An advanced look at the kinetic theory:   The assumptions describing an ideal gas make up the postulates of the kinetic theory:

1. An ideal gas is made up of a large number of gas molecules N each with mass m moving in random directions with a variety of speeds.
2. The gas molecules are separated from each other by an average distance that is much greater than the molecule's diameter.
3. The molecules obey laws of mechanics, interacting only when they collide.
4. Collisions between the walls of the container or with other gas molecules are assumed to be perfectly elastic.

Entropy disorder; the higher the temperature, the more disorder (or entropy) a substance has

Temperature Scales The most common scale is the Celsius (or Centigrade, though in the United States the Fahrenheit scale is common. Both of these scales use the freezing point and boiling point of water at atmospheric pressure as fixed points. On the Celcius scale, the freezing point of water corresponds to 0°C and the boiling point of water corresponds to 100°C. On the Farenheit scale, the freezing point of water is defined to be 32°F and the boiling point 212°F. It is easy to convert between these two scales by remembering that 0°C = 32°F and that 5°C = 9°F. The Kelvin scale is based upon absolute zero
(-273.15 °C), or 0 K.

Triple Point The triple point of water serves as a point of reference. It is only at this point (273.16 °K) that the three phases of water (gas, liquid, and solid) exist together at a unique value of temperature and pressure.

Temperature is a property of a system that determines whether the system will be in thermal equilibrium with other systems.

Molecular Interpretation of Temperature The concept that matter is made up of atoms in continual random motion is called the kinetic theory. We assume that we are dealing with an ideal gas. In an ideal gas, there are a large number of molecules moving in random directions at different speeds, the gas molecules are far apart, the molecules interact with one another only when they collide, and collisions between gas molecules and the wall of the container are assumed to be perfectly elastic. The average translational kinetic energy of molecules in a gas is directly proportional to the absolute temperature. If the average translational kinetic energy is doubled, the absolute temperature is doubled.

The relationship between Boltzmann's constant (k), Avogadro's number (N), and the gas constant (R) is given by:

An advanced look at the relationship between pressure and the kinetic theory:  The pressure exerted by an ideal gas on its container is due to the force exerted on the walls of the container by the collisions of the molecules with the walls of area A. The collisions cause a change in momentum of the gas molecules. These assumptions can be used to derive an expression between pressure and the average kinetic energy of the gas molecules. The pressure is directly proportional to the square of the average velocity. Since the average kinetic energy is directly proportional to the temperature, pressure is also directly proportional to the temperature (for a fixed volume).

The higher the temperature, according to kinetic theory, the faster the molecules are moving, on average.

rms speed The square root of the average speed speed in the kinetic energy expression is called the rms speed.

Heat with moles of gas   Typically, moles of gas are given instead of the mass of the gas. In that case, heat can be calculated using
           Q = n c DT
          where n is the number of moles
          c is the molar specific heat of the gas

Specific Heats of Gases   Since the volume of a gas changes significantly a change in temperature or a change in pressure, molar specific heats of gases are also expressed in terms of constant pressure or constant volume conditions. In the case of a constant volume process, the constant would be expressed as c_v; in the case of a constant pressure process, the constant would be expressed as c_p. Use these constants in Q = n c DT.

specific of heat of water: c = 4180 J/kg K (at a temperature of 15°C and a pressure of 1 atmosphere)

Please note, the units J/kg K are the same as J/kg °C

Specific Heat Capacity of Gases at Constant Pressure (C_p)   is defined as the amount of heat required to raise the temperature of one mole of a gas through 1 K at a constant pressure, or Q = nC_pDT

For monoatomic gases such as Ar, He, Ne, H, O, etc, C_P = 20.78 J/molK

Specific Heat Capacity of Gases at Constant Volume (C_V)   is defined as the amount of heat required to raise the temperature of one mole of a gas through 1 K at a constant volume, or Q = nC_VDT

For monoatomic gases such as Ar, He, Ne, H, O, etc, C_V = 12.47 J/molK

Notice that the specific heat of water is very high - higher than ice and steam. Water has a very high specific heat, meaning that it heats slowly and cools slowly.

The specific heat of a material yields information about how the material heats and cools. If you add ten joules of heat to two materials, the one with the lowest specific heat will show the greatest temperature change. If you cool two materials ten degrees, the material with the greatest specific heat loses the most energy.

Measurement of heat capacity (c) In an experiment, a substance is heated over a period of time. If V is the voltage, i is the current, Dt is the change in time in seconds, DT is the difference in temperature, and n is the number of moles, the specific heat capacity can be found by

Molar heat capacity The molar heat capacity is based upon the number of moles of the substance. Heat can be expressed in terms of molar heat capacity by:

Dulong-Petit Law The average molar heat capacities for all metals is approximately the same and is equal to about 25 J/mole K, or approximately 3 R. Thus the specific heat of a metal can be calculated using c = C/M, where M is the molecular mass of the substance.

Energy transfer mechanisms:

1. conduction (solids)-KE transfer due to collisions of particles; heat transfer occurs only when there is a difference in temperature
   • Thermal Conductivity It is found experimentally that the heat flow per unit of time (DQ/Dt)is proportional to the corss-sectional area of the object (A), the distance (d) between the two ends of the object, the temperatures of each end of the object (T₁ and T₂), and a proportionality constant, k, called the thermal conductivity of the substance.
     DQ/Dt = kA(T₁ - T₂)/d
   • Substances that have large values for k are good thermal conductors. Those with low values for k are good insulators.
2. convection (fluids)-KE transfer due to movements of fluids over large distances caused by different densities at different temperatures
3. radiation-energy transfer through a vacuum. Conduction and convection require the prescence of matter. Radiation consists of electromagentic waves.
   • Stefan-Boltzmann equation The rate at which an object radiates energy is proportional to the fourth power of the Kelvin temperature, T.
     
     DQ/Dt = seAT⁴
     where A is the area, s is the Stefan-Boltzmann constant which has the value of s=5.67 x 10^-8 W/m² K⁴, and e is the emissivity (a number between 0 and 1)

     Very black surfaces has emissivities close to 1 and very shiny surfaces have emissivities close to 0. A good absorber of radiation is also a good emitter of radiation.

When different parts of an isolated system are at different temperatures, heat will flow from the part at a higher temperature to that at the lower temperature until they are at thermal equilibrium

Objects are in thermal equilibrium when they are at the same temperature.

The three most common states of matter are solid, liquid, and gas. When heat is added to a substance, one of two things can occur. The temperature can increase or the material can change to a different state. There is a fourth state of matter - plasma. A plasma is a state of matter in which atoms are stripped of their electrons. In a plasma, atoms are separated into their electrons and bare nuclei.

When a material changes phases from solid to liquid or from liquid to gas, a certain amount of energy is absorbed (in the reverse process, the heat is given off). Let's look at ice (a solid) at a temperature of -5°. When heat is added to ice, its temperature increases until it reaches 0°. At this point, ice begins to melt--it changes its state from a solid to a liquid. The temperature remains constant at 0° until all the ice has melted. Now we have water at 0°. As heat is added to the water, its temperature increases until it reaches 100°. At this point, the water begins to boil, changing its state from liquid to gas. The temperature remains constant at 100° until all the water boils, turning into steam. Now we have steam at 100°. If you continue to add heat, the temperature of the steam begins to increase.

If energy is added to a system heating it and causing an increase in temperature, energy is positive; if energy is removed from a system cooling it and causing a decrease in temperature, energy is negative. If energy is added to a system causing a change in the state of matter from a solid to a liquid or from a liquid to a solid, that energy is positive. If energy is removed from a system causing a change in the state of matter from a gas to a liquid or from a liquid to a solid, that energy is negative.

At a phase change, the amount of heat given off or absorbed is found using:

Example: How much heat is added to 10 kg of ice at -20°C to convert it to steam at 120°C?

Using the above phase diagram, one sees that the ice absorbs heat and changes temperature until it reaches 0°C. The amount of heat absorbed is given by Q = mc_iceDT. At 0°C, ice changes phase, from the solid phase (ice) to the liquid phase (water). Heat must be added to change the phase. Using the phase diagram, one sees that no temperature change occurs until all the ice is melted into water. The amount of heat added is given by Q = mH_f. Once all the ice is melted into water, the temperature agains to rise as heat is added until it reaches 100°C. The amount of heat absorbed is given by Q = mc_waterDT. At 100°C, water changes phase, from the liquid phase (water) to the gaseous phase (steam). The amount of heat added to accomplish this phase change is given by Q = mH_v. Using the phase diagram, one sees that no temperature change occurs until all the water is converted into steam. Once all the water has been converted into steam, the temperature again begins to rise. The amount of heat added is given by Q = mc_steamDT.

The amount of heat added to change ice at -20°C to steam at 120°C is given by:

Experimentally determining the amount of heat added 

Sublimation The process whereby a solid changes directly to a gas without passing through the liquid phase.

Evaporation Evaportation can be explained in terms of the kinetic theory. The fastest moving molecules in a liquid escape from the surface, decreasing the average speed of those remaining. When the average speed is less, the absolute temperature is less. Thus evaporation, the escaping of the fastest moving molecules from the surface of a liquid, is a cooling process.

Boiling When the temperature of a liquid equals the point where the saturated vapor pressure equals the external pressure, boiling occurs.

Most substances expand when heated and contract when cooled. The exception is water. The maximum density of water occurs at 4°. This explains why a lake freezes at the surface, and not from the bottom up. If water at 0°C is heated, its volume decreases until it reaches 4°C. Above 4°C, water behaves normally and expands in volume as it is heated. Water expands as it is cooled from 4°C to 0°C and expands even more as it freezes. That is why ice cubes float in water and pipes break when the water inside of them freezes.

The change in length in almost all solids when heated is directly proportional to the change in themperatuer and to its original length. A solid expands when heated and contracts when cooled: The length of a material decreases as the temperature decreases; its length increases as the temperature increases. So a rod that is 2 m long expands twice as much as a rod which is 1 m long for the same ten degree increase in temperature.

A gas expands when heated and contracts when cooled: The volume of a gas decreases as the temperature decreases; its volume increases as the temperature increases.

Thermal Stress In many buildings and roads, the ends of a beam or other material are held rigidly fixed. If the temperature should change, large compressive or tensile forces develop, called thermal stresses. Elastic modulus can be used to calculate these thermal stresses.

AP Multiple Choice Questions on Heat

1. You should be able to use Q=mcDT to predict temperature changes. For example, an object's potential energy change is directly converted into heat. Or, frictional losses result in an increase in temperature of the surface.
2. Be able to interpret phase diagrams. Use them to predict melting (freezing) points, boiling points and the relationship between specific heats for the substance for different phases.
3. You do not need to know the formulas for length or volume expansion. You need to know that the heat transferred to a substance is directly proportional to its cross-sectional area and indirectly proportional to its length.
4. Be able to solve easy calorimetry problems to predict a final temperature of a mixture or the specific heat of a metal.
5. Be able to calculate the rate at which heat is transferred. Remember, this would be in the units of J/sec.
6. Be able to describe what happens to the temperature and to heat when a substance freezes (or melts) or boils (or condenses).
7. Be able to predict which would be the hotter substance (of unequal specific heats) when equal amounts of heat are added.
8. Be able to predict the appearance of an object when it is heated.

AP Free Response Questions on Heat

1. Be able to interpret phase diagrams. Use them to predict melting (freezing) points, boiling points, latent heats, and the specific heats for the substance for different phases.
2. All you need to know about entropy is that it is disorder. Entropy increases when the temperature increases.
3. Know that the temperature of a substance is directly proportional to its kinetic energy.
4. Be able to convert losses of energy to temperature increases using Q=mcDT.
5. They ask lab-type questions on heat. These are some of the ones that I have seen:
   • Use calorimetry data (which you graph) to determine the specific heat of a substance.
   • Use calorimetry data (which you graph) to determine the latent heat of a substance.
   • Be able to relate the mechanical equivalent of heat to thermal energy in a lab situation. In other words, if you increase the amount of potential energy (kinetic energy) of a substance, you do work. This is equal to Q=mcDT.
   • Or, you take temperature data over a period of time for a substance that is heating on a heater. Remember, if you know the wattage of the heater and how long the heater was on, you can use Electrical Energy=Vit to calculate energy. This energy would be transferred to the substance as heat.
   • Think about places in an experiment where heat could be lost, instead of all be absorbed by the substance.
   • Be able to graphically describe how temperature varies with time for different parts of an experiment in which a substance undergoes multiple phase changes.
6. Be able to solve calorimetry problems.

Physical quantities such as pressure, temperature, volume, and the amount of a substance describe the conditions in which a particular material exists. They describe the state of the mateterial and are referred to as state variables. These state variables are interrelated; one cannot be changed without changing the other.

The relationship between these variables can be described using an equation of state.

In physics, we use an ideal gas to repesent the material and thus simplifying the equation of state.

Ideal Gas Law The volume of a gas is proportional to the number of moles of the gas, n. The volume varies inversely with the pressure. The pressure is proportional to the absolute temperature of the gas. Combining these relationships yields the following equation of state for an ideal gas,

Ideal Gas Constant In SI units, R = 8.314 J/ mol K

Ideal Gas Real gases do not follow the ideal gas law exactly. An ideal gas is one for which the ideal gas law holds precisely for all pressures and temperatures. Gas behavior approximates the ideal gas model at very low pressures when the gas molecules are far apart and at temperatures close to that at which the gas liquefies.

pV-diagram A graph of pressure vs volume for a particular temperature for an ideal gas. Each curve, representing a specific constant temperature, is called an isotherm. The area under the isotherm represents the work done by the system during a volume change.

When a system undergoes a change of state from an initial state to a final state, the system passes through a series of intermediate staes. This series of states is called a path. Points 1 and 2 represent an initial state (1) with pressure P₁ and volume V₁ and a final state (2) with pressure P₂ and volume V₂. If the pressure is kept constant at P₁, the system expands to volume V₂ (point 3 on the diagram). The pressure is then reduced to P₂ (probably by decreasing the temperature)and the volume is kept constant at V₂ to reach point 2 on the diagram. The work done by the systemd during this process is the area under the line from state 1 to state 3. There is no work done during the constant volume process from state 3 to state 2. Or, the system might traverse the path state 1 to state 4 to state 2, in which case the work done is the area under the line from state 4 to state 2. Or, the system might traverse the path represented by the curved line from state 1 to state 2, in which case, the work is represented by the area underneath the curve from state 1 to state 2. The work is different for each path.

The work done by the system depends not only upon the initial and final states, but also upon the path taken.

Each of the laws of thermodynamics are associated with a variable. The zeroeth law is associated with temperature, T; the first law is associated with internal energy, U; and the second law is associated with entropy, S.

System any object or set of objects we are considering. A closed system is one in which mass is constant. An open system does not have constant mass. No energy flows into or out of a closed system which is said to be isolated.

Environment everything else

Thermal Equilibrium

Zeroth Law of Thermodynamics If two systems are in thermal equilibirum with a third system, they are in thermal equilibrium with each other.

The kinetic theory can be used to clearly distinguish between temperature and thermal energy. Temperature is a measure of the average kinetic energy of individual molecules. Thermal energy refers to the total energy of all the molecules in an object.

Internal Energy of an Ideal Gas The internal energy of an ideal gas only depends upon temperature and the number of moles of the gas (n).

Characteristics of an Ideal Gas:

1. An ideal gas consists of a large number of gas molecules occupying a negligible volume.
2. Ideal gas molecules have random motion.
3. Ideal gas molecules undergo elastic collisions with the walls of the container and with other gas molecules.
4. The temperature of an ideal gas is proportional to the kinetic energy of the gas molecules.

1^st law of thermodynamics   The total increase in the internal energy of a system is equal to the sum of the work done on the system or by the system and the heat added to or removed from the system. It is a restatement of the law of conservation of energy. Changes in the internal energy of a system are caused by heat and work.

AP changes for 2002 The 1^st law is being expressed in this form to be consistent with changes in the sign convention for work for the AP Physics Exam for 2002. Effective in 2002, the symbol W will represent the work done on a system rather than by a system. According to the College Board, this makes the sign convention consistent with that used for work in mechanics, as well as with the thermodynamic convention used in most chemistry and some physics textbooks.

The best way to remember the sign convention for work: if a gas is compressed (volume decreases), work is positive; if a gas expands (volume increases), work is negative. It is just like mechanics, if you (the environment) do work on the system, you would compress it. The work you do is considered to be positive.

1. Isothermal Process temperature (T) is constant. If there is no temperature change, there is no internal energy change.
   DU = 0
   Q = -W

   

   The curve shown represents an isotherm.

   Since the temperature is constant, no change in internal energy occurs. Internal energy changes only occur when there are temperature changes. At constant temperature, the pressure and volume of the system decrease as along the path state 1 to state 2. The amount of work is given by

   

   Example of an isothermal process: An ideal gas (the system) is contained in a cylinder with a moveable piston. Since the system is an ideal gas, the ideal gas law is valid. For constant temperture, PV=nRT becomes PV=constant. At point 1, the gas is at pressure P₁, volume V₁, and temperature T. A very slow expansion occurs, so that the gas stays at the same constant temperature. If heat Q is added, the gas must expand. As the gas expands, it pushes on the moveable piston, thus doing work on the environment (or negative work). At point w, the gas now has volume V₂ which is greater than V₁, pressure P₂ which is less than P₁, and temperature T. The amount of work done by the system on the environment during its expansion has the same magnitude as the amount of heat added to the system. The amount of work done is equal to the area under the curve.

   How to know if heat was added or removed in an isothermal process: if heat is added, the volume increases and the pressure decreases. Remember, pressure is determined by the number of collisions the gas molecules make with the walls of the container. If the volume increases at constant temperature, the gas molecules make fewer collisions with the walls of the container, and pressure decreases.

   Isobaric Process pressure (P) is constant. If pressure is kept constant, the work done during the process is given by

   W = - P DV

   DU = Q + W

   

   P is held constant, so the amount of work done is represented by the area underneath the path from 1 to 2. Typically, lab experiments are isobaric processes.

   Example of isobaric process: An ideal gas is contained in a cylinder with a moveable piston. The pressure experienced by the gas is always the same, and is equal to the external atmospheric pressure plus the weight of the piston. The cylinder is heated, allowing the gas to expand. Heat was added to the system at constant pressure, thus increasing the volume. The change in internal energy U is equal to the sum of the work done by the system on the environment during the volume expansion (negative work) and the amount of heat added to the system. The amount of work done is equal to the area under the curve.

   How to determine if heat was added or removed: in an isobaric process, heat is added if the gas expands and removed if the gas is compressed.

   How to tell if the temperature is increasing or decreasing: in an isobaric process, adding heat results in an increase in internal energy. If the internal energy increases, the temperature increases. Typically, volume expansions are small and all the heat added serves to increase the internal energy. In our graph, point 2 was at a higher temperature than point 1.

   Isochoric Process Volume (V) is constant. Since there is no change in volume, no work is done.

   W = 0

   DU = Q

   

   Since V is constant, no work is done. If heat is added to the system, the internal energy U increases; if heat is removed from the system, the internal energy U decreases. In the pV diagram shown, heat is removed along the path 1 to 2, thus decreasing the pressure at constant volume.

   Example of an isochoric process: An ideal gas is contained in a rigid cylinder (one whose volume cannot change). If the cylinder is heated, no work can be done even though enormous forces are generated within the cylinder. No work is done because there is no displacement (the system does not move). The heat added only increases the internal energy of the system.

   How to tell if heat is added or removed: in an isochoric process, heat is added when the pressure increases.

   How to tell if the temperature increases or decreases: since U=3/2 nRT, if the internal energy is increasing, then the temperature is increasing. In our diagram, point 1 is at a higher temperature than point 2.

   Aidabatic Process No heat (Q) is allowed to flow into or out of the system. This can occur if the system is well-insulated or the process happens quickly. (in other words, Q=0)

   DU = W

   The internal energy and the temperature decreases if the gas expands.

   

   In this well-insulated process shown, heat cannot transfer to the environment. The amount of work done is represented by the area under the path from state 1 to state 2. In this example, the volume increases along the path from state 1 to state 2, so work is done on the environment by the system (negative work). There is a decrease in internal energy U.

   Example of an adiabatic process: An ideal gas is contained in a cylinder with a moveable piston. Insulating material surrounds the cylinder, preventing heat flow. The ideal gas is compressed adiabatically by pushing against the moveable piston. Work is done on the gas (positive work). Remember, Q=0. The amount of work done in the adiabatic compression results in an increase in the internal energy of the system.

   This applet presents a simulation of four simple transformations in a contained ideal monoatomic or diatomic gas. The user chooses the type of transformation and, depending on the type of transformation, adds or removes heat, or adjusts the gas volume manually. The applet displays the values of the three variables of state P, V, and T, as well as a P-V or P-T graph in real time.

   How to tell if the temperature increases or decreases: since U=3/2 nRt, if the internal energy increases, the temperature increases. In our example, the final temperature would be greater. than the initial temperature. In our pV diagram, the temperature at point 1 is greater than the temperature at point 2.

Determining how entropy changes:  When dealing with entropy, it is the change in entropy which is important.

• In a reversible process (one in which there is no friction), if heat is added to a system, the entropy of the system increases, and vice versa. If entropy increases for the system, it must decrease for the environment by the same amount, and vice versa. For reversible processes, the total entropy (the entropy of the system plus the environment) is constant.
• In an irreversible process (those in the real world), the total entropy either is unchanged or increases.

Heat engines:

1. automobile engines-thermal energy from a high heat source is converted into mechanical energy (work) and exhaust is expelled
2. refrigerator-thermal energy is removed from a cold body (work is required) and transferred to a hot body (the room. Another example is a heat pump.

Drawing of a real engine showing transfer of heat from a high to a low termperature reservoir, performing work. The figure below shows the overall operation of a heat engine. During every cycle, heat Q_H is extracted from a reservoir at temperature T_H; useful work is done and the rest is discharged as heat Q_L to a reservoir at a cooler temperature T_L. Since an engine is a cycle, there is no change in internal energy adn the net work done per cycle equals the net heat transferred per cycle.

The purpose of an engine is to transform as much Q_H into work as possible. So...coffee can't spontaneously start swirling around because heat would be withdrawn from the coffee and totally transformed into work. A heat engine converts thermal energy into mechanical energy.

Drawing of a refrigerator showing transfer of heat from a low to a high temperature reservoir, requiring work. The purpose of a refrigerator is to transfer heat from the low-temperature to the high-temperature reservoir, doing as little work on the system as possible.

Efficiency of a heat engine The efficiency e of any heat engine is defined as the ratio of the work the engine does (W) to the heat input at the high temperature (Q_H).

Carnot (ideal) efficiency This is the theoretical limit to efficiency. It is defined in terms of the operating temperatures.

AP Multiple Choice Questions on Thermodynamics

1. Be able to perform simple Carnot efficiency calculations.
2. Be able to describe an ideal gas.
3. Be able to interpret PV diagrams.
   • Which part of the cycle has DU greater than zero?
   • Which part of the cycle has Q greater than zero?
   • Which part of the cycle W greater than zero?
   • In which part of the cycle is no work done?
   • At which part of the cycle is the gas at is highest temperature?
   • What is the net work done?
   • Which part of the cycle does the most work?
   • Compare the temperatures at different points in the cycle.
   • Which part of the cycle represents an adiabatic, isothermal, isochoric (isovolumetric), or isobaric process?
4. Be able to perform simple calculations using the ideal gas law or the combined gas law. For example, pressure is doubled and volume is quadrupled. What would be the relationship between the initial and final temperatures?
5. Know the characteristics of the processes.
6. Predict what would happen to temperature if kinetic energy is changed.
7. Be able to perform simple calculations using the first law of thermodynamics.

AP Free Response Questions on Thermodynamics

1. Be able to interpret PV diagrams.
   • Use ideal gas law (or combined gas law) to calculate the temperature at different points.
   • Sketch the PV diagram from information given.
   • Predict whether positive, negative, or no work is done for the cycle.
   • Calculate the work done for the cycle (or part of a cycle).
   • Calculate the heat absorbed or given off for a cycle (or part of a cycle).
   • Predict which segments absorb the most heat.
   • Predict if the cycle represents a heat engine or refrigerator.
2. Remember the equation for power (P=W/t or P=Fv) and that P=F/A.
3. Sometimes they ask about the rate at which heat is discarded by an engine. This can be found knowing that W=Q_H-Q_L
4. Be able to perform simple Carnot efficiency calculations.

AP Objectives-Heat & Temperature

AP Objectives-Kinetic Theory & Thermodynamics

States of Matter Notes

Thermal Energy Sample Problems

States of Matter Sample Problems

Thermal Energy Homework

States of Matter Homework

AP Heat Sample Problems

AP Thermodynamics Sample Problems