Thermal Properties of Matter
物质的热性质
Chapter 17 Temperature & Heat
Young & Freedman
1
1.
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Thermal Expansion
Linear Expansion
L0 is the original length
α is the coefficient of linear expansion
∆L is the expanded length
L is the linear dimension, it can be the thickness of the rod, the side of the square sheet,
or the diameter of a hole.
Note:
Some materials, such as wood or single crystals expand differently in different situation. We won’t consider
such cases.
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1.
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Thermal Expansion
The intermolecular force in a solid behave like a spring.
When the temperature increases, the energy and amplitude of the vibration also
increase.
As a result the average distance between atoms also increases.
As the atom get farther apart, every dimension increases.
Lennard-Jones potential
3
1.
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Thermal Expansion
The direct proportionality is not exact; it is approximately correct for sufficiently small
changes in temperature.
For a given material, α depend on the initial temperature T0 and the size of the temperature
interval ∆T.
We will ignore these complication here.
4
1.
Thermal Expansion
5
1.
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Thermal Expansion
Volume Expansion
• Experiments shows that if the temperature change ∆T is not too great (< 1000C)
V0 is the original volume
β is the coefficient of volume expansion
∆V is the expanded volume
It can be shown that
β = 3α
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1.
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Thermal Expansion
Area Expansion
Similarly, for isotropic materials (materials that expand equally in all directions)
∆A = γA0∆T
A0 is the original area
Γ is the coefficient of area expansion
∆A is the expanded area
It can be shown that
γ = 2α
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2.
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Peculiar Nature of Water
Most matter, expand when temperature increases and contract when temperature
decreases.
Water in the temperature range from 00C to 40C, decreases in volume with
increasing temperature.
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3.
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Thermal Stress 热应力
If we clamp the ends of rod rigidly to prevent expansion or contraction and then
change the temperature, thermal stresses develop.
The rod would like to expand or contract but the clamp won’t let it.
The resulting stresses may become large enough to strain the rod irreversibly or
even break it.
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3.
Young Modulus 杨氏模量
10
3.
Thermal Stress 热应力
4.
Specific Heat Capacity
The specific heat capacity (c) of a substance is the quantity of heat required to raise
the temperature of a unit mass of the substance by 1 Kelvin.
Heat Capacity (C’) of a substance is the
amount of heat required to raise the
temperature of the substance by 1 K
Q = C’∆T ⟹
C’ = mc
4.1
Solid
Determination of Specific Heat Capacity
4.1
Determination of Specific Heat Capacity
Liquid
M = Mass of calorimeter
m = mass of liquid
5.
Specific Latent Heat
5.
Specific Latent Heat
• The heat required to change a unit mass of a solid to a liquid at constant temperature
is known as the specific latent heat of fusion of the solid.
• The heat required to change a unit mass of a liquid to a gaseous phase at constant
temperature is known as the specific latent heat of vapourization of the liquid.
5.
Determination Specific Latent Heat of Fusion
5.
Determination Specific Latent Heat of Vaporization
5.1
Molar Heat Capacity
Molar heat capacity is the amount of heat needed to raise one mole of the substance by
1 Kelvin :
m : Total mass of the substance
n : number of moles
M : Molar mass
c : specific heat capacity
C : molar heat capacity
M : Molar mass
6.
Mechanism of Heat Transfer
The three mechanism of heat transfer are:
(i) Conduction 传导 (solid) – Through vibration of molecules
(ii) Convection 对流 (liquid & gaseous) – Through movement of molecules from one
region in the body to another in clusters.
(iii) Radiation 辐射 – Through electromagnetic radiation (infra-red) without a need of
medium.
6.1
Conduction 传导
Temperature Gradient
k = Thermal conductivity
6.1
Thermal Resistance
L
6.2
Convection 对流
Navier-Stokes Equation (Not in syllabus) – Fluid Mechanics 流体力学
Solving the Navier–Stokes existence and smoothness problem offers a $1
million prize from the Clay Mathematics Institute as part of its Millennium
Prize Problems.
6.2
Radiation 辐射 (Infra –red 红外线)
Emissivity
0≤𝑒≤1
k : Boltzmann constant; c : speed of light ; h : Planck constant
6.2
Radiation and Absorption
• While a body at absolute temperature T is radiating, its surrounding temperature Ts is also
radiating, and the body absorbs some of this radiation.
• The net rate of radiation from a body at temperature T with surroundings at temperature Ts is
• At thermal equilibrium T = Ts and the rates of radiation and absorption must be equal and the net
radiation is zero.
• A positive net H means a net heat flow out of the body.
• For conduction and convection the current depends on the temperature difference between the
two bodies.
Emissivity
0≤𝑒≤1
k : Boltzmann constant; c : speed of light ; h : Planck constant
6.2
Blackbody 黑体
Emissivity
0≤𝑒≤1
Inside thermoflask
Shiny body
6.2
Stefan-Boltzmann Law Derivation (Not in syllabus)
Emissivity
0≤𝑒≤1
The law can be derived from Planck’s law on blackbody surface:
k : Boltzmann constant;
c : speed of light ;
h : Planck constant
Ultraviolet Catastrophe
紫外灾变
where I is the intensity and 𝜐 is the frequency of radiation.
Note: Maclaurin expansion of exponential function.
𝑥2
𝑥3
𝑥𝑟
𝑒 𝑥 = 1 + 𝑥 + 2! + 3! + ⋯ + 𝑟! + ⋯
for all x
Classical radiation
intensity
7. Evaporation 蒸发 vs Boiling 沸腾
• Evaporation is a change of state from liquid to gas that takes place at the surface of a liquid and
at any temperature.
• The rate of evaporation increases with temperature.
• In a liquid, the molecules move in all directions at different speeds.
• They collide into each other and transfer of energy takes place.
• Some molecules gain energy and others lose energy.
• If a very fast molecule happens to be moving upwards when it is near the surface, it will have
enough energy to escape into the atmosphere and evaporation takes place.
• As a result, the remaining molecules in the liquid has lower average K.E. and thus a lower
temperature.
• Thus evaporation produces cooling.
• Perspiration in a human body helps to cool the body and maintain a stable body temperature.
• Evaporation of sweat removes heat from the body.
7. Evaporation 蒸发 vs Boiling 沸腾
• Boiling is a process where a liquid changes to a gas at the boiling point. It takes place throughout
the whole body of the liquid.
• During boiling, a change of phase from liquid to gas occurs beneath the liquid surface.
• The pressure of the gas in the bubble must exceed that of the surrounding water and atmospheric
pressure.
• The gas forms beneath the surface as bubbles which then buoyed upwards.
• Increase in atmospheric pressure will raise the boiling point. This is because the gas molecules
need more energy to overcome the bigger atmospheric pressure.
• Conversely lower the atmospheric pressure will decrease the boiling point.
• As a result, at high altitudes, water boils below 1000C and water boils above 1000C in a pressure
cooker.
7. Evaporation 蒸发 vs Boiling 沸腾
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Fig 13.11 shows a dish of water at room temperature inside a jar.
When the pressure inside the jar is slowly reduced by the vacuum pump, the water starts to boil.
On the other hand evaporation of the water cools the water left in the dish.
The water boils and at the same time its temperature drops due to evaporation until the freezing
point of water is reached.
• Boiling and freezing thus takes place simultaneously.
Cold Boiling!
8.
Newton’s Law of Cooling / Heating
Newton’s law of cooling states that the rate of temperature change of a body is directly proportional
to the temperature difference between the body and its surrounding, i.e.
𝑑𝑇
∝ ∆𝑇
𝑑𝑡
𝑇𝑜
𝑇 𝑡 = 𝑇𝑠 + 𝑇0 − 𝑇𝑠 𝑒 −𝑘𝑡
It can be shown that as a result
𝑇 𝑡 = 𝑇𝑠 + 𝑇0 − 𝑇𝑠 𝑒 −𝑘𝑡
T(t) is the body temperature at time t
T0 is the body initial temperature
Ts is the surrounding temperature
k is the cooling constant of proportionality
𝑇𝑠
8.
Newton’s Law of Cooling / Heating
In a similar way, it can be shown that the heating equation is
𝑇𝐸
𝑇𝑜
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