Chapter 4Macroscopic Parameters & Their Measurement

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!

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)

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!).

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!).

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:

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 .

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!

Figures (c), (d), (e), (f) 4 more of the many possible processes!

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

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.

Section 4.3: Temperature & Temperature Scales

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

)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

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

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.

Some Specific Heat Values

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

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!

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

• 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)

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.

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

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