$ gas in the mixture is the volume that the gas would have if all the other gases were removed from the mixture at a constant temperature and volume:

For a mixture of ideal gases, Amag's law holds:

Indeed, if we express \$ < nu> _i \$ from (6) and substitute it into (4), we obtain

The partial volume can be calculated by the formula:

The state parameters of a mixture of ideal gases obey the Mendeleev-Klaiperon equation in the following form:

where all the parameters in equation (9) relate to the mixture as a whole.

Or equation (9) is sometimes more convenient to write in this form:

where \$ R_= frac<< mu> _> = R sum limits ^ N_< frac<< mu> _i >> \$ is the specific gas constant of the mixture.

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Task: At 290 K, a \$ 1 m ^ 3 \$ vessel contains 0.5 \$ < cdot 10> ^ <-3> \$ kg of hydrogen and 0.10 \$ < cdot 10> ^ <-3> \$ kg of helium . Find the partial pressure of helium and the pressure of the mixture.

Find the number of moles for each component of the mixture using the formula:

then the number of moles of hydrogen in the mixture, if using the periodic table we find that the molar mass of hydrogen is \$ < mu> _= 2 cdot <10> ^ <-3> frac<кг><моль>\$:

We use the Mendeleev-Klaiperon equation to find the partial pressures of each component of the mixture:

Then the hydrogen pressure:

We calculate the partial pressure of hydrogen:

Similarly, we find the partial pressure of helium:

We find the pressure of the mixture as the sum of the pressures of its constituent components:

Therefore, the pressure of the mixture is equal to:

Answer: The partial pressure of helium is \$ 60.25 \$ Pa, the mixture pressure is \$ 662.75 \$ Pa.

Task: The gas mixture contains 0.5 kg \$ O_2 \$ and 1 kg \$ CO_2 \$. Determine the volume that the mixture of gases will take at a pressure of one atmosphere, if the gases are considered ideal. Take the temperature of the mixture equal to 300 K.

Find the mass of the gas mixture:

Find the mass components of the mixture \$ g_i \$:

We calculate the gas constant of the mixture:

The expression for the volume of the mixture obtained from the Mendeleev-Klaiperon equation:

Let us calculate the volume, given that p = 1 atm. = \$ <10> ^ 5Pa \$:

Answer: The mixture occupies a volume of \$ 0.9 m ^ 3. \$

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## Ideal gas mixtures

For an ideal gas, the partial pressure in the mixture is equal to the pressure that would be exerted if it occupied the same volume as the entire gas mixture at the same temperature. The reason for this is that, by definition, the forces of attraction or repulsion do not act between the molecules of an ideal gas, their collisions with each other and with the walls of the vessel are absolutely elastic, and the interaction time between the molecules is negligible compared to the average time between collisions. As far as the conditions of a real gas mixture are close to this ideal, the total pressure of the mixture is equal to the sum of the partial pressures of each gas in the mixture, as formulated by Dalton’s law. For example, given a mixture of ideal gas from nitrogen (N2), hydrogen (H2) and ammonia (NH3):

P = P N 2 + P H 2 + P N H 3 < displaystyle P = P _ << mathrm > _ <2>> + P_ << mathrm > _ <2>> + P_ << mathrm > _ <3> >>, where:

P N 2 < displaystyle P _ << mathrm > _ <2> >> = partial pressure of nitrogen (N2)

P H 2 < displaystyle P _ << mathrm > _ <2> >> = partial pressure of hydrogen (H2)

P N H 3 < displaystyle P _ << mathrm > _ <3> >> = partial pressure of ammonia (NH3)

## Ideal gas mixtures

The molar fraction of individual gas components in an ideal gas mixture can be expressed within the partial pressures of the components or moles of the components:

x i = P i P = n i n < displaystyle x _ < mathrm > = < frac >>

> = < frac >>>>

and the partial pressure of the individual gas components in an ideal gas can be obtained using the following expression:

P i = x i ⋅ P < displaystyle P _ < mathrm > = x_ < mathrm > cdot P>, where:

The mole fraction of an individual component in the gas mixture is equal to the volume fraction of this component in the gas mixture.

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