Thermal Physics
the number of molecules contained in one mole of any gas is Avogardo's number, Na = 6.02 X 10^23 particles/mol, so
n = N/Na where n is the number of moles and N is the number of molecules in the gas
we see that Avogardo's number is deliberately chosen to be the inverse of the
number of grams in an atomic mass unit in this way the atomic mass of an atom expressed in atomic mass units is numerically the same as the mass of an Avogardo's number of that kind of atom expressed in grams
the temperature at the triple point of water on the Kelvin scale occurs at 273.16°C therefore, the SI unit of temperature, on the Kelvin scale
occurs at 273.16 K therefore, the SI unit of temperature, the kelvin, is defined as 1/272.16 of the temperature of the triple point of water the figure shows the Kelvin temperatures for various physical processes and structures absolute zero has been closely approached but never achieved
thermal physics
the study of temperature, heat, and how they affect matter quantitative descriptions of thermal phenomena require careful definitions of the concepts of temperature, heat, and internal energy heat leads to changes in internal energy and thus to changes in temperature, which cause the expansion or contraction of matter
formal definition of heat
the transfer of energy between a system and its environment due to a temperature difference between them the symbol Q is used to represent the amount of energy transferred by heat between a system and its environment
convection
the transfer of energy by the movement of a substance natural convection forced convection
in general, metals are good conductors because
they contain large numbers of electrons that are relatively free to move through the metal and can transport energy from one region to another
the molecules interact only through short-range forces during elastic collisions
this assumption is consistent with the ideal gas model, in which the molecules exert no long-range forces on each other
thermometers are devices
used to measure the temperature of an object or a system when a thermometer is in thermal contact with a system, energy is exchanged until the thermometer and the system are in thermal equilibrium with each other for accurate readings, the thermometer must be much smaller than the system, so that the energy the thermometer gains or loses doesn't significantly alter the energy content of the system
for every gas
vmp < vav < vrms
the square root of v^2 is called the root-mean-square (rms) speed of the molecule
vrms = √v^2 = √3kbT/√m = √3RT/√M M is the molar mass in kilograms per mole, if R is given in SI units at a given temperature, lighter molecules tend to move faster than heavier molecules
natural convection
when the movement results from differences in density, such with air around fire airflow at a beach mixing that occurs as surface water in a lake cools and sinks
forced convection
when the substance is forced to move by a fan or pump, as in some hot air and hot water heating systems
relationship between changes in temperature on the Celsius and Fahrenheit scales
ΔTf = 9/5ΔTc
when the temperature decreases
T and Q are negative, and energy flows out of the system
when the temperature increases
T and Q are positive, corresponding to energy flowing into the system
Celsius temperatures in terms of Fahrenheit temperatures
Tc = 5/9(Tf - 32)
the relationship between Celsius and Kelvin scales
Tc = T - 273.15
the relationship between the Celsius and Fahrenheit temperature scale is
Tf = 9/5Tc + 32 for example, a temperature of 50.0°F corresponds to a Celsius temperature of 10.0°C and absolute temperature of 283 K
absolute zero is used as the basis for the
Kelvin temperature scale, which sets -273.15°C as its zero point (0 K) aka absolute temperature
latent heat of fusion
Lf is used when a phase change occurs during melting or freezing
latent heat of vaporization
Lv is used when a phase change occurs during boiling or condensing
in a gas thermometer, the pressure extrapolates to zero when the temperature is
-273.15°C this fact suggests that this particular temperature is universal in its importance, because it doesn't depend on the substance used in the thermometer in addition, because the lowest possible pressure is P = 0, a perfect vacuum, the temperature -273.15°C must represent a lower bound for physical processes we define this temperature as absolute zero
the temperature of the water-steam mixture is defined as
100°C, called the steam point or boiling point of water once the ends of the liquid column in the thermometer have been marked at these two points, the distance between marks is divided into 100 equal segments, each corresponding to a change in temperature of one degree Celsius
1 cal
4.186 J mechanical equivalent of heat
a mole is a number
the same number of particles is found in a mole of helium as in a mole of iron or aluminium this number is known as Avogardo's number
molar specific heat at constant volume
Cv = 3R/2
total kinetic energy of N molecules
KEtotal = N(mv^2/2) = 3NkbT/2 = 3nRT/2 total translational kinetic energy of a system of molecules is proportional to the absolute temperature of the system
total force on the wall
F = N/3(mv^2/d)
the most common temperature scale in use in the US is the
Fahrenheit scale it sets the temperature of the ice point at 32°F and the temperature of the steam point at 212°F
Avogardo's number
Na = 6.02 X 10^23 particles/mole
pressure of an ideal gas
P = F/A = F/d^2 = 1/3(Nmv^2/d^3) = 1/3(N/V)mv^2 P = 2/3(N/V)(mv^2/2) the pressure is proportional to the number of molecules per unit volume and to the average translational kinetic energy of a molecule
the rate of energy transfer by conduction through the rod
P = kA(Th - Tc)/L k, a proportionality constant that depends on the material, is called the thermal conductivity
Stefan's law
P = εσAT^4 the rate at which an object radiates energy is proportional to the fourth power of its absolute temperature P is the power in watts (or joules per second) radiated by the object, σ is the Stefan-Boltzmann constant, equal to 5.6696 X 10^-8 W/m^2 · K^4/m^2 · K^4, A is the surface area of the object in square meters, e is a constant called the emissivity of the object, and T is the object's Kelvin temperature the value of e can vary between zero and one, depending on the properties of the object's surface
temperature is proportional to average kinetic energy
PV = 2/3N(mv^2/2) PV = NkbT T = 2/3kb(mv^2/2) the temperature of a gas is a direct measure of the average molecular kinetic energy of the gas
ideal gas law
PV = nRT R is a constant for a specific gas that must be determined from experiments, whereas T is the temperature in kelvins each point on a P versus V diagram would represent a different state of the system experiments on several gases show that, as the pressure approaches zero, the quantity PV/nT approaches the same value of R for all gases for this reason, R is called the universal gas constant in SI units, where pressure is expressed in pascals and volume in cubic meters, R = 8.31 J/mol · K if the pressure is expressed in atmospheres and the volume is given in liters, then R = 0.0821 L · atm/mol · K using this value of R and equation, the volume occupied by 1 mol of any ideal gas at atmospheric pressure and at 0°C (273 K) is 22.4 L
we can rewrite the ideal gas law in terms of the total number of molecules as
PV = nRT = N/Na X RT or PV = NkbT where kb = R/Na = 1.38 X 10^-23J/K
energy Q needed to raise the temperature of a system of mass m by T
Q = mcT
the rate of energy transfer in homes is
Q/t = A(Th - TC)/ΣL/k = A(Th - Tc)/ΣR thickness of stagnant air must be considered (depends on the speed of the wind) British units can be converted by multiplying 0.1761
a gas with a larger molar specific heat requires
more energy to realize a given temperature change the size of the molar specific heat depends on the structure of the gas molecule and how many different ways it can store energy
Dewar flask
Thermos bottle designed to minimize energy transfer by conduction, convection, and radiation the insulated bottle can store either cold or hot liquids for long periods the space between the walls is evacuated to minimize energy transfer by conduction and convection the silvered surface minimizes energy transfer by radiation because silver is a very good reflector and has very low emissivity a further reduction in energy loss is achieved by reducing the size of the neck Dewar flasks are commonly used to store liquid nitrogen (boiling point 77 K) and liquid oxygen (boiling point 90 K)
internal energy
U the energy associated with the atoms and molecules of the system the internal energy includes kinetic and potential energy associated with the random translational, rotational, and vibrational motion of the particles that make up the system, and any potential energy bonding the particles together
internal energy U for a monatomic gas
U = 3nRT/2 for diatomic and polyatomic molecules, additional possibilities for energy storage are available in the vibration and rotation of the moleule
the expression for the internal energy of an ideal gas
U = 3nRT/2 valid only for monatomic gas
change in internal energy of an ideal gas
U = nCvT for ideal gases, this expression is always valid, even when the volume isn't constant the value of the molar specific heat, however, depends on the gas and can vary under different conditions of temperature and pressure
the work W done on a gas at constant pressure
W = -PV P is the pressure throughout the gas and V is the change in volume of the gas during the process if the gas is compressed, V is negative and the work done on the gas is positive if the gas expands, V is positive and the work done on the gas is negative
although we often picture an ideal gas as consisting of single atoms, molecular gases
exhibit ideal behavior at low pressures on average, effects associated with molecular structure have no effect on the motions considered, so we can apply the results of the following development to molecular gases as well as to monatomic gases
average kinetic energy per molecule
mv^2/2 = 3kbT/2
thermometer
a device calibrated to measure the temperature of an object
Maxwell velocity distribution
a system of gas at a given temperature will exhibit a variety of speeds
as an object radiates energy at a given rate, it also
absorbs radiation if it didn't the object would eventually radiate all its energy and its temperature would reach absolute zero the energy an object absorbs comes from its environment, which consists of other bodies that radiate energy if an object is at a temperature T and its surroundings at a temperature T0, the net energy gained or lost each second by the object as a result of radiation is Pnet = εσA(T^4 - T0^4)
thermal conduction
aka conduction the energy transfer process most closely with a temperature difference in this process the transfer can be viewed on an atomic scale as an exchange of kinetic energy between microscopic materials—molecules, atoms, and electrons—with less energetic particles gaining energy as they collide with more energetic particles
zeroth law of thermodynamics
aka law of equilibrium if objects A and B are separately in thermal equilibrium with a third object, then A and B are in thermal equilibrium with each other
radiation
all objects radiate energy continuously in the form of electromagnetic waves due to thermal vibrations of their molecules these vibrations create the orange glow of an electric stove burner, an electric space heater, and the coils of a toaster
the pressure P, volume V, temperature T and amount of n of gas in a container are related to each other by
an equation of state can be very complicated, but is found experimentally to be simple if the gas is maintained at a low pressure (or a low density)
ideal reflector
an object for which e = 0 absorbs none of the energy incident on it, reflecting it all
an ideal absorber
an object that absorbs all the light radiation incident on it, including invisible infrared and ultraviolet light aka black body because a room-temperature black body would look black because a black body doesn't reflect radiation at any wavelength, any light coming from it is due to atomic and molecular vibrations alone a perfect black body has emissivity e = 1 an ideal absorber is also an ideal radiator of energy the Sun, for example, is nearly a perfect black body this statement may seem contradictory because the Sun is bright, not dark; the light that comes from the Sun, however, is emitted, not reflected black bodies are perfect absorbers that look black at room temperature because they don't reflect any light all black bodies, except those at absolute zero, emit light that has a characteristic spectrum
what would happen to a substance if its temperature could reach 0 K?
as figure indicates, the substance would exert zero pressure on the walls of its container (assuming the substance doesn't liquefy or solidify on the way to absolute zero) we show that the pressure of a gas is proportional to the kinetic energy of the molecules of that gas according to classical physics, therefore, the kinetic energy of the gas would go to zero and there would be no motion at all of the individual components of the gas according to quantum theory, however, the gas would always retain some residual energy, called the zero-point energy, at that low temperature
the number of molecules in the gas is large, and the average separation between them is large compared with their dimensions
because the number of molecules is large, we can analyze their behavior statistically the large separation between molecules means that the molecules occupy a negligible volume in the container this assumption is consistent with the ideal gas model, in which we imagine the molecules to be pointlike
we can construce practical thermometers such as the thermometer
but these types of thermometers don't define temperature in a fundamental way
the molecules obey Newton's laws of motion, but as a whole they move randomly
by "randomly" we mean that any molecule can move in any direction with equal probability, with a wide distribution of speeds
thermals
created when a portion of the Earth reaches higher temperature than neighboring regions
conduction occurs only if there is a
difference in temperature between two parts of the conducting medium the temperature difference drives the flow of energy consider a slab of material of thickness x and cross-sectional area A with its opposite faces at different temperatures Tc and Th, where Th > Tc the slab allows energy to transfer from the region of higher temperature to the region of lower temperature by thermal conduction the rate of energy transfer, P = Q/t is proportional to the cross-sectional area of the slab and the temperature differeernce and is inversely proportional to the thickness of the slab P = Q/t ∝ AT/x
thermometers calibrated this way present problems when
extremely accurate readings are needed for example, an alcohol thermometer calibrated at the ice and steam points of water might agree with a mercury thermometer only at the calibration points because mercury and alcohol have different thermal expansion properties, when one indicates a temperature at 50°C, say, the other may indicate a slightly different temperature the discrepancies between different types of thermometers are especially large when the temperatures to be measured are far from the calibration points
suppose an ideal gas is confined to a cylindrical container with a volume that can be changed by moving a piston
first, when the gas is kept at a constant temperature, its pressure is inversely proportional to its volume (Boyle's law) second, when the pressure of the gas is kept constant, the volume of the gas is directly proportional to the temperature (Charle's law) third, when the volume of the gas is held constant, the pressure is directly proportional to the temperature (Gay-Lussac's law)
Celsius temperature sclae
formerly called the centigrade scale on the Celsius scale, the temperature of the ice-water mixture is defined to be zero degrees Celsius, written 0°C and called the ice point or freezing point of water
if a pan of water is heated on the burner of a stove, it is incorrect to say more heat is in the water
heat is the transfer of thermal energy, just as work is the transfer of mechanical energy when an object is pushed, it doesn't have more work; rather, it has more mechanical energy transferred by work similarly, the pan of water has more thermal energy transferred by heat
temperature is commonly associated with
how hot or cold an object feels when we touch it while our senses provide us with qualitative indications of temperature, they are unreliable and often misleading
a low-density approximates what is called an
ideal gas most gases at room temperature and atmospheric pressure behave approximately as ideal gases
specific heat
if a quantity of of energy Q is transferred to a substance of mass m, changing its temperature by T, the specific heat c of the substance is defined by c = Q/mT SI unit: Joule per kilogram-degree Celsius (J/kg · °C)
first law of thermodynamics
if a system undergoes a change from an initial state to a final state, then the change in the internal energy U is given by U = Uf - Ui = Q + W where Q is the energy exchanged between the system and the environment, and W is the work done on the system
two objects are in thermal contact
if energy can be exchanged between them
two objects are in thermal equilibrium
if they are in thermal contact and there is no net exchange of energy
phase change
in some cases the transfer of energy doesn't result in a change in temperature this can occur when the physical characteristics of the substance change from one form to another ex) solid to liquid (melting), liquid to gas (boiling), and a change in the crystalline structure of a solid
we showed that the internal energy of a monatomic ideal gas is associated with the translational motion of its atoms
in this special case, the internal energy is the total translational kinetic energy of the atoms; the higher the temperature of the gas, the greater the kinetic energy of the atoms and the greater the internal energy of the gas for more complicated diatomic and polyatomic gases, internal energy includes other forms of molecular energy, such as rotational kinetic energy and potential energy associated with molecular vibrations internal energy is also associated with the intermolecular potential energy ("bond energy") between molecules in a liquid or solid
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 pointlike, occupying negligible volume
one mole (mol) of any substance
is that amount of substance that contains as many particles (atoms, molecules, or other particles) as there are atoms in 12 g of the isotope carbon-12 taking carbon-12 as a test case, let's find the mass of an Avogardo's number of carbon-12 atoms a carbon-12 atom has an atomic mass of 12 u, or 12 atomic mass units one atomic mass unit is equal to 1.66 X 10^-24 g, about the same as the mass of a neutron or proton—particles that make up atomic nuclei the mass m of an Avogardo's number of carbon-12 is then given by m = Na(12 u) = 6.02 X 10^23 (12 u)(1.66 X 10^-24/u) = 12 g
the principle of conservation of energy for this isolated system requires that the net result of all energy transfers
is zero if one part of the system loses energy, another part has to gain the energy because the system is isolated and the energy has nowhere else to go when a warm object is placed in the cooler water of a calorimeter, the warm object becomes cooler while the water becomes warmer Qcold = -Qhot
when an object is in equilibrium with its surroundings
it radiates and absorbs energy at the same rate, so its temperature remains constant when an object is hotter than its surroundings, it radiates and absorbs energy at the same rate, so its temperature remains constant when an object is hotter than its surroundings, it radiates more energy than it absorbs and therefore cools
Boltzmann's constant
kb = R/Na = 1.38 X 10^-23J/K
calorimeters and calorimetry
one technique for measuring the specific heat of a solid or liquid is to raise the temperature of the substance to some value, place it into a vessel containing cold water of known mass and temperature, and measure the temperature of the combination after equilibrium is reached define the system as the substance and the water if the vessel is assumed to be a good insulator, so that energy doesn't leave the system, then we can assume the system is isolated vessels having this property is called calorimeters, and analysis performed using such vessels is called calorimetry
isobaric process
process in which the pressure remains constant during the expansion or compression
temperature
property that defines whether or not an object is in thermal equilibrium with other objects
two objects in thermal equilibrium with each other are at the
same temperature
all thermometers use of some physical property that changes with temperature and can be calibrated to make the temperature measurable
some of the physical properties used are (1) the volume of a liquid, (2) the length of a solid, (3) the pressure of a gas held at constant volume, (4) the volume of gas held at constant pressure, (5) the electric resistance of a conductor, and (6) the color of a very hot object
thermal conductivity
substances that are good conductors have large thermal conductivities, whereas good insulators have low thermal conductivities
because the size of a Celsius degree is the same as a kelvin, a temperature difference of 5°C is equal to a
temperature difference of 5 K the two scales differ only in the choice of zero point the ice point (273.15 K) corresponds to 0.00°C, and the steam point (373.15 K) is equivalent to 100.00°C
thermography
the amount of energy radiated by an object can be measured with temperature sensitive recording equipment via a technique called thermography an image of the pattern formed by varying radiation levels, called thermogram, is brightest in the warmest areas
a gas usually consists of a very large number of particles, so it's convenient to express
the amount of gas in a given volume in terms of the number of moles, n
PV diagram
the area under the graph in a PV diagram is equal in magnitude to the work done on the gas
latent heat
the energy Q needed to change the phase of a given pure substance is Q = +-mL where L, called the latent heat of the substance, depends on the nature of the phase change as well as on the substance unit J/kg
calorie
the energy necessary to raise the temperature of 1 g of water from 14.5°C to 15.5°C the Calorie, with a capital C, used in describing the energy content of foods, is actually a kilocalorie
British thermal unit (Btu)
the energy required to raise the temperature of 1 lb of water from 63°F to 64°F
heat
the exchange of energy between two objects because of differences in their temperature
a procedure based on two new points was adopted in 1954 by the International Committee on Weights and Measures
the first point is absolute zero the second point is the triple point of water, which is the single temperature and pressure at which water, water vapor, and ice can coexist in equilibrium this point is a convenient and reproducible reference temperature for the Kelvin scale; it occurs at a temperature of 0.01°C and a pressure of 4.58 mm of mercury
one thermometer, however, is more fundamental, and offers a way to define temperature and relate it directly to internal energy
the gas thermometer in a gas thermometer, the temperature readings are nearly independent of the substance used in the thermometer one type of gas thermometer is the constant-volume unit the behavior observed in this device is the variation of pressure with temperature of a fixed volume of gas when the constant-volume gas thermometer was developed, it was calibrated using the ice and steam points of water as follows (a different calibration procedure, to be discussed shortly, is now used) the gas flask is inserted into an ice-water bath, and mercury reservoir B is raised or lowered until the volume of the confined gas is at some value, indicated by the zero point on the scale the height h, the difference between the levels in the reservoir and column A, indicates the pressure in the flask at 0°C the flask is inserted water at the steam point, and reservoir B is readjusted until the height in column A is again brought to zero on the scale, ensuring that the gas volume is the same as it had been in the ice bath (hence the designation "constant-volume") a measure of a new value for h gives a value for the pressure at 100°C these pressure and temperature values are then plotted on a graph
Avogardo's number was chosen so that
the mass in grams of one Avogardo's number number of an element is numerically the same as the mass of one atom of the element, expressed in atomic mass units (u) this relationship is very convenient looking at the periodic table of the elements, we find that carbon has an atomic mass of 12 u, so 12 g of carbon consists of exactly 6.02 X 10^23 atoms of carbon the atomic mass of oxygen is 16 u, so in 16 g of oxygen there are again 6.02 X 10^23 atoms of oxygen the same holds true for molecules the molecular mass of hydrogen, H2, is 2 u, and there is an Avogardo's number of molecules in 2 g of molecular hydrogen
because there are 6.02 X 10^23 particles in one mole of any element
the mass per atom for a given element is m = molar mass/Na
assumptions of kinetic theory of gases
the number of molecules in the gas is large, and the average separation between them is large compared with their dimensions the molecules obey Newton's laws of motion, but as a whole they move randomly the molecules interact only through short-range forces during elastic collisions the molecules make elastic collisions with the walls all molecules in the gas are identical
Avogardo's number and the definition of a mole are fundamental to chemistry and related branch of physics
the number of moles of a substance is related to its mass by the expression n = m/molar mass where the molar mass of the substance is defined as the mass of one mole of that substance, usually expressed in grams per mole
