Chapter 21 Notes: Temperature, Scales, Thermal Expansion, and the Ideal Gas

21-1 TEMPERATURE AND THERMAL EQUILIBRIUM

• Temperature is defined rigorously by first establishing thermal equilibrium between systems.
• Thermal equilibrium: when two systems (A and B) are isolated from environment and from each other, in contact via a wall that may be adiabatic (thermally insulating) or diathermic (thermally conducting).
• Adiabatic wall: no energy or matter transfer; changes in one system do not affect the other.
• Diathermic wall: allows heat flow; when two systems are brought into contact, their properties change and eventually approach constant values; the systems are then in thermal equilibrium (same temperature).
• When direct contact is impractical, a third system C can be used to test for thermal equilibrium between A and B: if A and C are in equilibrium and B and C are in equilibrium with C, then A and B are in equilibrium with each other. This is the zeroth law of thermodynamics.
• Zeroth law (postulate):
  • “If systems A and B are each in thermal equilibrium with a third system C, then A and B are in thermal equilibrium with each other.”
• Importance of the zeroth law:
  • It underpins the concept of temperature and justifies using a thermometer (system C) to compare temperatures of A and B without direct contact.
• Temperature (concept):
  • When two systems are in thermal equilibrium, they have the same temperature.
  • Temperature is a scalar quantity; two systems in equilibrium have equal temperatures, though their pressures or volumes need not be equal.
• Practical interpretation of temperature vs human intuition:
  • Temperature is a fundamental property, not the subjective sense of hot/cold or energy transfer ability alone.
  • A cold object can feel colder because it can draw heat from the hand; this subjective feeling does not always reflect the actual temperature.
• Example illustrating the zeroth law caveat:
  • A and C attract B and C equally if C is a magnet but A and B may not attract each other; demonstrating that equivalent interactions with C do not guarantee equilibrium between A and B in all cases.

21-2 TEMPERATURE SCALES

• Temperature (T) is one of the seven base SI units (Kelvin as the base unit).
• Absolute zero on the Kelvin scale represents the lowest possible temperature; temperatures are measured in kelvins (K).
• Triple point of water defines the Kelvin degree size:
  • The Kelvin temperature at the triple point of water is defined as exactly
  • $T_{ ext{tr}} = 273.16 ext{ K} ag{21-1}$
• Kelvin scale: the degree is defined so that 1 K increment equals 1 °C increment.
• Conversion between Kelvin and Celsius:
  • $T_{C} = T - 273.15 ag{21-2}$
  • At the triple point, the Celsius temperature is 0.01 °C.
• Fahrenheit scale relation to Celsius:
  • $T<em>{F} = T</em>{C} + 32 ag{21-3}$
• Relationship among Kelvin, Celsius, and Fahrenheit scales is often illustrated (e.g., they coincide at -40 for the interval scales).
• Practical thermometer tests:
  • A thermometer (thermometer C) need not be marked with a scale; it can be compared by placing it in contact with systems and noting equal readings.
• General measurement relation for a “private” thermometer:
  • If the measurable thermometric property is X and the simplest linear relation is assumed,
  • $T^{*} = aX ag{21-5}$
  • Here, T* is a device-sensitive temperature reading, not necessarily the true Kelvin temperature.
  • Calibrate a by measuring X at the triple point of water to obtain X_tr, which yields the temperature relation
  • $T^{*}(X) = \frac{T<em>{ ext{tr}}}{X</em>r} X ag{21-6}$
  • In practice, measurements yield a device-specific scale that can be related to the true Kelvin temperature via calibration.
• Sample Problem 21-1 (illustrative):
  • If the resistance of a platinum coil increases by a factor of 1.392 between the triple point and the normal boiling point of water, the device-reading temperature for boiling water is
  • Using the linear relation with X = R and R_r the resistance at triple point,
  • $T^{*}(R) = \frac{R}{R<em>r} T</em>{ ext{tr}} = (1.329)(273.16 ext{ K}) <br />\neq 373.125 ext{ K}$
  • The result shown is $T^{*}(R) = 380.2 ext{ K}$, which differs from the true Kelvin temperature of boiling water (373.125 K). This illustrates the need for proper calibration or using a standard Kelvin-scale thermometer.
• The Constant-Volume Gas Thermometer (primary instrument for Kelvin temperatures):
  • Principle: For a fixed volume of a gas, pressure is proportional to temperature (P ∝ T) for ideal behavior.
  • A gas-filled bulb is alternately immersed in a triple-point bath and in the bath whose temperature is to be measured; the pressure is read on a manometer.
  • Step 1: Measure Ptr in the triple-point bath; then immerse in the unknown bath and measure p; compute provisional T* via the relation with p and Ptr, and plot T* vs p.
  • Step 2: Return bulb to the triple-point bath and reduce the gas amount (decrease density) to get a smaller Ptr; repeat measurements to obtain a new provisional T*, and plot again.
  • Extrapolate the T* vs p curve to p → 0 to obtain the Kelvin temperature T.
  • This extrapolation is based on the ideal-gas assumption and yields the relation
  • $T = T<em>{ ext{tr}}\, \frac{P}{P</em>r}\bigg|<em>{P</em>r o 0} ag{21-7}$
  • The ideal gas temperature scale is defined by taking the limit as Ptr → 0 for a constant volume; gases at very low densities approach the same temperature, independent of the gas species, allowing a universal Kelvin scale.
  • The lowest temperature measurable with a gas thermometer is about 1 K (for which low-pressure helium is used).
• Practical temperature scales and ITS-90 (briefly):
  • The International Temperature Scale uses a fixed set of fixed-points and interpolation/extrapolation procedures (ITS-90) for accurate Kelvin-scale calibration.
  • Primary fixed points (e.g., helium, hydrogen, neon, argon, mercury, water triple point, gallium, indium, tin, zinc, aluminum, silver, gold, copper) are tabulated for calibration; fixed points and interpolation schemes are refined roughly every two decades.
• Labrador of practical references:
  • Kelvin scale remains the fundamental temperature scale for physics; Celsius and Fahrenheit scales are convenient, practical scales for everyday use.
  • The Kelvin temperature is used directly in fundamental physics equations; Celsius/Fahrenheit are derived from Kelvin.
• Transitions and conversions between scales are essential for lab work and for comparing measurements across contexts.

21-3 MEASURING TEMPERATURES

• What makes a good thermometer:
  • It should be based on a thermometric property that varies with temperature and can be measured reproducibly.
  • Examples of thermometric properties include:
  • Volume of a liquid (e.g., mercury-in-glass thermometer)
  • Pressure of a gas kept at constant volume
  • Electrical resistance of a wire
  • Length of a strip of metal
  • Color of a lamp filament
• These properties provide device-specific temperature readings (private scales) that require calibration to relate to the Kelvin scale.
• Relationship of a measured property X to temperature:
  • $T^{*}(X) = aX ag{21-5}$
• Calibration concept:
  • Use a known fixed point (e.g., triple point of water) to determine a, and then relate T* to true Kelvin temperature through calibration.
• The constant-volume gas thermometer, discussed in detail in 21-3 and 21-4, serves as a practical physical realization of the Kelvin scale and demonstrates the underlying gas-law behavior that underpins the Kelvin definition.

21-4 THERMAL EXPANSION

• Thermal expansion basics:
  • Materials expand when heated; contraction occurs when cooled.
  • Applications and practical considerations include expansion joints in roadways and bridges, expansion loops in pipelines, and the behavior of dental fillings.
  • Thermometers and thermostats often rely on the differential thermal expansion of materials (e.g., bimetallic strips). See Fig. 21-7 (bimetallic strip) and Fig. 21-8 (coil thermometer).
• Microscopic picture (optional):
  • A crystalline solid behaves like atoms connected by springs; atoms vibrate more at higher temperatures, increasing the average interatomic separation and leading to expansion.
  • The potential-energy curve for two adjacent atoms is asymmetric about the equilibrium separation; as vibrational amplitude increases with temperature, the average separation increases, causing expansion.
  • If the potential were perfectly symmetric, average separation would remain at the equilibrium separation, and no thermal expansion would occur.
• Linear expansion (for solids):
  • If a linear dimension is L and the temperature change is ΔT, the fractional length change is approximately constant for small ΔT:
  • $\frac{ ext{d}L}{L} \frac{}{} = \bar{\boldsymbol{\boldsymbol{ ext{α}}}} ext{Δ}T ag{21-9}$
  • Equivalently:
  • $ext{Δ}L = ext{α}L ext{Δ}T ag{21-8}$
  • α is the coefficient of linear expansion (material dependent); its value is typically given as a constant over a modest temperature range.
• Isotropic solids: area and volume expansions are related to α:
  • $\frac{ ext{Δ}A}{A} = 2α ext{Δ}T ag{21-10}$
  • $\frac{ ext{Δ}V}{V} = 3α ext{Δ}T ag{21-11}$
• Coefficient of volume expansion for fluids:
  • Define β by ΔV = β V ΔT (21-12)
  • For liquids, β is relatively independent of temperature and typically ranges from about 200 × 10^-6 /°C to 1000 × 10^-6 /°C near room temperature.
  • For gases, β ≈ 1/T (with T in kelvins) for an ideal gas.
  • For gases at room temperature and constant pressure, β is about 3300 × 10^-6 /°C, which is much larger than typical liquids.
• Special case: water’s anomalous expansion curve
  • The volume expansion of water is non-monotonic near 4°C; below and above 4°C, water expands with temperature changes, while around 4°C it reaches a maximum density (minimum specific volume).
  • This behavior explains phenomena such as lakes freezing from the top (ice being less dense than water at corresponding temperatures) and has broad ecological implications.
• Isotropic vs. anisotropic expansion:
  • Isotropic solids (uniform expansion in all directions) share a single α that works across dimensions (length, width, diagonals, etc.).
  • Not all materials are perfectly isotropic; inhomogeneous or composite materials (e.g., bimetallic strips) can exhibit differential expansion that is exploited in devices (thermostats, temperature sensors).
• Problems and applications:
  • Problem examples illustrate how to design measurement setups and predict tolerances due to thermal expansion (e.g., steel ruler expansion, using Invar for precision metadata).
• Microscopic optional note (Fig. 21-9, 21-12) discusses how expansion arises from atomic interactions and vibrational energy, and how asymmetry in the potential-energy curve drives fractional changes in size with temperature.

21-5 THE IDEAL GAS

• Idea and motivation:
  • Real gases at sufficiently low density exhibit behavior that converges toward a common set of thermodynamic relationships; this limiting behavior defines the ideal gas.
  • The ideal gas is a useful abstraction because:
    1) Real gases at low densities approximate ideal-gas behavior, and
    2) The thermodynamic properties of an ideal gas are related in a simple, well-defined way.
• Conceptual setup to study gases and extrapolate to the ideal gas behavior (Fig. 21-13):
  • An insulated cylinder can be used to study gas properties, and by varying density (via pressure) one can extrapolate to the ideal-gas limit.
• Key ideas:
  • The ideal gas provides a simple, universal framework for relating pressure, volume, and temperature (PV = nRT in simple form, though detailed derivations are not shown in this excerpt).
  • The convergence of real gases toward ideal behavior at low density justifies using the ideal-gas model for foundational thermodynamics.
• From this section, the text emphasizes the utility of abstractions in physics (e.g., ideal gas, perfectly elastic collisions) and how they aid understanding and problem solving in thermodynamics and statistical mechanics.

Connections and Real-World Relevance

• Temperature and thermal equilibrium underpin all thermodynamic processes and the functioning of engines, refrigerators, climate systems, and even human perception of warmth.
• The zeroth law formalizes the operational concept of temperature and the construction of thermometers, which are essential for science, industry, and daily life.
• The Kelvin scale provides a universal, absolute temperature framework that is indispensable in fundamental physics and cross-scale comparisons.
• Temperature scales (Kelvin, Celsius, Fahrenheit) and their interrelationships are crucial for communication of scientific results across regions and disciplines.
• Thermal expansion is fundamental in engineering design (bridges, railways, pipelines, aircraft components) and in everyday devices (glass jars with lids, thermostats).
• The microscopic picture of thermal expansion links thermodynamics to statistical mechanics and solid-state physics, illustrating how macroscopic properties emerge from atomic interactions.
• The ideal gas provides a cornerstone model for understanding gas behavior, enabling simple, generalizable relationships that approximate real systems under appropriate conditions. It also serves as a bridge to more advanced kinetic theory and quantum statistics.

Key Formulas to Remember (LaTeX)

• Linear temperature relation for a device-reading thermometer:
  • $T^{*} = aX ag{21-5}$
• General device-reading temperature from a thermometric property X at triple point calibration:
  • $T^{*}(X) = \frac{T<em>{ ext{tr}}}{X</em>r} X ag{21-6}$
• Thermometric-temperature extrapolation for a constant-volume gas thermometer (Kelvin temperature):
  • $T = T<em>{ ext{tr}} \frac{P}{P</em>r} \bigg|<em>{P</em>r o 0} ag{21-7}$
• Kelvin temperature fixed point:
  • $T_{ ext{tr}} = 273.16 ext{ K} ag{21-1}$
• Celsius conversion:
  • $T_{C} = T - 273.15 ag{21-2}$
• Fahrenheit-Celsius conversion:
  • $T<em>{F} = T</em>{C} + 32 ag{21-3}$
• Linear expansion of a solid length:
  • $ext{Δ}L = ext{α}L ext{Δ}T ag{21-8}$
  • $\frac{ ext{Δ}L}{L} = ext{α} ext{Δ}T ag{21-9}$
• Areal and volumetric expansion for isotropic solids:
  • $\frac{ ext{Δ}A}{A} = 2 ext{α} ext{Δ}T ag{21-10}$
  • $\frac{ ext{Δ}V}{V} = 3 ext{α} ext{Δ}T ag{21-11}$
• Volume expansion for fluids:
  • $ext{Δ}V = ext{β}V ext{Δ}T ag{21-12}$
• Coefficient behavior for gases (approximate):
  • $\beta \nobreak o \frac{1}{T} ext{ (for ideal gas, with } T ext{ in K)}$

Note on Figures and Tables Mentioned

• Figure references (e.g., adiabatic vs. diathermic walls, constant-volume gas thermometer schematic and curves) illustrate the concepts described above.
• Figure 21-3 and Fig. 21-6 illustrate scale relations and thermal expansion phenomena (e.g., bimetallic strips, expansion of materials).
• Figure 21-9 and 21-12 provide microscopic illustrations of thermal expansion in solids (potential-energy curves) and the effect on lattice spacing.
• Table 21-2 and Table 21-3 contain fixed-point data and average coefficients of linear expansion for various materials; values vary with temperature ranges and material composition (e.g., steel, Invar, quartz).
• Table 21-1 lists representative Kelvin temperatures for selected systems (e.g., triple point of water, normal boiling and freezing points, extreme environments). The exact numerical values vary across sources and are presented in the textbook’s table.

Summary of Practical Takeaways

• Temperature is defined via thermal equilibrium and the zeroth law; a thermometer is a system that comes to equilibrium with the system of interest.
• The Kelvin scale is the fundamental temperature scale in physics, defined by the triple point of water at 273.16 K.
• Celsius and Fahrenheit are practical scales linked to Kelvin by defined relations; Kelvin lacks a degree symbol when reported.
• Temperature measurement can involve various thermometric properties; calibration is essential to relate device readings to true Kelvin temperatures.
• The constant-volume gas thermometer demonstrates a practical route to the Kelvin scale and shows how the ideal-gas limit provides a universal temperature reference.
• Thermal expansion is a key practical consideration in engineering and devices, governed by coefficients of linear, area, and volume expansion; solids and fluids behave differently, with water showing anomalous expansion near 4°C.
• The ideal gas concept provides a foundational, widely applicable model for real gases at low densities and underpins many thermodynamic relations.