Today’s Lecture
Thermodynamics
Temperature Scales
Specific Heat
Heat Conduction
Thermodynamics.
Study of temperature, heat and related macroscopic properties
of objects and matter.
Specifically macroscopic properties: temperature, heat,
pressure, internal energy…
What do we mean by “Any macroscopic property”?
Temperature, volume, pressure, electrical conductivity, length,
color, etc.
Why do we stress that we talk about MACROSCOPIC properties?
The properties that we have mentioned have no meaning on a
microscopic scale!
Thermodynamics.
Study of temperature, heat and related macroscopic properties
of objects and matter.
Specifically macroscopic properties: temperature, heat,
pressure, internal energy…
Temperature and Heat
Thermodynamic equilibrium: two systems are in thermodynamic
equilibrium with each other, when they are in thermal contact
and no change in any macroscopic property occurs with time.
What do we mean by thermal contact?
Heating one results in macroscopic changes in the other.
Two systems have the same temperature if they are in
thermodynamic equilibrium.
Two systems have the same temperature if they are in
thermodynamic equilibrium.
Zeroth law of thermodynamics:
If two systems A and C each are in thermodynamic equilibrium
with system B, then A and C are in thermodynamic equilibrium
with each other.
Sounds trivial, but gives a rationale for measuring temperature.
Zeroth law of thermodynamics:
If two systems A and C each are in thermodynamic equilibrium
with system B, then A and C are in thermodynamic equilibrium
with each other.
B
A
C
Liquid thermometers
Based on thermal
expansion of liquids.
The liquids are held in
little vessels at the
bottom.
At higher temperatures
they expand and climb
toward the top of
narrow capillaries.
Liquid thermometers
Coefficients of thermal expansion
ΔV / V
β=
ΔT
β
– fractional increase of
volume per one degree
Problems:
1. Liquid is a rather complex state of matter. Molecules in liquids are
held together by cohesive forces, different for different liquids.
2. The coefficients of thermal expansion vary a lot between liquids,
and may depend on temperature in an extreme fashion.
3. Liquids freeze at low temperatures and boil at high temperature. So
the ranges of operation of the liquid thermometers are restricted.
Gas Thermometer
Gas Thermometer
Gasses are much simpler than
liquids.
The molecules are moving freely
most of the time, and only once in
a while suffer short term collisions.
The collision events are still
different for different molecules….
BUT: when the gasses are rarified (low density) and the collisions are rare
behavior of different gasses in the gas thermometer becomes very much the
same – An Ideal Gas!
In order not to change the gas density, it is preferable to keep the gas
volume constant and to measure the gas pressure.
Gas Thermometer
Level of the mercury in
the right tube is varied to
keep the level in the left
tube constant.
Pressure of the gas is
measured as P =ρgh
The absolute temperature of the
system is defined as:
P
T = 273.16
P3
P3 is pressure of the gas at a special reference point called “triple
point”, which is unique and can be reproduced in every laboratory.
Gas Thermometer
To set the temperature scale we
need some convenient reference
points.
273.15 K, the same as 0°C is
the temperature of ice melting at
normal pressure;
373.15 K, the same as 100°C is
the temperature of water boiling at
normal pressure;
1K temperature difference is the same as 1°C temperature
difference, but 0K corresponds to absolute zero or -273.15 °C.
Temperature Scales
Fahrenheit
Celsius
Lord Kelvin
9
TF = 32° + TC
5
TK = 273.15 + TC
Temperature Scales
What is normal body temperature (98.6oF) on the
Celsius and Kelvin scales?
If you have a fever of 101.6oF, how much has
your temperature risen on each of these scales?
The point here is that a change in oC or K is the same!
Gas Thermometer
A constant volume gas thermometer
supports a 72.5mm column of mercury
when it is submersed in liquid N2 at
-196oC. What is the column height
when the thermometer is submersed
in molten Pb at 350oC?
From the expression for the absolute
temperature, we have:
Converting to absolute temperature, oC to K, we find:
Heat and Internal Energy
Heat is energy being transferred from one object to another
because of temperature difference alone.
Heat is energy in transit!
Internal energy relates to energetic contents of a body or a system.
Whereas
Heat is energy in transit!
Nevertheless, both heat, ΔQ, and internal energy, U, are measured in
the same energy units, Joules, J.
Heat can be transferred to a body (or a system) that would cause
growth of internal energy of the body, but not of its “heat”. Because
Heat is not a material or a form of matter.
Heat is energy in transit!
When pouring water
you transfer it from
one vessel to
another and you get
more water in the
second vessel.
In thermodynamics you transfer heat but you usually end up
having more or less internal energy.
What are common results of heat transfer?
1. Increase in temperature.
2. A phase transition (melting ice).
3. Mechanical work.
Cases #2 and #3 may not involve a change in temperature.
Case #1, no phase transition or work done.
How much does the temperature vary?
Heat capacity of the object C, measured in J/K;
tells you how many Joules of heat you need to transfer
to increase the temperature of the object by 1K.
ΔQ = CΔT
ΔQ
C=
ΔT
Heat capacity is an extrinsic parameter (depends on quantity):
When you bring two objects together, heat capacity of the system of the
two objects becomes the sum of the two individual heat capacities.
It is convenient to introduce specific heat, c, an intrinsic parameter,
which is heat capacity of a material per unit mass .
Specific heat is measured in J/(K⋅kg).
ΔQ = cmΔT
ΔQ
C
c= =
m mΔT
Heat capacity of a water ball is large due to both high
specific heat and large mass of the water inside the ball.
Specific heats, c, of some Common Materials
ΔQ = cmΔT
where
C
ΔQ
=
c=
m mΔT
Specific heat is measured in J/(K⋅kg).
Waiting to Take a Shower!
The water temperature in the water heater is only 18oC.
How much energy is required to heat its 150kg to 50oC?
The water heater can supply 5kW of power.
How long does it take to heat the water to 50oC?
Equilibrium Temperature.
Situation: Two objects with different temperatures, T1 and T2, are
brought into thermal contact and eventually reach thermal equilibrium at
a temperature T.
Heat ΔQ1 is transferred to Object 1; heat ΔQ2 is transferred to Object 2.
By energy conservation:
By definition of heat
capacity and specific heat:
m1c1T1 + m2 c2T2
T=
m1c1 + m2 c2
Equilibrium Temperature
As an example: a piece of copper at 300oC is dropped into 1.0kg
of water at 20oC. If the final equilibrium temperature is 25oC, what
was the mass of the copper?
Equating the energy gained by the water to that lost by the copper
Solving for mCu
Heat Transfer
1.Conduction
2. Convection
3. Radiation
Heat Conduction
A rectangular slab of thickness Δx and
with an area A.
The front side of the slab is at a
temperature T; the back side has a
somewhat different temperature,
T+ΔT.
We are trying to calculate the heatflow rate, the amount of heat flowing
Heat flows from the hotter to the
through the slab per unit time,
cooler side of the slab
H = ΔQ/Δt.
We expect H to be proportional to the area, A, of the slab, the
temperature difference, ΔT, between the back and the front and
inversely proportional to the thickness of the slab, Δx.
H should also depend of properties of the material the slab is made of…
Heat Conduction
Bringing all the parts together:
ΔT
H = − kA
Δx
Heat flows from the hotter to the
cooler side of the slab
The coefficient k reflects
specific properties of the
material of the slab and is
called thermal conductivity
H = ΔQ/Δt – heat-flow rate is measured in Joules/second, J/s,
or Watts, W. Thermal conductivity, k, is measured in W/m⋅K.
Thermal conductivities of different materials.
Best heat conductor – Copper; use it when you build heat sink, as a
material for pipes in your cooling system, a radiator.
Worst heat conductors are the best insulating materials – air,
fiberglass (layers in the walls of houses in cold regions), styrofoam
(cups for your hot coffee).
Heat Conduction – Example 1
ΔT
H = −kA
Δx
A lake with a flat bottom and steep
sides has a surface area of 1.5(km)2
and is 8m deep. In the summer the
surface is 30oC while the bottom is 4oC.
What is the rate of heat conduction in the lake?
What is the significance of the minus sign?
What about oC versus K?
Heat Conduction – Example 2
ΔT
H = − kA
Δx
An 8m x 12m house is built on a 23cm
thick concrete slab. What is the heatloss through the floor if the interior is
20oC while the ground is at 10oC?
Heat Conduction – Example 3
A pipe of length l and radius R1 is surrounded
by insulation of radius R2 and thermal
conductivity k. Find the expression for the heat
loss through the insulation when the fluid in the
pipe is at T1 and outside the insulation is at T2.
At equilibrium the heat flow is uniform as a function
of radius. Expressing this heat flow across a thin
layer of insulation at the radius r yields:
Integrating this
expression results
in the expression:
Heat Conduction – Example 3
A pipe of length l and radius R1 is surrounded
by insulation of radius R2 and thermal
conductivity k. Find the expression for the
thermal gradient, dT/dr, when the fluid in the
pipe is at T1 and outside the insulation is at T2.
From our solution for the heat flow we can write:
In this geometry the thermal gradient is not uniform. Note that the larger
the temperature difference the larger the thermal gradient and the larger
the heat flow.
Is the thermal gradient positive or negative?
Which direction does the heat flow??