3. THERMAL PHYSICS

3.1. From mechanics to thermal physics

Many concepts in thermal physics are based on mechanical concepts - for example temperature which is a measure of average kinetic energy. When mechanics is applied on millions and millions of atoms or molecules moving and colliding, it is often not possible to study every one in detail, but they are represented by "collective", more easily measurable quantities.

3.2. Temperature, internal energy and heat

Temperature

This is usually measured in degrees Celsius or ^oC where the freezing point of water is 0 ^oC and the boiling point is + 100 ^oC. We have taken observable physical phenomena for the substance water, and combined with something that changes in the interval between them (the height of a column of a liquid in a thermometer) a temperature scale could be defined. Though we have negative values on the Celsius scale temperature is a scalar - the negative sign does not give information about any direction, only about what value the "temperature" has compared to that of a chosen phenomenon (freezing or melting water).

In the Kelvin scale the size of a "degree" is the same as in Celsius, but the scale has been shifted to avoid negative numbers. The lowest possible temperature in the universe (more about why it is that later), about - 273 ^oC is 0 Kelvin = 0 K (not called 'degrees') and 0 ^oC is 273 K.

ex. 25 ^oC = (25 + 273) K = 298 K and 400 K = (400 - 273) ^oC = 127 ^oC

The higher the temperature, the more the atoms or molecules move. A more exact definition of temperature is that

temperature is proportional to average kinetic energy

[or (not necessary in IB) E_k,average = ½mv²_average = 3kT/2 where m = the mass of the atom or molecule, T = the temperature in Kelvins, k = the Boltzmann constant = 1.38 x 10^-23 JK^-1 ]

Thermal energy (= energy in the form of kinetic energy of the atoms in a material) can be transferred from one object to another in several ways, which means that one loses average kinetic energy (the temperature decreases, it cools) and another gains it (the temperature increases):

Thermal energy flows from an object with a higher temperature to one with a lower (always)

"Zeroeth law of thermodynamics

If two objects have the same temperature, then there is no flow of thermal energy between them and vice versa - if there is no flow of thermal energy, they must be at the same temperature. They are then in thermal equilibrium. That these objects (no flow of thermal energy and same temperature) are said to be equivalent is sometimes called the 0th law of thermodynamics.

Internal energy

The temperature is proportional to average kinetic energy of the atoms. But there are millions of them, and their total kinetic energy combined with their total potential energy (which they may have because there are forces between the atoms or molecules) is the total internal energy U.

Heat

Is the amount of thermal energy which in a certain situation flows from one object to another. Note that heat and internal energy have the unit Joule (J), while temperature has the unit Kelvin (K).

3.3. Solids, liquids and gases (and plasma)

States of matter

· solid: atoms closely packed often in some regular pattern (crystalline structure), most kinetic energy is in the form of vibrations

· liquid: atoms still rather closely packed, but their positions not fixed, there is no permanent pattern, most kinetic energy is vibrational, but some rotational and translational

· gas: atoms move freely, interact only briefly in collisions with each other and container walls, most energy is translational

· plasma: a state caused by extremely high temperature or pressure (or both), electrons separated from rest of atoms; found inside the sun or a nuclear explosion.

Exercise: Draw a sketch to illustrate translational, rotational and vibrational motion of a two-atom molecule

Maxwell-Boltzmann speed distribution (t03a)

In gases the speed of atoms (or molecules) follows a curve which looks like (but is not) an upside down parabola which is asymmetric so that it goes down "slower" on the right side, towards higher speeds.

Ex. Draw a speed distribution diagram for the gas in a container where there are 6 atoms: one with the speed 1 ms^-1, three with the speed 2 ms^-1, two with the speed 3 ms^-1. We can then calculate:

·        the most probable speed = the speed which the highest nr(#) of atoms have = the value on the y-axis where the peak is. Here v_mp = 2 ms^-1.

·        the average speed v_av, in the unit ms^-1 here = (1+2+2+2+3+3)/6 so v_av = 2.167 ms^-1

·        the root mean square speed: Ö[( 1² + 2² +2² +2² + 3² + 3²)/6] = Ö[31/6] = 2.27 so v_rms = 2.27 ms^-1.

Usually v_mp < v_av < v_rms .

[Draw into t03a the graphs of T₀ < T₁ and T₃>T₂ plus a separate graph of N as a function of v for the six atoms in the example.]

Force between atoms

The force between atoms is mostly electromagnetic, and the details of it depend on issues of chemistry not dealt with here. It generally attracts atoms to each other until they are at some short distance r₀ from each other - the "equilibrium separation" - around which they vibrate.

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· LEFT: the distance between (the center of = the nucleus of) the atom on the x-axis and the force on the y-axis. Negative force values for attractive, positive for repulsive. At r₀ the curve crosses the x-axis.

· RIGHT: Potential energy on the y-axis. Minimum at r₀ around which distance the atoms may vibrate (go back and forwards like a ball in a bowl) - further away the higher the average kinetic energy = the temperature. The curve is steeper towards smaller distances (it takes a lot of energy to bring atoms very close to each other, where they repel each other strongly), but more shallow towards higher distances. Consequence:

If the temp. = average kinetic energy is increased, the atom can more easily "roll uphill" away from the other atom => its average distance increases which is why materials generally expand when heated.

Note that the force curve is the gradient or slope of the potential energy curve (as in space physics).

Phase changes

When a solid melts to a liquid or a liquid boils to a gas, the heat (energy flowing in) goes to increase the potential energy, and therefore the average kinetic energy = the temperature remains constant.

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Note here that since P = W/t => W = Pt the supplied energy is directly proportional to time for a constant heating power.

3.4.* Thermal expansion

When a material expands due to increased temperature, the change in length is Dl = a * l₀ * DT where DT = the temperature change, l₀ the original length and a = a length expansion coefficient with the unit K^-1 . This results in a new length l = l₀ + a * l₀ * DT = l₀ ( 1 + a * DT ). In the same way, a change in volume leads to a new volume V = V₀(1 + g * DT) where g = a volume expansion coefficient. It can be shown that g = approximately 3a.

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3.5. Specific heat capacity and specific latent heat

Specific heat capacity c

If the amount Q of heat (unit J) flows into an object with the mass m, its temperature will change with DT (same in Celsius and Kelvin). The size of the temperature change also depends on the substance, which is represented by the specific heat capacity c. We have:

Q = mcDT [DB p.6]

where c is in the unit Jkg^-1K^-1

Heat capacity C

If we have an object which is made of several materials it may be easier to define a constant for this object, the heat capacity C with the unit JK^-1 using

Q = CDT [not in DB]

If the object is made of one substance, the relation between C and c is C = mc.

Substances have different c-values because a certain mass of the substance may contain different numbers of atoms and molecules with different masses; and the chemical forces between these are more or less strong (ex. the hydrogen bonds between water molecules give them a high c-value).

Specific heat capacity is measured in a calorimeter, a vessel with good insulation against heat flow in or out (like a thermos flask) designed so that a liquid inside it can be stirred and temperature measured.

· Electric method:

An amount of electric energy, Q = W = E = VIt (voltage x current x time) is supplied to a calorimeter with the heat capacity C_c containing the mass m_l of the liquid causing an increase of temperature = DT. The specific heat capacity c_s of the sample is then the unknown in:

energy released = energy absorbed

Q = mc_sDT + C_cDT (solve this for c_s)

· Mixing method:

The same calorimeter contains the mass m₁ of the liquid at the temperature T₁ and the mass m₂ of the same liquid at the higher temperature T₂ is inserted; after stirring the temperature stabilizes at T_mix . We can then solve this for c_s :

energy released by cooled liquid = energy absorbed by heated liquid + same by heated calorimeter (which is at same initial temperature as m₁)

m₂c_sDT₂ = m₁c_sDT₁ + C_cDT_c

m₂c_s(T₂ - T_mix) = m₁c_s(T_mix - T₁)+ C_c(T_mix - T₁)

Solve this for c_s.

· Other methods:

To find the specific heat capacity c_s for an unknown solid sample with mass m₂, we can heat it to a known temperature T₂ (e.g. by keeping it in boiling water or in an oven set at a certain temperature for some time) and then insert it into the calorimeter which now contains m₁ of some liquid with an already known specific heat capacity c_k (e.g. water) at T₁. The equation from the mixing method now becomes:

m₂c_s(T₂ - T_mix) = m₁c_k(T_mix - T₁)+ C_c(T_mix - T₁)

which is then solved for c_s .

Specific latent heat L

When a substance is melting/freezing or boiling/condensing the temperature does not change, but heat energy flows in or out of it. Examples:

·        hot water vapour at 100 ^oC causes a worse burn than liquid water at the same temperature, since heat is given first when the vapour condenses to a liquid and then when the 100 ^oC water cools to 37 ^oC.

·        when the temperature in the winter falls below 0 ^oC the lakes and seas do not immediately freeze - first they must be cooled to 0 ^oC and then more heat must flow out to freeze it,

For both freezing/melting and boiling/condensing we can use:

Q = mL [DB p. 6]

where for the first we have L_f = specific latent heat of "fusion" (for melting or solidification) in the unit 1 Jkg^-1 and for the second L_v = specific latent heat of vaporization (for boiling or condensation) in the same unit. These can be measured using:

· Electric method (L_v) :

A vessel contains the mass m₁ of a liquid heated using a heat source with a known power (which can be found from relevant electrical quantities, P = VI, to be explained later). The vessel is placed on an electronic scale, tared (zeroed) to show the mass of the liquid only, not the mass of the vessel (the heater can be one immersed in the liquid). While the liquid is being heated to its boiling point, the mass decreases to m₂ because of evaporation. When the boiling starts, the mass is recorded and a stopwatch started. After the time t the mass has decreased further to m₃ and the L_v can be found from:

energy supplied = energy absorbed

Pt = VIt = (m₂ - m₃)L_v which is then solved for L_v.

· "Mixing" method (L_f)

A calorimeter with the heat capacity contains m₁ of a liquid (with known specific heat capacity c) at T₁ and into it m₂ of the solid form of the same substance (e.g. ice if the liquid was water) at its melting temperature is inserted, which causes the temperature to drop to T_mix (above the melting point of the substance - otherwise take more and/or warmer liquid). The L_f can be found from:

energy released (by the liquid) = energy absorbed (by the solid)

m₁c(T₁ - T_mix) = m₂L_f (solve for L_f)

· Other calculations:

If a mass m of a substance (e.g. ice) at the temperature T_initial, which is lower than its melting point is heated until it has turned into gas, the energy needed is:

Q_total = mc_ice(T_melt - T_initial) + mL_f + mc_water(T_boil - T_melt) + mL_v

where the terms are: heat to warm the ice to the melting point, heat to melt the ice, heat to warm the water from the melting to the boiling point, and heat to vaporize the water.

3.6. Evaporation

"Boiling" means that the liquid is turning to gas everywhere - in a kettle of boiling water bubbles of water vapor are formed at the bottom, and stay gaseous while they rise to the surface.

But if we leave a glass of water uncovered the water will eventually "evaporate" - turn to gas - even at room temperature. The reason for this is that some of the molecules (at the surface) have high enough speeds and kinetic energies to break away from the forces between molecules keeping them in the liquid. The room temperature only says what the average kinetic energy is. When these fast-moving molecules are gone, the average kinetic energy decreases => the liquid is cooled, and then heat flows from its surroundings into it. This is why it feels colder to have wet clothes than dry ones.

The rate of this evaporation depends on several factors, like:

·        what liquid it is (what its L_v -value is): some liquids with low values evaporate quickly and therefore cool quickly; heat flows fast into them from the environment (alcohol or acetone on the skin feels colder than water, although all substances may have been taken from containers at room temperature).

·        the temperature: evaporation takes place at all temperatures, but faster with higher temperature (if you water the lawn, do it in the evening, in the day more of it will evaporate before it gets into the ground).

·        how much (for example) water vapor already is in the air: in a dry climate, sweat evaporates quickly, but in the jungle it stays on the skin. (Humidity-high water content)

This can be used to measure the humidity in air with a psychrometer (a dry thermometer and one with a moist gauze; the temperature difference can be used to find the relative air humidity).

3.7. Transporting thermal energy : conduction, convection, radiation

Conduction

If you put one end an iron rod into the fire, it will soon feel hot in the other end. This is because the heated atoms (or electrons in a metal) have higher average E_k some higher average speed, and in a series of collisions this Ek is spread through the rod. Different materials conduct heat faster or slower; try putting silver, steel and plastic spoons in a cup of hot tea.

[Not in the IB program anymore, but useful background information for investigations:

P = Q/t = (-)kA DT/Dx

where P = power of heat transfer = amount of energy Q transported in time t depends on k = thermal conductivity values for different materials, A = the cross section area of the conductor (that is, the area through which heat is conducted), DT = temperature difference between hot and cold end, Dx = length of rod or thickness of the material.].

Convection

In the Mexican gulf, ocean water is heated and flows in the "Gulfstream" to along Coastal America making the climate warmer than it would otherwise be. In a smaller scale, houses with central heating have an oil or natural gas burner where water is heated and pumped through the rooms where it is radiated out (see below). This means that heat is transported not by collisions within a material but by transporting the material itself (which then should have a high specific heat capacity, which is why water is suitable).

Radiation

The sun can heat the earth without being in touch with it (conduction) or letting material flow from the earth to the sun (some small amounts of particles do flow, causing aurora borealis, but this does not significantly heat the planet). This energy is transported in the form of electromagnetic (EM) radiation, which is explained more in the Waves section. For now: many types of radiation are of this sort, including light, infrared and radio waves.

The earth can then radiate some heat out in space in the night (unless the atmosphere only lets sunlight through, and not the radiation from earth = greenhouse effect).

[The reason for this is that the typical wavelength for the radiation depends on the temperature of the object which is radiating (a rather hot metal emits invisible radiation, a hotter one red light, an even hotter white light), and the sun surface is much hotter than the earth's; this is explained more in connection with the l_max = k/T, k = 2.90 x 10^-3 m, formula from Astrophysics. Molecules in the atmosphere may stop radiation of some wavelengths better than others]

·        Shiny objects radiate heat less than black or dark ones; the same goes for absorbing radiation. This is why thermos flasks are shiny.

·        The hotter an object gets, the higher the power of radiation (radiated energy per unit time), but this is not directly proportional to the temperature T, rather to T⁴.

· The larger the area radiating, the higher the power. This is why motorcycle engines are shaped to increase the area in contact with air.

[See the L = sAT⁴ formula from Astrophysics where L is a type of power.]

3.8. Ideal gas law

Pressure

If some force F (which can be the result of gaseous atoms colliding with the surface or other) acts on a surface with the area A (perpendicular to the surface) the pressure p (a scalar quantity) is

p = F/A [DB p.6]

has the unit 1 Nm^-2 = 1 pascal = 1 Pa. Other units : 1 bar = 100 000 Pa, 1 millibar = 1mb = 100 Pa, 1 atmosphere = 1 atm (about ordinary air pressure) = 1013 mb = 101.3 kPa.

1 atm = 1.01 x 10⁵ Nm^-2 =101 kPa = 760 mmHg [DB p. 2]

Macroscopic gas laws

· Boyle's law: If you open a container with gas under high pressure, it fills the room and therefore the volume V increases and the pressure drops. Increased V gives smaller p, so p = k/V for some constant k

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This gives a hyperbola in a p-V graph (compare to y = 1/x, y = 2/x etc.)

· Charles law: If an amount of gas is heated it expands, ex. heating the air in a hot air balloon. When the same mass of air gets a higher volume V the density decreases; this is why hot air is "lighter" than cold and rises upwards. Higher T gives higher V so V = kT for some other constant

 t07b

This gives a straight line in a V-T graph. Since there can be no negative volume, the point where the V-graph hits the T-axis is the lowest possible temperature: 0 K = -273^oC.

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· Gay-Lussac's - Pressure or Admonton law: If an amount of gas is heated and is in a rigid container so the volume cannot increase, the pressure will rise. Ex. if you throw a spray can in the fire the gas in it is heated and the pressure increases until it explodes (do not do this at home!). Higher T gives higher p so p = kT for some constant k so :

This gives a straight line in a p-T graph. Since there can be no negative pressure, the point where the p-graph hits the T-axis is the lowest possible temperature: 0 K = -273^oC.

All these laws can be summed up in one formula: Known as the “Universal Gas Law”

pV/T = constant which means p₁V₁/T₁ = p₂V₂/T₂ = p₃V₃/T₃ = ... as long as the amount of gas is the same

If the amount of gas changes - some of leaks out or some is inserted - then we must take into account how many atoms or molecules we have, which is done using the chemical quantity amount of substance = n in the unit mole. For a mass m (exceptionally here in the unit g, not kg !!!) of gas with the molar mass M in the unit gmol^-1 we have:

n = m/M

The amount of substance n in moles is related to the number N of atoms or molecules we have via Avogadros number N_A = 6.02 x 10²³ :

N = n N_A

It can be found that the ideal gas law is pV/T = nR or :

pV = nRT [DB p.6]

where the ideal gas constant R = 8.31 JKmol^-1

All this is based on a model of an ideal gas, which means:

·        many small gas atoms are assumed to move in straight lines in random directions

·        they change direction only when colliding with each other and the container walls

·        the collisions are assumed to be perfectly elastic = not only momentum but also kinetic energy is conserved (important!)

Under such assumptions, the gas laws can also be supported theoretically, ex.

* if you increase the volume, the atoms have a longer distance to move between collisions with the wall => fewer collisions happen in a certain time which means less force acts on a specific area of the walls => the pressure has decreased (p = k/V)

* if the gas is heated, then the average kinetic energy of the atoms goes up, therefore the average speed goes up, which means that we either get more collisions and higher pressure (p = kT) or, if we keep the pressure constant, the volume must increase V = kT).

3.9. Thermodynamics (first law): heat and work

First law of thermodynamics

We now study the flow of energy or work between a thermodynamical system - some object or device or amount of gas or liquid that we investigate - and its surroundings. The basic rule is that

Energy does not appear from nowhere or disappear into nothing

or the principle of energy conservation (conservation = the same totally before and after.) To formulate it mathematically we use these quantities:

DQ = thermal energy transferred (positive when into the system, negative out) of the system

DU = change in internal energy (positive when increases, negative for decrease)

DW = work done (positive when done by the system, negative when done on the system)

We then have:

DQ = DU + DW [DB p.6]

This rule may be easier to understand in the mathematically equivalent form

DU =DQ - DW

·        U is the total internal energy, the sum of all kinetic and potential energies of the atoms in the system we investigate

· DU is the change in this. What (work or energy) comes into the system and what goes out of it must either balance out (be zero when added) or result in a change in U which is positive if more work or energy comes than goes out, otherwise negative

Take a case where the U is constant => DU = 0 (which means that the temperature is constant), for example a steam engine where heat is flowing in and the engine does work on the wheels to move a train. If DU = 0, this can go on for hours without overheating the engine or having it cool off.

·        since the DQ is positive for energy flowing in, the formula is suitable if the work "flowing out" (being done by the system on something else) is subtracted. With a minus sign in the formula but a positive value for the DW, the sum can be 0.

[It would have been possible to agree on a different sign system and let DU = DQ + DW with the simple sign rule for both Q and W that everything "into" the system is positive and everything "out of" it is negative]

3.10. Thermodynamic processes

Work done by a moving piston

Suppose we have a gas at the pressure p in a cylindrical container with a mobile wall (ex. a car engine cylinder with a piston).

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·        since p = F/A we have the force F = pA on the mobile wall

·        when it is moved the distance s by F, the work W = Fs is done

·        the change in the volume of the gas is then DV = As

·        W = Fs = pAs = pDV if p is constant, or if we use the symbol DW for W:

DW = pDV [DB p.6]

Isobaric process

In a pV-diagram we have V on the x-axis and p on the y-axis.

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The work done in a process is the area under its graph in a pV-diagram

If p is constant then the process is isobaric ("same pressure" recall the alternative pressure unit 1 bar).

·        The graph is a horizontal line, the area is a rectangle.

Isochoric process

If the volume is constant, then the process is isochoric (the piston remains in the "same place", compare to "choreography" - describing how dancers move)

·        The graph is a vertical line, the area under = the work done = 0

Isothermal process

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If the temperature is kept constant, the process is isothermal, and since pV = nRT we get p = nRT/V which with constant T gives p = constant/V which gives a hyperbola graph (compare to y = 1/x, y = 2/x etc).

·        The graph is a hyperbola, the work done = the area under it, found using integration or numeric approximation.

 [Integration gives that if the volume increases from V₁ to V₂ the work is W = nRT ln (V₂/V₁) ]

Other processes

The types of processes mentioned here are special cases - in real engines the processes may show some other curves in the pV-diagram, where the area under the graph would be the work done - often only found with numeric integration.

Adiabatic process

One process which is not isobaric, isochoric or isothermal is the adiabatic process.

A process is adiabatic if no heat (energy) Q flows into or out of the system

Recall that DQ = DU + DW where now Q or as we may call it DQ = 0 giving DU = - DW or just

DU = - W for adiabatic process

Using the earlier mentioned sign rules this means than we can have:

 [The sign rules were:

DQ = thermal energy transferred (positive when into the system, negative out)

DU = change in internal energy (positive when increases, negative for decrease)

DW = work done (positive when done by the system, negative when done on the system)]

· adiabatic expansion, where the gas in the piston does the work W on something else, so W is positive, and DU is negative, which means that the total kinetic energy must go down, so the average kinetic energy goes down, so the temperature goes down

("joules go out of the gas as work but none come in