BIT1002 Heat

Cooling of a coffee container from Alain Perronnet http://www.ann.jussieu.fr/~perronnet

Macroscopic Heat Heat capacity Latent Heat Thermal Expansion Heat transfer

Thermal Properties of Matter

• Heat Capacities and specific heats
• Latent Heat
• Thermal Expansion

• Experimentally, if we supply heat to a solid, we see is a curve like

Specific Heat

Specific heat is heat capacity for 1 kg.

• e.g. water requires 4.169 KJ/kg to raise temp. of water 1⁰C, so spec. heat of water is 4169 J kg^-1C^-1..
• No need for solid spec. heat to be the same as liquid.
• General defn. is \color{red}{C = \frac{{\delta Q}}{{\delta T}}} . Energy goes into raising Internal Energy U.
• We actually learn more by looking at the energy required for a single molecule: hence molar heat capacity (heat capacity/g mole) is
• \color{red}{ c = \frac{1}{n}\frac{{\delta Q}}{{\delta t}}}
  where n is the number of molecules Spec. heat c = C/molecular weight
• e.g. for He, C = 12.5 J mole^{-1 0}C^-1 , c = 12.5/4 = 3.1 J g^-1 = 3100 J kg^{-1 0}C^-1
• Note (Important for a gas) this requires that the volume is held constant: (i.e. it is C_v )

Latent Heat:

Heat required to change phase

Note that at certain temperatures the addition of heat doesn't change the temp, but does change the material from a solid to a liquid (Latent heat of fusion) a liquid to a gas (Latent Heat of Vaporisation).

• e.g. water
• L_fusion = 333 KJ kg^{-1 0}C^-1
• L_boiling =2255 KJ kg^{-1 0}C^-1
• e.g. 30 g Ice at -10⁰C is added to 100 g Scotch at 20⁰C. What is the final temperature, and how much of the ice has melted? (Assume Scotch is water!)

Thermal Expansion

Most materials expand when heated: Arises because the force-law between atoms is not symmetrical. Hot (i.e. energetic) atoms will be slightly further apart on average than cold ones. \color{red}{ \frac{{\delta L}}{L} = \alpha \delta T}

• α= 10^-5 K^-1 for most metals
• Volume coef of expansion = 3 α (why?) Most materials have α roughly indep of T

• Water has a bizarre behaviour,being most dense at 4°C. Note water is also peculiar in that it expands when frozen ⇒ severe difficulties for biological systems.

• Does it run fast or slow at 30⁰C.
• By how much (α = 2x10^-6)

Heat Transfer

Three basic mechanisms

• Conduction : all materials, inefficient for gases
• Radiation : Works in vacuum, best at high temps
• Convection : liquids and gases only
• and Evaporation

Conduction

• Energetic atoms in one place share energy with cooler neighbours Note that it takes time for energy to be transferred.

• Heat flow \color{red}{H = \frac{{\delta Q}}{{\delta t}}}. Would guess
  • Heat flow gets reduced if distance is increased \color{red}{H \propto \frac{1}{L}}
  • Heat flow gets increased if area is increased: \color{red}{H \propto A}
  • Heat flow gets increased if temp diff. is increased: \color{red}{H \propto \delta T}
• Which gives ries to
  \color{red}{ H = \frac{Q}{t} = \frac{{kA\left( {T_1 - T_0 } \right)}}{L}}
  k is thermal conductivity
• k ∼ 300 for metals

• If the outside wall is rock (k = .8), 20 cm thick and area is 15 m², what is the heat loss?
• How will this change if 5 cm of fibreglass is added?

  k ∼ .08 for fibreglass.

• R value is L/k: useful since R values can be added
• Units are K W^-1

Radiation

• Black-body radiation: the radiation emitted by all hot bodies is (almost) exactly the same. Must measure temperature in degrees absolute (K).

  T(K) = T(oC) + 273
  

• so that room temperature (~20^oC) is ~ 300 K. For stars T ~ 6000 K ⇒ 20,000 K so doesn't matter whether we use K or ^oC!
• Two fundamental laws: Stefan- Boltzmann law

  Total Energy \color{red}{U = \sigma T^4 ,\sigma = 5.67 \times 10^{ - 8} Jm^{ - 2} K^{ - 4} }

• Wien's law:
  Wavelength of peak \color{red}{\lambda _{\max } = \frac{B}{T},B = 2.9 \times 10^ {- 3} mK^{ - 1} }; i.e. as we heat up objects, they go

  black → red → orange → yellow     → white

  

• What does this look like?

Putting these together gives black-body curve:

• How much energy do you take in per day?
• You appear to radiate 10 times as much energy as you take in each day in food. Why is the answer stupid?
  1. You actually wear clothes, and this stops the radiation.
  2. Everything around you is at almost exactly the same temperature as you are, so it radiates back almost the same amount of energy.
  3. You can't use conservation of energy in this way: in particular the energy you take in via food is quite different from the energy that you radiate.
  4. This would be true at night, when there is no radiation hitting you from the sun.
• Note that this assumes that the body is a perfect radiator: a black-body.
• e.g. what is λ_max for a human?

• Sun is 7x10⁵ km in radius, temp. of 5800 K. How much energy does it radiate?
• How much is absorbed by earth?
• What temp. would earth be at to re-radiate this?

• Radiation and the Greenhouse effect

• The Earth is not a perfect B-B absorber/radiator: the atmosphere is opaque to some wavelengths, in particular I-R is strongly absorbed around 1-10µM.

• This is where a BB at 300⁰K peaks.
• Sun has temp. of 5800⁰K: where does it peak?
• i.e. earth's atmosphere is transparent to incident solar radiation, opaque to terrestrial, so the Earth stays at higher temp.
• This is the "Greenhouse Effect" however, note that greenhouses do not work by the Greenhouse effect!
• Increasing CO₂ and (in particular) organic gases such as methane increase I.R. absorption

Convection:

• Note that hot fluids have lower density, hence will rise in a gravitational field (Archimedes). Hence steady circulation set up, which removes heat from centre

  RIchard Pogge Ohio State

• In still air, H = q A δT, where q depends on everything.

• For naked humans q = 7.1 W m^-2 K^-1 e.g. what is heat loss if T = -30⁰C?

• In reality, almost every process invloves all of the effects: e.g. waiting for your coffee to cool

• Note real heat transfer requires computers
• Finally we want to look at thermodynamics