Thermal Energy Kinetic Theory Linear Expansion - Heat

Thermal Expansion

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 0C is heated, its volume decreases until it reaches 4C. Above 4C, water behaves normally and expands in volume as it is heated. Water expands as it is cooled from 4C to 0C 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 temperature 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. Holes in materials also expand or contract with the material, if a material gets larger, the hole also gets larger.

DL = a L DT

where L is the length of the material

a is the coefficient of linear expansion

DT is the temperature change in C

Thermal Properties of Matter

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. Where V0, P0, and T0 represent the initial state of the material and V, P, and T represent the final states of the material. .

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,

PV = nRT

Where T is measured in Kelvin and R is the ideal gas constant

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.

Kinetic theory of gases

particles in a hot body have more kinetic energy than those in a cold body; as temperature increases, kinetic energy increases. If the temperature of rises, the gas molecules move at greater speeds. If the volume remains the same, the hotter molecules would be expected to hit the walls of the container more frequently than the cooler ones, resulting in a rise in pressure.

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

measure of an objects kinetic energy; temperature measures how hot or how cold an object is with respect to a standard

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 0C and the boiling point of water corresponds to 100C. On the Farenheit scale, the freezing point of water is defined to be 32F and the boiling point 212F. It is easy to convert between these two scales by remembering that 0C = 32F and that 5C = 9F. 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.

KEav = 1/2 mvav2 = 3/2 kT

where T is the temperature in Kelvin and k is Boltzmann's constant

k = 1.38 x 10-23 J/K

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

k = R/N

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).

PV = 2/3 N (1/2 mvav2)

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.

vrms = (3RT/M)1/2

where R is the ideal gas constant, T is temperature in Kelvin, and M is the molecular mass in units of kg/mol

 

Heat(symbol is Q; SI unit is Joule)

amount of thermal energy transferred from one object to another due to temperature differences (we will learn in thermodynamics why heat flows from a hot to a cold body).

 

Temp and Internal Energy

 

T = Temperature related to the average KE of the molecules in a substace


U = Internal Energy (Thermal Energy) the total KE of the molecules.

= # molecules * avg KE

= N * KEavg

which simplifies to U = 3/2 n R T

 

 

Mechanical Equivalent of Heat

James Joule described the reversible conversion of heat energy and work. The calorie is defined as the amount of energy needed to raise the temperature of one gram of water at 14.5 one degree Celcius. The SI unit for work and energy is the Joule.

1 calorie = 4.186 J

1000 calories is equal to 1 food Calorie

 

Heat transfer

There are three basic ways in which heat is transferred. In fluids, heat is often transferred by convection, in which the motion of the fluid itself carries heat from one place to another. Another way to transfer heat is by conduction, which does not involve any motion of a substance, but rather is a transfer of energy within a substance (or between substances in contact). The third way to transfer energy is by radiation, which involves absorbing or giving off electromagnetic waves.

Convection

Heat transfer in fluids generally takes place via convection. Convection currents are set up in the fluid because the hotter part of the fluid is not as dense as the cooler part, so there is an upward buoyant force on the hotter fluid, making it rise while the cooler, denser, fluid sinks. Birds and gliders make use of upward convection currents to rise, and we also rely on convection to remove ground-level pollution.

Forced convection, where the fluid does not flow of its own accord but is pushed, is often used for heating (e.g., forced-air furnaces) or cooling (e.g., fans, automobile cooling systems).

Conduction

When heat is transferred via conduction, the substance itself does not flow; rather, heat is transferred internally, by vibrations of atoms and molecules. Electrons can also carry heat, which is the reason metals are generally very good conductors of heat. Metals have many free electrons, which move around randomly; these can transfer heat from one part of the metal to another.

The equation governing heat conduction along something of length (or thickness) L and cross-sectional area A, in a time t is:

Q/t = k A ∆T / L

(Q/t) = H = rate of heat loss or gain

k is the thermal conductivity, a constant depending only on the material, and having units of J / (s m C).

Copper, a good thermal conductor, which is why some pots and pans have copper bases, has a thermal conductivity of 390 J / (s m C). Styrofoam, on the other hand, a good insulator, has a thermal conductivity of 0.01 J / (s m C).

Example

Consider what happens when a layer of ice builds up in a freezer. When this happens, the freezer is much less efficient at keeping food frozen. Under normal operation, a freezer keeps food frozen by transferring heat through the aluminum walls of the freezer. The inside of the freezer is kept at -10 C; this temperature is maintained by having the other side of the aluminum at a temperature of -25 C.

The aluminum is 1.5 mm thick. Let's take the thermal conductivity of aluminum to be 240 J / (s m C). With a temperature difference of 15, the amount of heat conducted through the aluminum per second per square meter can be calculated from the conductivity equation:

Radiation

The third way to transfer heat, in addition to convection and conduction, is by radiation, in which energy is transferred in the form of electromagnetic waves. We'll talk about electromagnetic waves in a lot more detail later in the year; an electromagnetic wave is basically an oscillating electric and magnetic field traveling through space at the speed of light. Don't worry if that definition goes over your head, because you're already familiar with many kinds of electromagnetic waves, such as radio waves, microwaves, the light we see, X-rays, and ultraviolet rays. The only difference between the different kinds is the frequency and wavelength of the wave.

Note that the radiation we're talking about here, in regard to heat transfer, is not the same thing as the dangerous radiation associated with nuclear bombs, etc. That radiation comes in the form of very high energy electromagnetic waves, as well as nuclear particles. The radiation associated with heat transfer is entirely electromagnetic waves, with a relatively low (and therefore relatively safe) energy.

Everything around us takes in energy from radiation, and gives it off in the form of radiation. When everything is at the same temperature, the amount of energy received is equal to the amount given off. Because there is no net change in energy, no temperature changes occur. When things are at different temperatures, however, the hotter objects give off more energy in the form of radiation than they take in; the reverse is true for the colder objects.

Heat transfer in general

We've looked at the three types of heat transfer. Conduction and convection rely on temperature differences; radiation does, too, but with radiation the absolute temperature is important. In some cases one method of heat transfer may dominate over the other two, but often heat transfer occurs via two, or even all three, processes simultaneously.

A stove and oven are perfect examples of the different kinds of heat transfer. If you boil water in a pot on the stove, heat is conducted from the hot burner through the base of the pot to the water. Heat can also be conducted along the handle of the pot, which is why you need to be careful picking the pot up, and why most pots don't have metal handles. In the water in the pot, convection currents are set up, helping to heat the water uniformly. If you cook something in the oven, on the other hand, heat is transferred from the glowing elements in the oven to the food via radiation

 

Thermodynamics

 

Thermodynamics

study of properties of thermal energy

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

If two objects at different temperatures are placed in thermal contact (so that the heat energy can transfer from one to the other), the two objects will reach the same temperature, or become in 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.

Internal or Thermal Energy(symbol is U; unit is J)

sum of all the energy an object possesses; it cannot be measured; only changes in internal energy can be determined

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).

U = 3/2 nRT

where R is the ideal gas constant, R = 8.315 J/mol K

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.

1st 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.

DU = Q + W

where Q is the heat added to the system and W is the net work done on the system. In other words, heat added is positive; heat lost is negative. Work done on the system (an example would be compression of a gas) is positive; work done by the system (an example would be expansion of a gas) is negative.

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.

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.

pv diagram

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.

 

Processes

1.      Isothermal Process temperature (T) is constant. If there is no temperature change, there is no internal energy change.

DU = 0

Q = -W

pv diagram for isothermal process

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.

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.

2.                  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

pv diagram for isobaric process

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.

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

W = 0

DU = Q

pv diagram for isochoric process

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.

4.                  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.

pv diagram for adiabatic process

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.

2nd law of thermodynamics This law is a statement about which processes can occur in nature and which cannot.

  1. In natural processes, heat cannot flow from a cold to a hot substance
  2. Natural processes increase the entropy of the universe; with time, disorder cannot become order
  3. A heat engine cannot convert all its heat to mechanical energy. No machine is ever 100% efficient.

The second law of thermodynamics explains things that don't happen:

  1. Air molecules fill the room evenly, instead of all moving to one corner.
  2. A spoon reaches an equilibrium temperature, instead of one end being cold and one end being hot.
  3. Coffee, swirling in your cup, will eventually stop swirling. The coffee doesn't spontaneously cool down and start to swirl around.

It is not possible to reach absolute zero (0 K). Since heat can only flow from a hot to a cold substance, in order to decrease the temperature of a substance, heat must be removed and transferred to a "heat sink" (something that is colder). Since there is no temperature less than absolute zero, there is no heat sink to use to remove heat to reach that temperature.

entropy (S)

disorder, chaos.

DS = Q / T

where T is the Kelvin temperature

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 QH is extracted from a reservoir at temperature TH; useful work is done and the rest is discharged as heat QL to a reservoir at a cooler temperature TL. 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.

heat engine

The purpose of an engine is to transform as much QH 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.

refrigerator

There is no perfect refrigerator because it is not possible for heat to flow from one body to another body at a higher temperature with no other change taking place. The purpose of a heat pump or a refrigerator is to convert mechanical energy into thermal energy.

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 (QH).

e = W / QH

or, e = (QH - (QL) / QH

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

eideal = (TH - (TL) / TH