CHEM0821 – Elementary Physical Chemistry . Dr Daniel Stone d.stone@leeds.ac.uk Unit 1.3 Dalton’s Law of Partial Pressures . In an inert (unreactive) mixture of ideal gases, each gas behaves as if it were the only gas present. In air, which is a mixture of ~4/5 N2 and ~1/5 O2 by volume, ~4/5 of the total air pressure is due to N2 and ~1/5 of the total air pressure is due to O2. The contribution each gas makes to the total pressure is called its partial pressure. The partial pressure of a gas is the pressure it would exert if it alone occupied its container. It is measured in units of pressure. Dalton’s Law of Partial Pressures states that in a mixture of gases which do not react chemically the total pressure is the sum of the partial pressures of the components. For a mixture of gases A and B, we can write this as: Partial Pressures and Mole Fraction . The partial pressure of a gas in a mixture depends on the total pressure and on the mole fraction of that gas. Mole fractions are a way to express the composition of a mixture. The mole fraction of a component in a mixture represents the number of moles of a given component divided by the total number of moles of the mixture. Mole fraction is a unitless (dimensionless) quantity. For a mixture of gases A and B we can define the mole fractions by: From Avogadro’s Hypothesis, it follows that: Note that the sum of mole fractions for all species must be equal to 1. Example . A container holds 2 mol of N2, 0.2 mol of O2 and 5 mol of H2 at a total pressure of 5 atm. What are the mole fractions and partial pressures of each gas? Diffusion . During our consideration of Dalton’s Law of Partial Pressures we’ve made an underlying assumption that the composition of the gas mixture is uniform throughout – i.e. that we have a homogeneous (well-mixed) mixture. How does this mixing occur? Diffusion is the process by which particles (i.e. atoms and molecules) mix with one another due to the random motion of the particles. In gases, diffusion is rapid owing to the rapid movement of the gas molecules. Although the movement of individual gas molecules is random, there is a net movement of molecules from a region of high concentration to regions of low concentration (i.e. down a concentration gradient) which ultimately leads to a homogeneous mixture. The rate of diffusion increases with increasing molecular speed. Diffusion also takes places in liquids, again leading to homogeneous mixtures, and also occurs (much more slowly) in solids. Diffusion allows gases to pass through porous substances or through small apertures and enables gas exchange in the lungs, uptake of small molecules into cells, and is the basis for processes such as kidney dialysis. Brownian Motion . Brownian motion is the random motion of particles (such as dust, aerosols or pollen grains) suspended in a fluid (gas or liquid), resulting from collisions with the fast moving molecules in the fluid. It was first observed by the botanist Robert Brown in 1827. Brown noticed that pollen grains suspended in water appeared to move around in the water in a random way when viewed under a microscope. Albert Einstein published a paper in 1905 that explained the observed Brownian motion in detail, demonstrating that the pollen grains were being moved as a result of collisions with individual water molecules. This work served as convincing evidence for the existence of atoms and molecules. The collisions and resulting random movement of the molecules also give rise to diffusion. Einstein’s 1905 paper on Brownian motion is his most cited paper! More than for his papers on special relativity or his work on photons. Effusion . Effusion is the movement of a gas through a very small aperture from a region of high pressure to a region of low pressure. For effusion to occur, individual molecules pass through the aperture without colliding with any other molecules. The diameter of the aperture must therefore be smaller than the mean distance travelled a molecule travels before colliding with another molecule (the mean free path) for effusion to occur. If the diameter of the aperture is larger than the average distance between molecular collisions, then movement of the gas through the aperture will occur by diffusion. The rates of diffusion and effusion increase with increasing temperature, since both depend on the motion of molecules, and the molecular speed depends on temperature. Graham’s Law of Effusion . The rate at which a gas effuses through an aperture is dependent on its molecular speed. A gas that is moving faster is more likely to effuse through an aperture than a slower moving gas. At a constant temperature and pressure, the molecular speed, and thus the rate of effusion, of a gas is inversely proportional to the square root of its molecular (or molar) mass. This can be extended to consider the relative rates of two gases and summarised by Graham’s Law of Effusion, which states that at constant temperature and pressure the relative rate at which two gases effuse is equal to the square root of the inverse ratio of their molecular (or molar) masses. Example . Place the following gases in order of increasing rate of effusion: SO 2, O2, CO2, N2, H2 What is the ratio of the effusion rates of O2 and CO2? The Kinetic Theory of Gases . Previously we saw that a law is a statement of experimental results but doesn’t necessarily provide an explanation for observations. - A law is a concise statement of experimental results, it need not include an explanation of them. - A hypothesis is an explanation of experimental findings by means of a concept or model. - A well-established hypothesis becomes known as a theory. We have now looked at several gas laws, and in some cases we have started to consider some explanations of the observations. We will now look in more detail at the kinetic theory of gases which attempts to explain the behaviours and properties of gases described by the gas laws. The kinetic theory of gases is based upon three main assumptions. We will consider each one in turn. The Kinetic Theory of Gases – Assumption 1 . Gas molecules are in constant random motion. - Molecules move in straight lines until they collide with each other or the walls of the container. - Except during collisions, the molecules do not interact (i.e. there are no intermolecular bonds or forces between molecules). - Collisions with the walls of the container exert pressure. The Kinetic Theory of Gases – Assumption 2 . All collisions are perfectly elastic. - Collisions between molecules result in no loss of kinetic energy. - Kinetic energy may be transferred from one molecule to another on collision, but it cannot be transformed to other forms of energy such as heat or sound. The Kinetic Theory of Gases – Assumption 3 . The volume of the gas molecules themselves is negligible compared to the volume in which the gas is contained. - A gas consists of very small particles (i.e. atoms or molecules), each with non-zero mass. - The total volume of the individual gas molecules is negligible compared to the volume of the container. - The distance between molecules is many times the diameter of a molecule. - Most of the ‘space’ in a gas is empty. Unit 1.3 Summary . Partial pressure – The contribution each gas in a mixture makes to the total pressure. Dalton’s Law of Partial Pressures – In a mixture of gases which do not react chemically the total pressure is the sum of the partial pressures of the components. Mole fraction – The number of moles of a component in a mixture divided by the total number of moles present. Diffusion – The process by which particles mix with one another due to the random motion of the particles. The rate of diffusion increases with increasing molecular speed. Effusion – The movement of a gas through a very small aperture from a region of high pressure to a region of low pressure. The rate of effusion increases with increasing molecular speed. Graham’s Law of Effusion – At constant temperature and pressure, the relative rate at which two gases effuse is equal to the square root of the inverse ratio of their molecular (or molar) masses. Kinetic Theory of Gases – Explains the behaviours and properties of gases described by the gas laws. Assumes that gas molecules are in constant random motion, that all collisions are perfectly elastic, and that the volume of the gas molecules themselves is negligible compared to the volume in which the gas is contained.
