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Chapter 4Macroscopic Parameters & Their Measurement
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The Laws of Thermodynamics: Overview• 0th Law: Defines Temperature (T)
• 1st Law: Defines Energy (Internal Energy Ē & Mechanical Work W)
• 2nd Law: Defines Entropy (S)
• 3rd Law: Gives a Numerical Value to Entropy (At low T)
NOTE! These laws are UNIVERSALLY VALID for systems at equilibrium.
They cannot be circumvented for such systems!
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Chapters 4 & 5:• In these chapters, we have a
Purely Macroscopic Discussion of the consequences of
The 4 Laws of Thermodynamics.
• The focus is on measurements of various macroscopic parameters:
Work (W)Internal Energy (Ē)Heat (Q) Temperature (T) Entropy (S)
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Section 4.1: Work (W) & Internal Energy (Ē)• From Classical Mechanics, in principle, we know how to measure Macroscopic,
Mechanical Work (W):• Simply put, such a measurement would change an external parameter x of the system & observe the
resulting change in the mean generalized force <X>. (In what follows, Make the Replacement <X> → X(x)). For a quasi-static, infinitesimal change, the infinitesimal work done is defined as:
đW = X(x)dx.• Then, from the observed change in X(x) as a function of x, the macroscopic work done is the
integral:
W = ∫đW = ∫X(x)dx.The limits are xi → xf, where xi & xf are the initial & final x in the process.
• Of course, as we’ve discussed,
The Work W Depends on the Process(depends on the path in the X – x plane!).
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Example: Work Done by Pressure with a
Quasi-static Volume Change Vi Vf
• If the volume V is the external parameter, the mean generalized force is the mean pressure <p> = p(V). So, for a quasi-static volume change, the work done is the integral:
W = ∫đW = ∫p(V)dVThe limits are Vi → Vf.
• Again, The Work W Depends on the Process (depends on the path in the p – V plane!).
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AdV
P
dx
F
dx
dWF PA
PAdxdW PdV
The work W done by the gas in expanding the cylinder from V1 to V2: 2
112
V
VPdVW
1Vo
P
V2V
'11
2
The work W done by an expanding gas is equal to the area of the region under the curve in a PV diagram and clearly depends on the path taken.
Example A gas in a cylindrical chamber
with a pistonThe force on the piston:
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o
P
V
2
1
2V1V
If a gas is allowed to complete a cycle, has net work been done?
The net work W done by a gas in a complete cycle is equal to the pink area of the region enclosed by the path . If the cycle is clockwise on the PV diagram, the gas does positive work .
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Note: There are many possible ways to take the gas from an initial state i to final state f. the work done is, in general, different for each. This is consistent with the fact that đW is an inexact differential!
Figures (a) & (b) are only 2 of the many possible processes!
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Figures (c), (d), (e), (f) 4 more of the many possible processes!
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Some Thermodynamics Terminology• A Process is a change of a system from some initial state to
some final state.• The Path is the intermediate steps between the initial state
and the final state. • Isobaric: A process done at constant pressure: p1 = p2
• Isochoric: A process done at constant volume, V1 = V2. • Isothermal: A process done at constant temperature, T1=T2 • Adiabatic: A process where Q = 0, that is, no heat is
exchanged. • Free Expansion: A process where Q = W = ΔĒ = 0• Cyclic: A process where the initial state = the final state.
Section 4.2: Heat (Q) & The 1st Law of Thermodynamics
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First Law of Thermodynamics
ΔĒ = Ēf – Ēi = Q - W For an infinitesimal, quasi-static process, this becomes
dE = đQ - đW The mean internal energy Ē of a system tends to increase if energy is added as heat Q and tends to decrease if energy is lost as work W done by the system.
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Section 4.3: Temperature & Temperature Scales
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TemperatureThe Triple Point of Water
erature)point temp-(triple 16.2733 KT
The Constant – Volume Gas Thermometer
CpT
ghpp 0
p is the pressure within the gas & C is a constant.
p0 is the atmospheric pressure, ρ is the density of the mercury in the manometer
33 CpT p3 is the measured gas pressure
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)lim)(16.273(3
0 p
pKT
gas
A temperature with a gas thermometer is
al)(provision ))(16.273()(33
3 p
pK
p
pTT
The Celsius and Fahrenheit Scales
00 95 FC
015.273TTC
0325
9 CF TT
FC 00 320
TC represents a Celsius temperature and T a Kelvin temperature
The relation between the Celsius and Fahrenheit scales is
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The Heat Capacity of a substance is defined as:
Cy(T) (đQ/dT)y
The subscript y indicates that property y of the substance is held constant when Cy is measured
The Specific Heat per kilogram of mass m:
mcy(T) (đQ/dT)y
The Specific Heat per mole of υ moles:
υcy(T) (đQ/dT)y
Section 4.4: Heat Capacity & Specific Heat
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Heat CapacityThe heat capacity is obviously different for every substance:
Requires more heat to cause a rise in temperature
Substance CCopper 0.384
Wax 0.80Aluminum 0.901
Wood 2.01Water 4.18
The heat capacity also depends on temperature, the volume & other system parameters.
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Some Specific Heat Values
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The First Law of Thermodynamics: đQ = dĒ + đW
The Second Law of Thermodynamics: đQ = TdS dS = Entropy Change
Combining these gives: TdS = dĒ + đW• Using this result with the definition of Heat Capacity
with constant parameter y:
Cy(T) (đQ/dT)y
gives the general result:
Cy(T) = T(S/T)y
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The First Law of Thermodynamics: đQ = dĒ + đW• If the volume V is the only external parameter đW = pdV. So, under constant volume conditions: đQ = dĒ The Heat Capacity at Constant Volume has the form:
CV(T) (đQ/dT)V = (Ē/T)V
• However, if the Pressure p is held constant, the First Law must be used in the form đQ = dĒ + đW The Heat Capacity at Constant Pressure has the form:
Cp(T) (đQ/dT)p
NOTE!! Clearly, in general, Cp ≠ CV
Further, in general, Cp > CV
Cp & CV are very similar for solids & liquids, but very different for gases, so be sure you know which one you’re using if you look one up in a table!
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Heat Capacity for Constant Volume Processes (Cv)
• Heat is added to a substance of mass m in a fixed volume enclosure, which causes a change in internal energy, Ē. So, from the 1st Law:
Q = Ē2 - Ē1 = Ē = mCvT
Heat Qaddedm m
Tinsulation
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• Heat is added to a substance of mass m held at a fixed pressure, which causes a change in internal energy, Ē, AND
some work pV. Q = Ē + W = mCpT
Heat Qadded
T
m m
x
Heat Capacity for Constant Pressure Processes (Cp)
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Experimental Heat Capacity
Experimentally, it is easier to add heat at constant pressure than at constant volume. So, tables typically report Cp for various materials.
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Calorimetry ExampleSimilar to Reif, pages 141-142
• A technique to Measure Specific Heat is to heat a sample of material, add it to water, & record the final temperature.
• This technique is known as Calorimetry.– Calorimeter = A device in which this
heat transfer takes place.• The system of the sample + water is isolated• Conservation of Energy requires
that the heat energy Qs leaving the sample equals the heat energy that enters the water, Qw. This gives:
Qs + Qw = 0
A Typical Calorimeter
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Qs + Qw = 0 (1)Sample Properties:
Mass = ms. Initial Temperature = Ts. Specific Heat = cs (cs = unknown)
Water Properties:Mass = mw. Initial Temperature = Tw. Specific Heat = cw (cs = 4,286 J/(kg K))
Final Temperature (sample + water) = Tf
• Put Qs = mscs(Tf – Ts ) & Qw = mwcw(Tf – Tw) into (1):mscs(Tf – Ts ) + mwcw(Tf – Tw) = 0
• Solving for cs gives:
• Technically, the mass of the container should be included, but if mw >> mcontainer it can be neglected.
w w f ws
s s f
m c T Tc
m T T