Chapter 17: Thermal Behavior of Matter Equations of state State variables and equations of state Variables that describes the state of the material are called state variables: pressure, volume, temperature, amount of substance The volume V of a substance is usually determined by its pressure p, temperature T, and amount of substance, described by the mass m In a few cases the relationship among p, V, T and m (or n) is simple enough to be expressed by an equation called the equation of state For complicated cases, we can use graphs or numerical tables.

Equations of state (cont’d) Ideal gas An ideal gas is a collection of atoms or molecules that move randomly and exert no long-range forces on each other. Each particle of the ideal gas is individually point-like, occupying a negligible volume. Low-density/low-pressure gases behave like ideal gases. Most gases at room temperature and atmospheric pressure can be approximately treated as ideal gases.

Equations of state (cont’d) Definition of a mole One mole (mol) of any substance is that amount of the substance that contains as many particles (atoms, or other particles) as there are atoms in 12 g of the isotope carbon-12 12C. This number is called Avogadro’s number and is equal to 6.02 x 1023. One atomic mass unit (u) is equal to 1.66x10-24 g. The mass m of an Avogadro’s number of carbon-12 atoms is : The mass per atom for a given element is: 1.66x10-24=1/6.02x1023

Equations of state (cont’d) Definition of a mole (cont’d) The same number of particles is found in a mole of a substance. Atomic mass of hydrogen 1H is 1 u, and that of carbon 12C is 12 u. 12 g of 12C consists of exactly NA atoms of 12C. The molecular mass of molecular hydrogen H2 is 2u, and NA molecules are in 2 g of H2 gas. Molar mass of a substance The molar mass of a substance is defined as the mass of one mole of that substance, usually expressed in grams per mole. Number of moles The number of moles of a substances n is: m : mass of the substance

Equation of state (cont’d) Ideal gas equation (Equation of state for ideal gas) Boyle’s law When a gas is kept at a constant temperature, its pressure is inversely proportional to its volume. Charles’s law When the pressure of a gas is kept constant, its volume is directly proportional to the temperature. Gay-Lussac’s law When the volume of a gas is kept constant, its pressure is directly proportional to the temperature. p : pressure, V : volume, T : temperature in K R : universal gas constant 8.31 J/(mole K) 0.0821 L atm/(mol K) 1 L (litre) = 103 cm3 = 10-3 m3 Ideal gas equation: The volume occupied by 1 mol of an ideal gas at atmospheric pressure and at 0oC is 22.4 L

Equations of state (cont’d) The ideal gas equation total mass = # of moles times molar mass # of moles gas const. Ideal-gas equation: An ideal gas is one for which the above equation holds precisely for all pressure and temperatures. density constant for constant mass

Equations of state (cont’d) The ideal gas equation (cont’d) The condition called standard temperature and pressure (STP) for a gas is defined to be a temperature of 0 oC=273.15 K and a pressure of 1 atm = 1.013 x 105 Pa. Example 18.1 If you want to keep a mole of an ideal gas in your room at STP, how big a container do you need?

Equations of state (cont’d) The van der Waals equation The ideal gas equation can be derived from a simple molecular model that ignores: the volumes of the molecules themselves the attractive forces between them The van der Waals equation makes corrections for these short-comings. reduces volume of gas due to finite size of molecules The attractive intermolecular forces reduces the pressure by gas A constant that depends on the attractive intermolecular forces. A constant that represents the size of a gas molecule

Equations of state (cont’d) pV diagram and phases Non-ideal gasses behave differently because the attractive forces between molecules become comparable to or greater than the kinetic energy of motion ideal gas behavior pV=const The molecules are pulled closer together and the volume V is less than for an ideal gas const.temp. c is called the critical point For T<Tc a gas will liquify under increased pressure For T>Tc a gas will not liquify under pressure

Molecular properties of matter Molecules and phases of matter All familiar matter is made up of molecules. The smallest size of a molecule is about of order of 10-10 m. Larger molecules may have a size of 10-6 m. The force between molecules in a gas varies with distance r between molecules. The major source of the force in this case is electro- magnetic interaction between them. A molecule that has energy E>E0 (see below) can move around as depicted in the figure. A molecule with E>0 (see below) can escape as in gaseous phase of matter. U(r) r In solids molecules vibrate around fixed point more or less. r E<0 E0 A molecule in liquid has slightly higher energy than in solid.

Kinetic-molecular model of an ideal gas Assumptions Molecules are featureless points, occupy negligible volume. Total number of molecules (N) is very large. Molecules follow Newton’s laws of motion. Molecules move independently making elastic collisions. No potential energy of interaction (no bonding).

Kinetic-molecular model of an ideal gas Molecule of mass m moving with speed vx in negative-x direction in a cube of side l collides elastically with the wall. -vx vx vx Elastic collision: y-component of velocity does not change before and after the collision. Impulse: Time between collisions with the same wall: Force exerted by molecule on the wall: Force exerted by N molecules with different speeds:

Kinetic-molecular model of an ideal gas (cont’d) Mean square velocity The mean-square velocity is defined as follows: The total force exerted on the wall by N particles:

Kinetic-molecular model of an ideal gas (cont’d) Velocities in random directions Velocities in random directions: (true for any vector) If the velocities have random directions: The total force on the wall:

Kinetic-molecular model of an ideal gas (cont’d) Ideal gas law and mean velocity Pressure on the wall due to molecular impact Ideal Gas Law kB = Boltzmann’s Constant Mean translational kinetic energy is proportional to T

Kinetic-molecular model of an ideal gas (cont’d) The mean kinetic energy and ideal gas Compare with ideal gas equation: average translational kinetic energy kB Avogadro’s number Ideal Gas Law kB = Boltzmann’s Constant Mean translational kinetic energy is proportional to T

Kinetic-molecular model of an ideal gas (cont’d) RMS speed Molecular speed (root-mean-square speed) At a given temperature, gas molecules of different mass m have the SAME average kinetic energy but DIFFERENT rms speeds.

Kinetic-molecular model of an ideal gas (cont’d) Collision between molecules Consider N spherical molecules with radius r, and suppose only one molecule is moving. r r 2r v r When it collides with another molecule the distance between centers is 2r. r The moving molecule collides with any other molecule whose center inside a cylinder of radius 2r. vdt In a short time dt a molecule with speed v travels a distance vdt, during which time it collides with any molecule that is in the cylindrical volume of radius 2r and length vdt. The volume of the cylinder is 4pr2vdt. There are N/V molecules per unit volume.

Kinetic-molecular model of an ideal gas (cont’d) Collisions between molecules (cont’d) The number of molecules dN with centers in the cylinder: r The number of collisions per unit time: r 2r v r r If there are more than one molecule moving, it can be shown that: vdt The average time between collision (the mean free time): The average distance traveled between collision (the mean free path):

Heat capacities Heat capacities of ideal gases (~ monatomic gasses) Change of translational kinetic energy due to change in temperature: Heat input needed for a change in temperature: heat capacity (specific heat) at const.volume From dK=dQ, For an ideal gas

Heat capacities (cont’d) Theorem of equipartition of energy Each quadratic term in the expression of the average total energy of a particle in thermal equilibrium with its surrounding contributes on the average (1/2)kT to the total energy. OR Each degree of freedom contributes an average energy of (1/2)kT. Translational kinetic energy comprises 3 terms (degree of freedom): Rotational energy of diatomic molecule has 2 degree of freedom:

Heat capacities (cont’d) Heat capacities of gases in general

Heat capacities (cont’d) Heat capacities of gases in general (cont’d) However, which degree of freedom is available depends on the temperature. Furthermore, in case of diatomic molecules, for example, two more degrees of freedom are possible from two possible modes of vibration. diatomic molecule

Heat capacities (cont’d) Heat capacities of solids Consider a crystalline solid consisting of N identical atoms (monatomic solid). Each atom is bound to an equi- librium position by interatomic forces. Each atom has three degrees of freedom, corresponding to its three components of velocity. In addition each atom acts as 3D harmonic oscillator because of the potential created by interatomic forces – three more degrees of freedom. 6 degrees of freedom

Phase Diagram for Water Phases of matter Phase equilibrium A transition from one phase to another ordinarily takes place under conditions of phase equilibrium between two phases, and for a given pressure this occurs at only one specific temperature. Phase diagram Phase Diagram for Water All three phases exist in equilibrium at the triple point. (273.16 K and 610 Pa for water)

Phases of matter (cont’d) Phase diagram (cont’d) Water has an Unusual Property Most substances contract when transforming from a liquid to a solid (e.g. carbon dioxide). Water is unusual in that it expands upon freezing (solid-liquid interface curve has a negative slope). Ice floats on liquid water. water Carbon dioxide

Phases of matter (cont’d) Phase diagram (cont’d)

Exercises Problem 1 A hot-air balloon stays aloft because hot air at atmospheric pressure is less dense than cooler air at the same pressure. If the volume of the balloon is 500 m3 and the surrounding air is at 15.0oC, what must the temperature of the air in the balloon be for it to lift a total load 290 kg (in addition to the mass of the hot air)? The density of air at 15.0oC and atmospheric pressure is 1.23 kg/m3. Solution The density of the hot air must be where is the density of the ambient air and m is the load. The density is inversely proportional to the temperature, so

Exercises Problem 2 The vapor pressure is the pressure of the vapor phase of a substance when it is in equilibrium with the solid or liquid phase of the substance. The relative humidity is the partial pressure of water vapor in the air divided by the vapor pressure of water at that same temperature, expressed as a percentage. The air is saturated when the humidity is 100%. a) The vapor pressure of water at 20.0oC is 2.34 x 103 Pa. If the air temperature is 20.0oC and the relative humidity is 60%, what is the partial pressure of water vapor in the atmosphere ( the pressure due to water vapor alone)? b) Under the conditions of part (a), what is the mass of water in 1.00 m3 of air? (The molar mass of water is 18g/mol.) Solution (a) (b)

Exercises Problem 3 Modern vacuum pumps make it easy to attain pressures of order 10-13 atm in the laboratory. At a pressure of 9.00 x 10-14 atm and an ordinary temperature (say T=300 K), how many molecules are present in a volume of 1.00 cm3? Solution

Exercises Problem 4 Solution A balloon whose volume is 750 m3 is to be filled with hydrogen at atmospheric pressure (1.01 x 105 Pa). a) If the hydrogen is stored in cylinders with volumes of 1.90 m3 at a gauge pressure of 1.20 x 106 Pa, how many cylinders are required? b) What is the total weight (in addition to the weight of gas) that can be supported by the balloon if the gas in the balloon and the surrounding air are both at 15.0oC? The molar mass of hydrogen (H2) is 2.02 g/mol. The density of air at 15.0oC and atmospheric pressure is 1.23 kg/m3. Solution The absolute pressure of the gas in a cylinder is: At atmospheric pressure, the volume of hydrogen will increase by a factor of so the number of cylinders is: (b) The difference between the weight of the air displaced and the weight of hydrogen is: