Dalton's Law of Partial Pressures

Dalton's law of partial pressures states that the total pressure exerted by a mixture of gases is the sum of partial pressure of each individual gas present. Each gas is assumed to be an ideal gas.

P_total = P₁ + P₂ + P₃ + . .

Where P₁ and P₂ are the partial pressure of gas 1and gas 2 in the mixture. Since each gas behaves independently, the ideal gas law can be used to calculate the pressure of that gas if we know the number of moles of the gas, the total volume of the container, and the temperature of the gas.

Each gas exerts the same pressure they would exert if they were in the container alone. For example, if a mixture of gases at 298 K in a 1.00 L container consists of 1.00 g of H₂, 1.50 g of N₂, and 2.00 g of Br₂. The partial pressure of each gas can be written

The total pressure can be expressed as

The total pressure in this example is 13.8 atm.

Note the following observations about gas mixtures are important to remember.

Each gas occupies the entire volume of the container.

The gases will mix homogeneously.

The gases should not react (no chemical reaction should occur between the gases in the mixture).

The type of gas has no bearing on the partial pressure of the gas.

How can two or more gases in a closed container each occupy the entire volume of its container?

Answer: Ideal gases are considered to be point particles, the volume of each gas molecule is considered so small, it is not important. Ideal gases do not attract or repel other gases, therefore the intermolecular forces among the gase molecules is not important. Because gases consist mostly of empty space, the distance between gas molecules is large enough so that each gas molecule does not "see" another gas molecule. This is another reason why there are no interactions with other particles.

The partial pressure of a gas can also be expressed in terms of the mole fraction of each gas.

There is a relationship between the partial pressure of a gas, the total pressure of the system and the mole fraction of the gas.

Collecting a Gas Over Water

Applications of Dalton's Law: Collecting a gas over water

Let's say you want to generate H₂(g) by reacting zinc metal with hydrochloric acid

Zn(s) + 2HCl(aq) ---> ZnCl₂(aq) + H₂(g)

We can set up an apparatus to collect the gas.

When you collect a gas over water, you fill a test tube with water and invert it under the water so that the entire tube stays filled with water when it is held upright. The gas you are collecting is bubbled into the test tube and the gas displaces the water.

Here the pressure of the gases inside the test tube is less than atmospheric pressure.

When the level of the water inside the test tube equals the level of the water in the container, then

P_atmospheric = P_{inside the test tube}

You can adjust the level of the gas inside the test tube with the level of the water in the contaner so that the pressure of the gas inside the test tube is equal to the atmospheric pressure.

If the gas level inside the test tube is more than the level of the water in the container, then the pressure inside the test tube is greater than atmosphereic pressure.

When a gas is collected over water, inside the test tube there will be some water vapor. What you collect is a mixture of the gas you produced and water vapor. The total pressure in the test tube is due to the partial pressure of the gas collected and the partial pressure of the water vapor.

P_total = P_gas + P_H2O

You can measure P_tot and can look up in a table the vapor pressure of water, P_wataer at the temperature of the water in the experiment.

Temp., °C	Vapor pressure of water, mmHg
5.0	6.543
10.0	9.209
15.0	12.79
20.0	17.54
35.0	42.18
50.0	92.15
80.0	355.1
100.0	760.0

The vapor pressure of water increases as the temperature of the water increases. At 100°C, the vapor pressure of water equals atmosphereic pressure.

If we collected 310.0 mL of gas over water at a pressure of 738 mmHg and a temperature of 20.0°C, what is the partial pressure of H₂(g) in the test tube?

P_total = P_H2 + P_H2O

P_total - P_H2O = P_H2

738 mmHg - 17.54 mmHg = 721 mm Hg = P_H2

721 mm Hg x 1.00 atm/760.0 mmHg = 0.949 atm = P_H2

How much zinc metal must have reacted with excess HCl(aq) to produce this much H₂(g)?

Kinetic Molecular Theory of Gases

1. Gases move in a continuous, random, straight-line, motion.

2. Volume of molecules are negligible compared to the volume of the entire container

3. Collisions between molecules are elastic

4. The average kinetic energy of any gas is proportional to the temperature (in units of Kelvin).

5. No attractive forces exist between gas molecules

The average kinetic energy of a gas is proportional to the temperature. So that if we compare two gases, say He and NH₃, at the same temperature, each gas will have the same average KE.

av KE_gasA = av KE_gasB

(1/2)m_Av_A² = (1/2)m_Bv_B²

If we compare He and NH₃, m_He is less than the m_NH3

therefore the average speed of the He molecules must be greater than the average speed of the NH₃ molecules.

In order for the av KE of each gas to be equal, the lighter HE molecules travel faster (more collisions), yet they hit the walls of the container with a lighter force than the NH₃ molecules. The NH₃ molecules are traveling a bit slower (not as many collisions), but they hit the walls of the container with more force - the two factors of speed and mass of each gas balance each other.

We need to discuss the fact that at a specific temperature, all of the gas molecules do not have the same speed. In fact, there is a distribution of speeds.

As the temperature is increased the average speed of a sample of O₂ molecules increases. There is a difference bewteen average speed and root-mean-square speed.

Note that <v²> does not equal <v>²!!!

Use the root mean square speed! ---> v_rms = (<v²>)^(.5) (an exponent to one-half is the same operation as a square root radical).

Also that at a specific temperature, different gases have different average speeds.

Consider a mixture of Helium and Neon at some temperature, T.

This says that if we compare the average speed of two gases at the same temperature, the av KE is the same. The only way for this to be true since the masses are different is for the light molecules to be moving at a faster average speed than the heavier molecules. If an increase in temperature occurs, the number of molecules stays constant, the molecules go faster, the average kinetic energy increases.

Graham's Law of Effusion

Graham's Law of Effusion states that the rate of effusion of a gas is inversely proportional to the square root of its molecular weight.

Effusion refers to the movement of gas molecules from an enclosed space into a evacuated space. The key concept to underlying Graham's Law is the following. The faster the speed of a molecule, the faster it will effuse. When comparing two different gases at the same temperature, on average, a lighter gas molecule moves faster than a heavier gas molecule.

We can define the kinetic energy of a single gas moelcule as

KE = (1/2)mv²_molecule

At a given temperature, different gases have the same average kinetic energy.

average KE_gasA = average KE_gasB

(1/2)m_Au_A² = (1/2)m_Bu_B²

Since the mass of gas A is different than the mass of Gas B, in order for this relationship to be true, the speed of the lighter molecule must be greater than the speed of the heavier gas molecule. In this case, the speed of the gas molecules is the root-mean-square speed of the molecules.

Does it makes sense that rate of effusion is inversly proportional to the molecular weight of the gas molecule? We can compare the rate of effusion of two gases, at the same temperature, from a contanier with a small hole in it into an evacuated space.

Which molecule, the lighter ones or the heavier ones, will effuse at a faster rate?

(animation of this process). Comparing two gases at the same pressure and temperature, lighter molecules effuse faster than heavier molecules.

Diffusion refers to the movement of gas molecules (Gas A) from a space where the gas A molecule are concentrated to a space or through a space in which there other gas molecules (Gas B) are present (a less concentrated space for Gas A).

(animation of this process).

Comparing two gases at the same pressure and temperature, lighter molecules diffuse faster than heavier molecules.

The rate of effusion can be expressed in units of cm/sec. Compare the rate of effusion of hydrogen gas to oxygen gas. Hydrogen gas is lighter than oxygen gas therefore we should expect H₂ to effuse at a faster rate than O₂. Assuming equal temperature and pressure of the two gases, we can set up a ratio of the rates of effusion.

The rate of effusion is directly proportional to therms speed of the molecules.

The rate of effusion of hydrogen gas is 4 times as fast as the rate of effusion for oxygen gas (compared at the same temperature).

We can determine the molecular weight of a gas using Graham's Law of Effusion.

A sample of HI gas (MW = 128) effuses at 0.0962 cm/sec. A sample of butylamine gas effuses at 0.126 cm/sec. What is the molecular weight of butylamine?

Real World Application

Uranium for the Manhattan Project was purified in Oak Ridge, Tennessee using Graham's Law of Effusion. UF₆ consists of a mixture of ²³⁵UF₆ and ²³⁸UF₆. Only pure ²³⁵U could be used for the atomic bomb. But, ²³⁵U is a small fraction of the U in UF₆.

²³⁵U represents 0.7% of uranium isotopes

²³⁸U represents 99.3% of uranium isotopes

To obtain pure ²³⁵U, UF₆ gas was sent through a series of gas effusion tubes.

The mass of ²³⁵UF₆ < ²³⁸UF₆ but the v_rms of ²³⁵UF₆ > ²³⁸UF₆ . The ²³⁵UF₆ effuses faster than ²³⁸UF₆ , therefore the ²³⁵UF₆ was separated from the ²³⁸UF₆.