Thermodynamics/Temperature And Heat

Temperature

 It is the measure of how hot or cold an object is, relative to a chosen standard. It is also a measure of the kinetic energy of the molecules that make up the object.

 2 bodies are in thermal contact with each other if energy exchange can occur between them in the absence of mechanical work done by one on the other.

 Thermal equilibrium = situation in which 2 objects in thermal contact with each other fail to have any net energy exchange due to a difference in their temperature.

The zeroth law of thermodynamics

“If objects A and B are separately in thermal equilibrium with a third object C, then A and B are in thermal equilibrium with each other and they have the same T”.

Thermometers and temperature scales

• A thermometer is a device used to define and measure the temperature (designated as "T") of a system. It makes use of the change in some physical property with T.

For example: 

- the change in volume (designated as "V") of a liquid in a rigid container such as a glass tube.

- the change in dimensions of a solid , such as the length of a metal rod.

- the change in pressure (designated as "P") of a gas at constant volume such as inside a rigid walled container ( like a bottle).

- change in V of a gas at constant P.

- the change in the electrical resistance (designated as "R") of an electical conductor.

- the change in color of a very hot "body" (solid, liquid or gas) as it is heated or cooled.

• T scale = scale with 2 standard T called “the fixed points” having arbitrary no.

• Degree = equal divisions dividing the interval between the 2 standard T.

• Familiar T scales:

1) Celcius (centigrade) scale

Basis: Freezing and boiling points of water.

Lower fixed point: 0°C = T of melting of ice.

Upper fixed point: 100°C = T of boiling of water.

2) Fahrenheit scale

Basis: Freezing and boiling points of a mixture composed of equal amounts of ice and salt table.

Lower fixed point: 0°F = freezing point of the salt solution.

Relations with Celsius grade: Melting of ice = 32°F Boiling of water = 212°F

3) Kelvin (absolute) scale

Basis: Molecular motion of atoms.

Lower fixed point: 0 K = absolute zero at which molecular motion almost vanish Zero point vibrational energy = KE of atoms at absolute zero.

Relations with Celsius grade: Melting of ice = 273 K Boiling of water = 373 K

Equations

TC = TK – 273

TC = 5/9 (TF – 32)

Thermocouples

 It is a thermometer formed of 2 junctions, each formed by 2 different metals A and B.

 One of the 2 junctions is placed in the material whose T is to be measured.

 The other junction is maintained at a constant reference T.

 When reference T is different from the T of the rest of the junction, an emf appears between the 2 end wires and can be measured by a voltmeter.

 Emf α ∆T. Thus unknown T can be measured.

 Advantages:

o Small mass that enables it to quickly reach thermal equilibrium with the material being probed.

o Capable of detecting T ranging from -180°C to 1500°C, depending on materials used in the junctions.

Heat and internal energy

• Quantity of heat = thermal energy that flows from one body to another because there is a T difference between them.

• Internal energy U = KE (from the motion of the molecules forming the system) + PE (from the intermolecular forces).

• For ideal gas, molecules are not subject to attractive forces, so PE = 0. U is thus completely KE and only depends on absolute T of the gas and not its V or P.

U = KE = 3/2 N k T = 3/2 n NA k T = 3/2 n R T

Where NA = Avogadro’s no. (no. of molecules in 1 mole). n = no. of moles in the gas. R = universal gas constant.

Unit of heat and the mechanical equivalent of heat

 Unit of heat = calorie = amount of heat needed to raise T of 1 gm of water by 1°C, from 14.5°C to 15.5°C.

 Mechanical equivalent of heat J = no. of joules equivalent to 1 calorie.

1 cal = 4.186 Joule = 4.186 x 10 ^ 7

J = W / H (Joule / cal)

Heat capacity, specific heat and molar specific heat

• Heat capacity = amount of heat needed to raise T of body by 1°C.

Heat capacity = Q / ∆T

• Specific heat (s) = amount of heat needed to raise T of unit mass of the material by 1°C. It is also the heat capacity per unit mass.

s = Q / (m ∆T)

Q = m s ∆T

• This law is only valid if the material gains or loses heat but stays in the same phase.

• If ∆T is not too great, s is constant. If not, then s varies and we must take an infinitesimal T change: dT.

Q = m ∫ s dT

• Molar specific heat (c) = amount of heat needed to raise T of 1 mole of the material by 1°C.

c = Q / (n ∆T)

• If c is a function of T (varies), we take an infinitesimal T change: dT

Q = n ∫ c dT

Latent heat

 Latent heat of transformation (L) = amount of heat needed to change unit mass of a substance from one phase to another at constant T and P.

L = Q / m

Q = m L

 Types: latent heat of fusion (L_f), of vaporization (L_v), and of sublimation (L_s).

 Latent heat does not result in a change in T. It involves a rearrangement of molecules to accomplish a phase change. E.g.

Latent heat of vaporization L_v

Molecules in a liquid are close together, and forces between them are stronger than in a gas. Thus, work must be done on the liquid against these attractive forces in order to separate molecules. This amount of energy is L_v.

Latent heat of fusion L_f

Molecules in a solid are fixed in their position. Thus, work must be done to increase the amplitude of vibration of atoms about their equilibrium position and overcome this attractive binding force. This amount of energy is L_f

 L_v is much larger than L_f because the distance between atoms in the gas phase is much larger than that in liquid or solid. So more work is required to vaporize a given mass than to melt it.

Methods of heat transfer

a) By conduction:

 It is the method by which heat is transferred in solids.

 Energy is transferred without transfer of mass of the medium.

 Energy transfer is due to vibration of atoms and motion of e. When one part of the material is heated, KE of molecules in this part increases. Due to molecular vibrations, KE is transferred to neighboring molecules and energy increases.

 Metals are good conductors of heat because they contain large no. of free e that can move through the metal and transport energy from one region to another.

∆Q = k A ∆t (∆T / L)

Where ∆Q = amount of heat, k = coefficient of thermal conductivity (watt/m . K), A = area of contact, ∆t = time interval, ∆T = T difference, L = distance between hot and cold bodies, (∆T / L) = temperature gradient = change in T per unit length of material (K / m).

 Rate of flow (H) = amount of heat conducted per unit time.

H = (∆Q / ∆t) = k A (∆T / L) = A [∆T / (L / k)] = A (∆T / R)

Where R = L / k = R value = resistance of the material to heat flow.

b) By convection:

 It is the method by which heat is transferred in liquids and gases.

 Hot parts of a fluid have lower density than cold parts. So, the hot fluid moves upward and transfer heat from one part to another.

 Thus, heat transfer is due to movement of a heated substance.

∆Q / ∆t = q A ∆T

Where q = coefficient of thermal convection (watt/m2　　. K). It depends on the orientation and shape of body and the wind speed.

c) By radiation:

 It is the method by which heat is transferred in vacuum (e.g. heat from sun).

 Heat radiation is an IR radiation belonging to electromagnetic spectrum.

λ f = c E = h f = h c / λ

Wien’s displacement law

The wavelength corresponding to the peak emission in various black body spectra as a function of temperature

Wien's displacement law shows us that the peak of the curve of a blackbody radiation "displaces" to the left as temperature is increased.

Black body = ideal system that absorbs all radiation incident on it. A hot blackbody of temp. T emits a spectrum of radiation of different λ with various intensity.

λ max = B / T

Where B = 2.898 x 10^-3 m °K. Thus, hotter bodies emit shorter λ.

Stefan’s law of radiation

 “The amount of heat emitted by radiation per unit time from a body is proportional to the fourth power of its absolute T”.

P = Q / ∆t = σ e A T^4

Where P = emitted radiation rate, σ = Stefan’s constant = 5.67 x 10^-8 W/m2　　K, e = emissivity of surface.

 For black body, e = 1 (perfect emitter and absorber but poor reflector).

 For white shiny surfaces, e = 0 (perfect reflector but poor emitter).

 Thus, e is large for dark rough surfaces and small for light smooth ones.

 When a body of temp. T is in a medium of temp T0, net P emitted or received is

P net = P out – P in = σ e A (T – T0)

 When the object is in thermal equilibrium with its surroundings, it radiates and absorbs heat at the same rate, so its T remains constant. When it is hotter than its surroundings, it radiates more energy than it absorbs, so it cools.

 Let T of body be changed by dT and P be changed by dP. Thus,

dP = 4 σ e A T dT

 Relative change in the rate of heat radiation is

dP / P = 4 (dT / T)

Thus, the relative change in the rate of heat radiated from a body is 4 times the relative change in its absolute T.