Chapter 4 Macroscopic 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!

Purely Macroscopic Discussion 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)dV The limits are Vi → Vf. Again, The Work W Depends on the Process (depends on the path in the p – V plane!).

Example A gas in a cylindrical chamber with a piston The force on the piston: The work W done by the gas in expanding the cylinder from V1 to V2: 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.

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 .

Figures (a) & (b) are only 2 of the many possible processes! 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!

Section 4.2: Heat (Q) & The 1st Law of Thermodynamics 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.

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

The Triple Point of Water Temperature The Triple Point of Water The Constant – Volume Gas Thermometer 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 p3 is the measured gas pressure

The Celsius and Fahrenheit Scales A temperature with a gas thermometer is The Celsius and Fahrenheit Scales TC represents a Celsius temperature and T a Kelvin temperature The relation between the Celsius and Fahrenheit scales is

Section 4.4: Heat Capacity & Specific Heat 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

Heat Capacity Substance C Copper 0.384 Wax 0.80 Aluminum 0.901 Wood The heat capacity is obviously different for every substance: Substance C Copper 0.384 Wax 0.80 Aluminum 0.901 Wood 2.01 Water 4.18 The heat capacity also depends on temperature, the volume & other system parameters. Requires more heat to cause a rise in temperature

Some Specific Heat Values

Cy(T) = T(S/T)y Cy(T)  (đQ/dT)y 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) insulation DT Heat Q added m m 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 = DĒ = mCvDT

Heat Capacity for Constant Pressure Processes (Cp) Heat Q added DT m Dx 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 = DĒ + W = mCpDT

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 Example Similar 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

mscs(Tf – Ts ) + mwcw(Tf – Tw) = 0 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.