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Heat and Thermal Energy

Heat  an energy transfer that occurs because of a difference in temperature.

Internal Energy  the energy a substance has because of its temperature.

Energy may be transferred between two objects without heat flow.

Example: rubbing two coins together. Both internal energies are increased due to
mechanical work but both remain in thermal equilibrium throughout.

Units of Heat

    Calorie  the amount of heat energy required to raise the temperature of 1 gram
     of H2O from 14.50C to 15.50C. A food calorie is 103 "physics" calories

    BTU - the amount of heat energy required to raise the temperature of 1 lb. of
     H2O from 630F to 640F.

    Joule  the S.I. unit of heat and work.

1 calorie = 4.186 Joules = 3.9  10-3 Btu

Example: A student eats a 1000 food calorie meal consisting of pizza, soda, and
snacks. How many 100kg clean & jerks (225 lbs.) must they do to work off this meal
just by lifting weights (neglecting metabolism)?

Work required is:

1000 food cal  103 cal = 1  106 calories = 4.186  106 J

The work done lifting weights is against gravity  w = mgh. For n repetitions
through some height h, w = nmgh. Assume h = 2.0 meters.

           4.186  10 6 J
n=                                 2136reps!
     (100kg )(9.8m  s - 2 )(2.0m)

How far would you have to run uphill to burn off these calories (neglecting
metabolism)?

4271 meters!
Heat Capacity and Specific Heat

   The heat capacity (C) of a substance is the amount of heat energy required to
    raise the temperature of a given mass of the substance by some amount. Heat
    capacity varies from substance to substance.

       The specific heat (C/m) of a substance is the heat capacity per unit mass of
        the substance.

The heat transferred between a substance and its surroundings not in thermal
equilibrium may be expressed in terms of specific heat as (if the temperature
range is small):
                                  Q = mcT

Calorimetry  measuring specific heat

Conservation of energy requires that for objects in thermal contact in an isolated
system any heat leaving a hot object must equal the heat entering the cold object.

One method for measuring the specific heat of a substance is to heat it to a known
temperature, place it in an insulated vessel filled with water (c = 4186 J/kg0C) of
known mass and temperature, and measure the temperature of the water after
equilibrium has been reached.

heat gained = -heat lost

                                                                  mw c w (T f - Twi )
             mw cw (T f - Twi ) = - m x c x (T f - Txi )  c x =
                                                                   m x (Txi - T f )

Example: Cooling a hot ingot (Tf = 22.40C)

ingot          mx = 0.05kg             Txi = 2000C                cx = ?

water          mw = 0.4kg              Twi = 200C                 cw = 4186 J/kg0C

mw cw (T f - Twi ) = mx c x (Txi - T f )  c x = 453 J kg 0C

According to CRC tables, this closely matches iron @ 448 J/kg0C.
Example: A silver bullet of mass 2 grams (0.002kg) impacts a well insulated wall at
200 m/s. How much does the temperature of the bullet increase upon impact?

w = KE = Q = mct

                                        40 J
KE = 40 J = mct  t =                                    = 85.50 C
                              (2  10 kg )(234 J kg C )
                                     -3            0
Latent Heat

   Normally the transfer of heat to a substance from its environment results in a
    change in temperature of the substance.
   When a substance undergoes a change in phase it is possible to add or extract
    quite a bit of heat to or from a substance without changing its temperature.

Example Phase Changes

   solid  liquid
   liquid  gas
   crystal A  crystal B

All phase changes involve a change in internal energy known as the heat of
transformation. The energy required to change the phase of a mass m of a given
substance is:

                                      Q = mL

where L is the latent heat of the substance (J/kg).

Heat of fusion (Lf) is used when the phase change is from solid to liquid, and heat
of vaporization (Lv) is used when the phase change is from liquid to vapor, a.k.a.
heats of solidification and condensation. Heats of vaporization are normally much
greater than heats of fusion.

Phase changes involve dramatic adjustment in long range order.

   Solids  each atom in a solid can translate, vibrate, or rotate about a fixed
    equilibrium position. Rotational states tend to be very low energy while
    vibrational states tend to be a little higher. As heat is added to a solid
    substance the vibrations of atoms become large enough in amplitude that the
    attractive forces between atoms are overcome so that they no longer occupy
    fixed positions.
   Liquids  less long range order. Atoms or molecules exist in small groups weakly
    bonded to each other.
   Gasses  No long range order.

More work is required to vaporize a substance than to melt it because the average
distance between atoms/molecules is greater.

                     W = Fscos, as s increases so does W Q.
Example: The heat required to convert 1 gram of ice at 300C to steam at
1200C.

   1. The heat required to change the temperature of the ice from
   300C  00C, no phase change, is Q = mct.

   Q = (mct)ice = (10-3kg)(2090 J/kg0C)(300C) = 62.7 J

   Total heat added = 62.7 Joules

   2. When the ice reaches 00C it begins to melt. The ice/water mixture remains
      at 00C even though heat is being added because of the phase change. This
      continues until all of the ice melts. The heat required for the mixture to
      complete the phase change is Q = mLf.

   Q = mLf = (10-3 kg)(3.33  105 J/kg) = 333 J

   Total heat added = 395.7 Joules

   3. The heat required to change the temperature of the water from
   00C  1000C, no phase change, is Q = mct.

   Q = (mct)water = (10-3kg)(4.19  103 J/kg0C)(1000C) = 419 J

   Total heat added = 813.7 Joules

   4. When the water reaches 00C it begins to boil. The water/steam mixture
      remains at 1000C even though heat is being added because of the phase
      change. This continues until all of the water is converted to steam. The heat
      required for the mixture to complete the phase change is Q = mLv.

   Q = mLv = (10-3 kg)(2.26  106 J/kg) = 2260 J

   Total heat added = 3073.7 Joules

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