Units of Chapter 17 Ideal Gases Kinetic Theory Solids and Elastic Deformation Phase Equilibrium and Evaporation Latent Heats Phase Changes and Energy Conservation
17-1 Ideal Gases Gases are the easiest state of matter to describe, as all ideal gases exhibit similar behavior. An ideal gas is one that is thin enough, and far away enough from condensing, that the interactions between molecules can be ignored.
If the volume of an ideal gas is held constant, we find that the pressure increases with temperature: 17-1 Ideal Gases
If the volume and temperature are kept constant, but more gas is added (such as in inflating a tire or basketball), the pressure will increase: 17-1 Ideal Gases
Finally, if the temperature is constant and the volume decreases, the pressure increases: 17-1 Ideal Gases
Combining all three observations, we write where k is called the Boltzmann constant:
17-1 Ideal Gases Rearranging gives us the equation of state for an ideal gas: Instead of counting molecules, we can count moles. A mole is the amount of a substance that contains as many elementary entities as there are atoms in 12 g of carbon-12.
17-1 Ideal Gases Experimentally, the number of entities (atoms or molecules) in a mole is given by Avogadros number: Therefore, n moles of gas will contain molecules.
17-1 Ideal Gases Avogadros number and the Boltzmann constant can be combined to form the universal gas constant and an alternative equation of state:
17-1 Ideal Gases The atomic or molecular mass of a substance is the mass, in grams, of one mole of that substance. For example, Helium: Copper: Furthermore, the mass of an individual atom is given by the atomic mass divided by Avogadros number:
17-1 Ideal Gases Boyles law, which is consistent with the ideal gas law, says that the pressure varies inversely with volume. These curves of constant temperature are called isotherms.
17-1 Ideal Gases Charless law, also consistent with the ideal gas law, says that the volume of a gas increases with temperature if the pressure is constant.
17-1 Ideal Gases In this photograph, the balloon was inflated at room temperature and cooled with liquid nitrogen. The decrease in volume of the air in the balloon is obvious.
17-2 Kinetic Theory The kinetic theory relates microscopic quantities (position, velocity) to macroscopic ones (pressure, temperature). Assumptions: N identical molecules of mass m are inside a container of volume V ; each acts as a point particle. Molecules move randomly and always obey Newtons laws. Collisions with other molecules and with the walls are elastic.
17-2 Kinetic Theory Pressure is the result of collisions between the gas molecules and the walls of the container. It depends on the mass and speed of the molecules, and on the container size:
17-2 Kinetic Theory Not all molecules in a gas will have the same speed; their speeds are represented by the Maxwell distribution, and depend on the temperature and mass of the molecules.
17-2 Kinetic Theory We replace the speed in the previous expression for pressure with the average speed: Including the other two directions, Therefore, the pressure in a gas is proportional to the average kinetic energy of its molecules.
17-2 Kinetic Theory Comparing this expression with the ideal gas law allows us to relate average kinetic energy and temperature: The square root of is called the root mean square (rms) speed.
17-2 Kinetic Theory Solving for the rms speed gives:
17-2 Kinetic Theory The rms speed is slightly greater than the most probable speed and the average speed.
17-2 Kinetic Theory The internal energy of an ideal gas is the sum of the kinetic energies of all its molecules. In the case where each molecule consists of a single atom, this may be written:
17-3 Solids and Elastic Deformation Solids have definite shapes (unlike fluids), but they can be deformed. Pulling on opposite ends of a rod can cause it to stretch:
17-3 Solids and Elastic Deformation The amount of stretching will depend on the force; Y is Youngs modulus and is a property of the material:
17-3 Solids and Elastic Deformation Another type of deformation is called a shear deformation, where opposite sides of the object are pulled laterally in opposite directions.
17-3 Solids and Elastic Deformation As expected, the deformation is proportional to the force. S is the shear modulus.
17-3 Solids and Elastic Deformation Finally, if a solid is uniformly compressed, it will shrink.
17-3 Solids and Elastic Deformation Here, the proportionality constant, B, is called the bulk modulus.
17-3 Solids and Elastic Deformation The applied force per unit area is called the stress, and the resulting deformation is the strain. They are proportional to each other until the stress becomes too large; permanent deformation will then occur.
17-4 Phase Equilibrium and Evaporation If a liquid is put into a sealed container so that there is a vacuum above it, some of the molecules in the liquid will vaporize. Once a sufficient number have done so, some will begin to condense back into the liquid. Equilibrium is reached when the numbers remain constant.
17-4 Phase Equilibrium and Evaporation The pressure of the gas when it is in equilibrium with the liquid is called the equilibrium vapor pressure, and will depend on the temperature.
17-4 Phase Equilibrium and Evaporation The vaporization curve determines the boiling point of a liquid: A liquid boils at the temperature at which its vapor pressure equals the external pressure. This explains why water boils at a lower temperature at lower pressure – and why you should never insist on a 3-minute egg in Denver!
17-4 Phase Equilibrium and Evaporation This curve can be expanded. When the liquid reaches the critical point, there is no longer a distinction between liquid and gas; there is only a fluid phase.
17-4 Phase Equilibrium and Evaporation The fusion curve is the boundary between the solid and liquid phases; along that curve they exist in equilibrium with each other. Almost all materials have a fusion curve that resembles (a); water, due to its unusual properties near the freezing point, follows (b).
17-4 Phase Equilibrium and Evaporation Finally, the sublimation curve marks the boundary between the solid and gas phases. The triple point is where all three phases are in equilibrium. This is shown on the phase diagram below.
17-4 Phase Equilibrium and Evaporation A liquid in a closed container will come to equilibrium with its vapor. However, an open liquid will not, as its vapor keeps escaping – it will continue to vaporize without reaching equilibrium. As the molecules that escape from the liquid are the higher-energy ones, this has the effect of cooling the liquid. This is why sweating cools us off.
17-4 Phase Equilibrium and Evaporation If we look at the Maxwell speed distributions for water at different temperatures, we see that there is not much difference between the 30° C curve and the 100° C curve. This means that, if 100° C water molecules can escape, many 30° C molecules will also.
17-4 Phase Equilibrium and Evaporation This same evaporation process can cause a planet to lose its atmosphere – some molecules will have speeds exceeding the escape velocity. The evaporation process will be faster for lighter molecules and for less massive planets.
17-5 Latent Heats When two phases coexist, the temperature remains the same even if a small amount of heat is added. Instead of raising the temperature, the heat goes into changing the phase of the material – melting ice, for example.
17-5 Latent Heats The heat required to convert from one phase to another is called the latent heat. The latent heat, L, is the heat that must be added to or removed from one kilogram of a substance to convert it from one phase to another. During the conversion process, the temperature of the system remains constant.
17-5 Latent Heats The latent heat of fusion is the heat needed to go from solid to liquid; the latent heat of vaporization from liquid to gas.
17-6 Phase Changes and Energy Conservation Solving problems involving phase changes is similar to solving problems involving heat transfer, except that the latent heat must be included as well.
Summary of Chapter 17 An ideal gas is one in which interactions between molecules are ignored. Equation of state for an ideal gas: Boltzmanns constant: Universal gas constant: Equation of state again: Number of molecules in a mole is Avogadros number:
Summary of Chapter 17 Molecular mass: Boyles law: Charless law: Kinetic theory: gas consists of large number of pointlike molecules Pressure is a result of molecular collisions with container walls
Summary of Chapter 17 Molecules have a range of speeds, given by the Maxwell distribution Relation of kinetic energy to temperature: Relation of rms speed to temperature:
Summary of Chapter 17 Internal energy of monatomic gas: Force required to change the length of a solid: Force required to deform a solid:
Summary of Chapter 17 Pressure required to change the volume of a solid: Applied force per area: stress Resulting deformation: strain Deformation is elastic if object returns to its original size and shape when stress is removed
Summary of Chapter 17 Most common phases of matter: solid, liquid, gas When phases are in equilibrium, the number of molecules in each is constant Evaporation occurs when molecules in liquid move fast enough to escape into gas phase Latent heat: amount of heat required to transform from one phase to another Latent heat of fusion: melting or freezing
Summary of Chapter 17 Latent heat of vaporization: vaporizing or condensing Latent heat of sublimation: sublimation or condensation directly between gas and solid phases When heat is exchanged within a system isolated from its surroundings, the energy of the system is conserved