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Learning Objectives
- To determine the contribution of each component gas to the total pressure of a mixture of gases
In our use of the ideal gas law thus far, we have focused entirely on the properties of pure gases with only a single chemical species. But what happens when two or more gases are mixed? In this section, we describe how to determine the contribution of each gas present to the total pressure of the mixture.
Partial Pressures
The ideal gas law assumes that all gases behave identically and that their behavior is independent of attractive and repulsive forces. If volume and temperature are held constant, the ideal gas equation can be rearranged to show that the pressure of a sample of gas is directly proportional to the number of moles of gas present:
\[P=n \left(\dfrac{RT}{V}\right) = n \times \rm const. \label{10.6.1} \]
Nothing in the equation depends on the nature of the gas—only the amount.
With this assumption, let’s suppose we have a mixture of two ideal gases that are present in equal amounts. What is the total pressure of the mixture? Because the pressure depends on only the total number of particles of gas present, the total pressure of the mixture will simply be twice the pressure of either component. More generally, the total pressure exerted by a mixture of gases at a given temperature and volume is the sum of the pressures exerted by each gas alone. Furthermore, if we know the volume, the temperature, and the number of moles of each gas in a mixture, then we can calculate the pressure exerted by each gas individually, which is its partial pressure, the pressure the gas would exert if it were the only one present (at the same temperature and volume).
To summarize, the total pressure exerted by a mixture of gases is the sum of the partial pressures of component gases. This law was first discovered by John Dalton, the father of the atomic theory of matter. It is now known as Dalton’s law of partial pressures. We can write it mathematically as
\[\begin{align} P_{tot} &= P_1+P_2+P_3+P_4 \ldots \\[4pt] &= \sum_{i=1}^n{P_i} \label{10.6.2} \end{align} \]
where \(P_{tot}\) is the total pressure and the other terms are the partial pressures of the individual gases (up to \(n\) component gases).
For a mixture of two ideal gases, \(A\) and \(B\), we can write an expression for the total pressure:
\[\begin{align} P_{tot} &=P_A+P_B \\[4pt] &=n_A\left(\dfrac{RT}{V}\right) + n_B\left(\dfrac{RT}{V}\right) \\[4pt] &=(n_A+n_B)\left(\dfrac{RT}{V}\right) \label{10.6.3} \end{align} \]
More generally, for a mixture of \(n\) component gases, the total pressure is given by
\[\begin{align} P_{tot} &=(P_1+P_2+P_3+ \; \cdots +P_n)\left(\dfrac{RT}{V}\right)\label{10.6.2a} \\[4pt] &=\sum_{i=1}^n{P_i}\left(\dfrac{RT}{V}\right)\label{10.6.2b} \end{align} \]
Equation \(\ref{10.6.2b}\) restates Equation \(\ref{10.6.3}\) in a more general form and makes it explicitly clear that, at constant temperature and volume, the pressure exerted by a gas depends on only the total number of moles of gas present, whether the gas is a single chemical species or a mixture of dozens or even hundreds of gaseous species. For Equation \(\ref{10.6.2b}\) to be valid, the identity of the particles present cannot have an effect. Thus an ideal gas must be one whose properties are not affected by either the size of the particles or their intermolecular interactions because both will vary from one gas to another. The calculation of total and partial pressures for mixtures of gases is illustrated in Example \(\PageIndex{1}\).
Example \(\PageIndex{1}\): The Bends
Deep-sea divers must use special gas mixtures in their tanks, rather than compressed air, to avoid serious problems, most notably a condition called “the bends.” At depths of about 350 ft, divers are subject to a pressure of approximately 10 atm. A typical gas cylinder used for such depths contains 51.2 g of \(O_2\) and 326.4 g of He and has a volume of 10.0 L. What is the partial pressure of each gas at 20.00°C, and what is the total pressure in the cylinder at this temperature?
Given: masses of components, total volume, and temperature
Asked for: partial pressures and total pressure
Strategy:
- Calculate the number of moles of \(\ce{He}\) and \(\ce{O_2}\) present.
- Use the ideal gas law to calculate the partial pressure of each gas. Then add together the partial pressures to obtain the total pressure of the gaseous mixture.
Solution:
A The number of moles of \(\ce{He}\) is
\[n_{\rm He}=\rm\dfrac{326.4\;g}{4.003\;g/mol}=81.54\;mol \nonumber \]
The number of moles of \(\ce{O_2}\) is
\[n_{\rm O_2}=\rm \dfrac{51.2\;g}{32.00\;g/mol}=1.60\;mol \nonumber \]
B We can now use the ideal gas law to calculate the partial pressure of each:
\[P_{\rm He}=\dfrac{n_{\rm He}RT}{V}=\rm\dfrac{81.54\;mol\times0.08206\;\dfrac{atm\cdot L}{mol\cdot K}\times293.15\;K}{10.0\;L}=196.2\;atm \nonumber \]
\[P_{\rm O_2}=\dfrac{n_{\rm O_2} RT}{V}=\rm\dfrac{1.60\;mol\times0.08206\;\dfrac{atm\cdot L}{mol\cdot K}\times293.15\;K}{10.0\;L}=3.85\;atm \nonumber \]
The total pressure is the sum of the two partial pressures:
\[P_{\rm tot}=P_{\rm He}+P_{\rm O_2}=\rm(196.2+3.85)\;atm=200.1\;atm \nonumber \]
Exercise \(\PageIndex{1}\)
A cylinder of compressed natural gas has a volume of 20.0 L and contains 1813 g of methane and 336 g of ethane. Calculate the partial pressure of each gas at 22.0°C and the total pressure in the cylinder.
- Answer
-
\(P_{CH_4}=137 \; atm\); \(P_{C_2H_6}=13.4\; atm\); \(P_{tot}=151\; atm\)
Mole Fractions of Gas Mixtures
The composition of a gas mixture can be described by the mole fractions of the gases present. The mole fraction (\(\chi\)) of any component of a mixture is the ratio of the number of moles of that component to the total number of moles of all the species present in the mixture (\(n_{tot}\)):
\[\chi_A=\dfrac{\text{moles of A}}{\text{total moles}}= \dfrac{n_A}{n_{tot}} =\dfrac{n_A}{n_A+n_B+\cdots}\label{10.6.5} \]
The mole fraction is a dimensionless quantity between 0 and 1. If \(\chi_A = 1.0\), then the sample is pure \(A\), not a mixture. If \(\chi_A = 0\), then no \(A\) is present in the mixture. The sum of the mole fractions of all the components present must equal 1.
To see how mole fractions can help us understand the properties of gas mixtures, let’s evaluate the ratio of the pressure of a gas \(A\) to the total pressure of a gas mixture that contains \(A\). We can use the ideal gas law to describe the pressures of both gas \(A\) and the mixture: \(P_A = n_ART/V\) and \(P_{tot} = n_tRT/V\). The ratio of the two is thus
\[\dfrac{P_A}{P_{tot}}=\dfrac{n_ART/V}{n_{tot}RT/V} = \dfrac{n_A}{n_{tot}}=\chi_A \label{10.6.6} \]
Rearranging this equation gives
\[P_A = \chi_AP_{tot} \label{10.6.7} \]
That is, the partial pressure of any gas in a mixture is the total pressure multiplied by the mole fraction of that gas. This conclusion is a direct result of the ideal gas law, which assumes that all gas particles behave ideally. Consequently, the pressure of a gas in a mixture depends on only the percentage of particles in the mixture that are of that type, not their specific physical or chemical properties. By volume, Earth’s atmosphere is about 78% \(N_2\), 21% \(O_2\), and 0.9% \(Ar\), with trace amounts of gases such as \(CO_2\), \(H_2O\), and others. This means that 78% of the particles present in the atmosphere are \(N_2\); hence the mole fraction of \(N_2\) is 78%/100% = 0.78. Similarly, the mole fractions of \(O_2\) and \(Ar\) are 0.21 and 0.009, respectively. Using Equation \ref{10.6.7}, we therefore know that the partial pressure of N2 is 0.78 atm (assuming an atmospheric pressure of exactly 760 mmHg) and, similarly, the partial pressures of \(O_2\) and \(Ar\) are 0.21 and 0.009 atm, respectively.
Example \(\PageIndex{2}\): Exhaling Composition
We have just calculated the partial pressures of the major gases in the air we inhale. Experiments that measure the composition of the air we exhale yield different results, however. The following table gives the measured pressures of the major gases in both inhaled and exhaled air. Calculate the mole fractions of the gases in exhaled air.
| Inhaled Air / mmHg | Exhaled Air / mmHg | |
|---|---|---|
| \(P_{\rm N_2}\) | 597 | 568 |
| \(P_{\rm O_2}\) | 158 | 116 |
| \(P_{\rm H_2O}\) | 0.3 | 28 |
| \(P_{\rm CO_2}\) | 5 | 48 |
| \(P_{\rm Ar}\) | 8 | 8 |
| \(P_{tot}\) | 767 | 767 |
Given: pressures of gases in inhaled and exhaled air
Asked for: mole fractions of gases in exhaled air
Strategy:
Calculate the mole fraction of each gas using Equation \(\ref{10.6.7}\).
Solution:
The mole fraction of any gas \(A\) is given by
\[\chi_A=\dfrac{P_A}{P_{tot}} \nonumber \]
where \(P_A\) is the partial pressure of \(A\) and \(P_{tot}\) is the total pressure. For example, the mole fraction of \(\ce{CO_2}\) is given as:
\[\chi_{\rm CO_2}=\rm\dfrac{48\;mmHg}{767\;mmHg}=0.063 \nonumber \]
The following table gives the values of \(\chi_A\) for the gases in the exhaled air.
| Gas | Mole Fraction |
|---|---|
| \({\rm N_2}\) | 0.741 |
| \({\rm O_2}\) | 0.151 |
| \({\rm H_2O}\) | 0.037 |
| \({\rm CO_2}\) | 0.063 |
| \({\rm Ar}\) | 0.010 |
Exercise \(\PageIndex{2}\)
Venus is an inhospitable place, with a surface temperature of 560°C and a surface pressure of 90 atm. The atmosphere consists of about 96% CO2 and 3% N2, with trace amounts of other gases, including water, sulfur dioxide, and sulfuric acid. Calculate the partial pressures of CO2 and N2.
- Answer
-
\[P_{\rm CO_2}=\rm86\; atm \nonumber \]
\[P_{\rm N_2}=\rm2.7\;atm \nonumber \]
(Video) Gas mixtures and partial pressures 10.6
Summary
The partial pressure of each gas in a mixture is proportional to its mole fraction. The pressure exerted by each gas in a gas mixture (its partial pressure) is independent of the pressure exerted by all other gases present. Consequently, the total pressure exerted by a mixture of gases is the sum of the partial pressures of the components (Dalton’s law of partial pressures). The amount of gas present in a mixture may be described by its partial pressure or its mole fraction. The mole fraction of any component of a mixture is the ratio of the number of moles of that substance to the total number of moles of all substances present. In a mixture of gases, the partial pressure of each gas is the product of the total pressure and the mole fraction of that gas.
FAQs
How do you find the partial pressure of a gas mixture? ›
There are two ways to calculate partial pressures: 1)Use PV = nRT to calculate the individual pressure of each gas in a mixture. 2)Use the mole fraction of each gas to calculate the percentage of pressure from the total pressure assignable to each individual gas.
What is the partial pressure of a mixture of 10.0g of Ne and 10.0g of AR have a total pressure of 1.6 atm? ›Question: A mixture of 10.0g of Ne and 10 g Ar have a total pressure of 1.6atm. What is the partial pressure of Ne? the answer 1.1 atm.
What pressure would a gas mixture in a 10.0 L tank exert? ›Calculate the pressure: P ≈ 46.6 atm So, the pressure exerted by the gas mixture in the 10.0 L tank would be approximately 46.6 atm.
What is partial pressure formula? ›As has been mentioned in the lesson, partial pressure can be calculated as follows: P(gas 1) = x(gas 1) * P(Total); where x(gas 1) = no of moles(gas 1)/ no of moles(total). As you can see the above formulae does not require the individual volumes of the gases or the total volume.
What is the total pressure in a 10.0 L cylinder which contains 0.4 g? ›P=0. 4926 atm.
How is the partial pressure of a gas in a mixture calculated how is the rate of effusion of a gas calculated? ›The partial pressure of a gas in a mixture can be calculated by multiplying its mole fraction by the total pressure. The pressure of a gas that has been collected by water displacement is determined by subtracting the vapor pressure of water at that temperature from the total pressure of the sample.
What is the total pressure in a mixture of 8g of di oxygen and 4g of di hydrogen? ›Hence, the total pressure of the mixture is 56.025 bar.
What is the volume of gas at pressure 21 into 10 raise to 4? ›The volume of a gas at pressure 21×104N/m2 and temperature 27∘C is 83 litres.
What does the pressure of 5.0 L of gas increase? ›The pressure of 5.0 L of gas increases from 1.50 atm to 1240 mmHg.
What is the pressure of a gas when its volume is 750ml? ›Expert-Verified Answer
P2 = ? Thus the pressure is 1.33 atm.
What is the difference between partial pressure and total pressure? ›
Partial pressure is the component of total pressure associated with a specific gas species, while the total pressure is the sum of partial pressures for all gas species contributing in a particular location where the pressure is measured.
What is the relationship between total pressure and partial pressure? ›Dalton's law of partial pressures is a gas law which states that the total pressure exerted by a mixture of gases is equal to the sum of the partial pressures exerted by each individual gas in the mixture.
How to calculate pressure? ›Since pressure is defined as the force per unit area, its formula is expressed as P = F/A, where P is pressure, F is force, and A is the area by which the force is applied perpendicularly.
What is partial pressure for dummies? ›In a mixture of gases, each gas contributes to the total pressure of the mixture. This contribution is the partial pressure. The partial pressure is the pressure the gas if the gas were in the same volume and temperature by itself.
What is partial pressure ratio? ›Dalton's law of partial pressures states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of the individual gases. The ratio of partial pressure of a gas to the total pressure is equal to the mole fraction of the gas.
What is the value of partial pressure? ›At sea level without supplemented inspired oxygenation, the alveolar oxygen partial pressure (PAO2) is: PAO2 = (760 - 47) 0.21 - 40 / 0.8 = 99.7 mm Hg.
How do you answer partial pressure? ›Partial pressure of each gas (Pi) depends on the mole fraction (Xi) of each gas and the total pressure of the gas mixture, Pi = Xi Ptotal where Xi = ni / ntotal, ni is the number of moles of gas i and ntotal is the total number of moles of all the gases in the mixture.
Is partial pressure a percentage? ›The concept of partial pressure comes from the fact that each specific gas contributes a part of the total pressure and that part is the partial pressure of that gas. It is really pretty much like taking a percentage or fraction of the total to describe all the parts.
How do you find the final pressure of two gases? ›The final volume is the sum of the initial volumes. If the gases are ideal, the final pressure is the same as the initial pressure, and the partial pressures are PA=nART/(VA+VB) and PB=nBRT/(VA+VB).
What is the pressure in a 20.0 L cylinder? ›A 20.0-L volume of an ideal gas in a cylinder with a piston is at a pressure of 3.0 atm.
Which gas cylinder has a pressure of 14.9 atmosphere? ›
L.P.G. cylinder can withstand a pressure of 14.9 atmosphere. The pressure gauge of the cylinder indicates 12 atmosphere at 27∘C Due to a sudden fire in the building the temperature rises.
What is the volume of 10.0 g of oxygen gas at STP? ›Answer and Explanation: Let's assume that oxygen gas behaves ideally at STP. Hence, the volume of 10.0 grams of oxygen gas at STP will be 6.99 L.
What is an example of a gas gas mixture? ›- Gas-Gas mixture: Two examples are air and perfumes, helium and hydrogen present in air balloons.
What is real gas mixture partial pressure? ›The partial pressure pi of component i in a real gaseous mixture is the pressure exerted by molecules i, i.e., the contribution of the molecules i to the force per unit area that the molecules of the real gas mixture exert on the wall.
What is the total pressure of a gaseous mixture of 2.8 g? ›The total pressure of a gaseous mixture of 2.8gN2, 3.2gO2, and 0.5gH2 is 4.5atm.
What is the total pressure of a mixture of 6.4 grams of oxygen and 5.6 grams of nitrogen? ›⇒12×1200⇒600 mm. Q. The total pressure of a mixture of 6.4 grams of oxygen and 5.6 grams of nitrogen present in a 2 litre vessel is 1200 mm.
What will be the pressure exerted by a mixture of 3.2 g of methane and 4.4 g of carbon dioxide contained in a 9 dm3 flask at 270c? ›Hence, the total pressure exerted by the mixture is 8.314×104Pa.
What is 10 l of gas at STP? ›At STP, the temperature is 273 K or 0 ∘ C and pressure is 1 atm. We may calculate the moles of gas in a 10 liter sample as, 22.4 L = 1 mol 1 L = 1 22.4 mol 10 L = 10 × 1 22.4 mol = 0.45 moles of gas.
What will happen if volume of gas increased by 10 percent? ›Volume is inversely proposed to pressure based on the boyle's law. So, to increase the volume of a gas by 10%, the pressure of the gas should be decreased by 10%.
What is the volume of a gas at 27 and 760mm pressure? ›The volume of a gas at 27^oC and 760mm Hg is 500cc .
Will increasing pipe size increase gas pressure? ›
The flow of fluid through an enlargement (increase in pipe diameter) results in a decrease in velocity and consequently, a pressure rise.
What if gas pressure is too high? ›Gas Pressure
If the pressure is too high, your furnace can overheat, and internal components can get damaged. If the pressure is too low, your furnace's efficiency will decrease, and the combustion process will create more condensation inside your furnace's combustion chamber.
Typical high pressures cylinders range from 2,200psi – 2,600psi. 3,000psi and 6,000psi cylinders are also available for specific applications.
What is the pressure of a gas mixture in a 10.0 L tank if it were composed of 48.5 g He and 94.6 g co2 at 398 K? ›Instant Text Answer
Calculate the pressure: P ≈ 46.6 atm So, the pressure exerted by the gas mixture in the 10.0 L tank would be approximately 46.6 atm.
P2 = P1V1V2 = 0.860⋅atm×11.2⋅L 15.0⋅L ≅0.6⋅atm .
How do you convert gas pressure to volume? ›For example, if you want to calculate the volume of 40 moles of a gas under a pressure of 1013 hPa and at a temperature of 250 K, the result will be equal to: V = nRT/p = 40 × 8.31446261815324 × 250 / 101300 = 0.82 m³ .
What are examples for partial pressure? ›This general property of gases is also true in chemical reactions of gases in biology. For example, the necessary amount of oxygen for human respiration, and the amount that is toxic, is set by the partial pressure of oxygen alone.
What is the formula for gas pressure? ›The equations describing these laws are special cases of the ideal gas law, PV = nRT, where P is the pressure of the gas, V is its volume, n is the number of moles of the gas, T is its kelvin temperature, and R is the ideal (universal) gas constant.
What's the difference between total versus partial pressures of a gas mixture? ›The pressure exerted by each gas in a gas mixture (its partial pressure) is independent of the pressure exerted by all other gases present. Consequently, the total pressure exerted by a mixture of gases is the sum of the partial pressures of the components (Dalton's law of partial pressures).
What does partial pressure depend on? ›The partial pressure of an individual gas is equal to the total pressure multiplied by the mole fraction of that gas. Because it is dependent solely on the number of particles and not the identity of the gas, the Ideal Gas Equation applies just as well to mixtures of gases as it does to pure gases.
What happens when partial pressure increases? ›
1) When the partial pressure of any of the gaseous reactants or of the products is increased, the position of equilibrium is shifted so as to decrease its partial pressure. This is usually achieved by favoring the reaction in which there is decrease in the number of moles of gaseous components.
What are 5 examples of pressure? ›- IN SCHOOLS :- there is a very heavy bag . ...
- Through breathing.
- Aircraft fly because of the air pressure on the wings.
- Toy balloon inflates because of air pressure inside.
- Bullet fired from a gun is driven by gas pressure .
Solution. First, calculate the partial pressure for H2O by subtracting the partial pressure of H2 from the total pressure. Then, write K (equilibrium constant expression) in terms of activities. Remember that solids and pure liquids are ignored.
What is the partial pressure of each component of the mixture of gases? ›According to Dalton's law of partial pressures, the total pressure by a mixture of gases is equal to the sum of the partial pressures of each of the constituent gases. The partial pressure is defined as the pressure each gas would exert if it alone occupied the volume of the mixture at the same temperature.
How is partial pressure of a gas in a mixture calculated how is the rate effusion of a gas calculated? ›The partial pressure of a gas in a mixture can be calculated by multiplying its mole fraction by the total pressure. The pressure of a gas that has been collected by water displacement is determined by subtracting the vapor pressure of water at that temperature from the total pressure of the sample.
What is the partial pressure of O2 at equilibrium? ›0.2 atm will be the partial pressure of O2.
What is the total pressure of a mixture of two gases each? ›According to Dalton's law of partial pressure, the total pressure of a mixture of two gases is the sum of the partial pressure.
What is an example of a gas mixture? ›- Gas-Gas mixture: Two examples are air and perfumes, helium and hydrogen present in air balloons.
What is the partial pressure of n2 in the mixture? ›N2 gas is 78% of the air we breath which is on average 1 atm total pressure. So the partial pressure of N2 of air at 1 atm pressure is 0.78 atm.
What is ideal gas mixture pressure? ›For a mixture of ideal gases, the total pressure exerted by the mixture equals the sum of the pressures that each gas would exert on its own. This observation, known as Dalton's law of partial pressures, can be written as follows: P (total) = P ₁ + P ₂ + P ₃ + ...
What is the difference between gas pressure and partial pressure? ›
Answer: Partial pressure is the pressure exerted by an object, whereas vapour pressure is the pressure exerted by a gas.
How does partial pressure affect gas? ›The greater the partial pressure of a gas, the more of that gas will dissolve in a liquid, as the gas moves toward equilibrium. Gas molecules move down a pressure gradient; in other words, gas moves from a region of high pressure to a region of low pressure.
What is the relationship between partial pressure and total pressure? ›Dalton's law of partial pressures is a gas law which states that the total pressure exerted by a mixture of gases is equal to the sum of the partial pressures exerted by each individual gas in the mixture.
What is the pressure of a gas? ›The pressure of a gas is then a measure of the average linear momentum of the moving molecules of a gas. The pressure acts perpendicular (normal) to the wall; the tangential (shear) component of the force is related to the viscosity of the gas. Let us look at a static gas; one that does not appear to move or flow.