Heat and Temperature

Zeroth Law of Thermodynamics: If two bodies A and B are in thermal equilibrium, and A and C are also in thermal equilibrium, then B and C are also in thermal equilibrium.

Implications: • Establishes temperature as a physical property • Allows us to assign equal temperatures to bodies in thermal equilibrium • Forms the theoretical basis for thermometry • Creates a transitive relationship among bodies in equilibrium • Enables objective temperature comparison via a third reference body

Example: If thermometer reads 25°C in two different water samples, we know they are in thermal equilibrium with each other.

Anomalous expansion of water: • Volume decreases as temperature increases from 0°C to 4°C • Coefficient of volume expansion (γ) is negative in this range • Volume reaches minimum at 4°C (density maximum) • Above 4°C, water expands normally with increasing temperature

Mathematical description: • At 0-4°C: $γ = \frac{1}{V}\frac{dV}{dT} < 0$ • At 4°C: $γ = 0$ (inflection point) • Above 4°C: $γ > 0$ (normal behavior)

Ecological significance: • Water at 4°C sinks to bottom of lakes in winter • Forms a stable layer at bottom with constant 4°C temperature • Ice forms at top (less dense) and insulates water below • Creates thermal refuge for aquatic organisms at lake bottoms • Allows marine life to survive in deep water during freezing conditions • Prevents lakes and ponds from freezing solid

This property is crucial for aquatic life survival in cold climates.

Adiabatic Wall: • Prevents any heat flow between separated systems • Maintains temperature differences indefinitely • Perfect thermal insulator (idealized concept) • Example: Perfect thermos or vacuum gap

Diathermic Wall: • Allows rapid heat transfer between systems • Enables thermal equilibrium to be reached quickly • Perfect thermal conductor • Example: Thin metal sheet

Significance: • Fundamental concepts for defining thermodynamic boundaries • Adiabatic walls enable isolation of systems energetically • Diathermic walls allow controlled thermal interactions • Used to define thermal equilibrium conceptually • Necessary for formulating Laws of Thermodynamics • Allow design of systems with controlled heat flow paths • Pure ideal cases against which real materials are compared

These concepts provide the theoretical framework for analyzing heat transfer between thermodynamic systems.

Thermal Equilibrium: • State where no heat transfer occurs between bodies in contact • Systems maintain constant thermal properties over time • No temperature gradients exist between systems • Energy has reached stable distribution between bodies

Zeroth Law establishes temperature as valid physical property by: • Creating a transitive relationship among equilibrated systems • If A ≡ B (in equilibrium) and A ≡ C, then B ≡ C • Proves equilibrium is an equivalence relation (symmetrical, reflexive, transitive) • Allows assigning a single numerical value (temperature) to each equilibrium state • Demonstrates temperature is an intrinsic property, not dependent on particular measuring device • Shows temperature is a property that determines direction of heat flow

This law logically precedes the First and Second Laws (hence "Zeroth"), as it establishes the concept of temperature itself as physically meaningful.

A compensated platinum resistance thermometer solves the "cold lead problem" (connecting wires at different temperatures than the sensing element) through:

• Using compensating wires (XX) that match resistance properties of connecting wires (YY)
• Arranging the circuit so lead wire resistance appears in both arms of the Wheatstone bridge
• Placing compensating wires in the third arm while platinum coil and its leads occupy the fourth arm

The compensation method:

1. Lead wires (YY) connecting to the platinum coil have resistance Rc
2. Compensating wires (XX) with identical resistance Rc are placed in the opposite arm
3. Both sets of wires experience the same temperature gradient
4. The equal resistances in opposite arms cancel each other's effects

Significance:

• Eliminates measurement errors caused by temperature gradients along connecting wires
• Enables accurate measurements across wide temperature ranges
• Allows the sensing element to be placed in remote locations
• Makes the thermometer suitable for high-precision scientific and industrial applications

Without compensation, lead wire resistance would introduce significant systematic errors proportional to wire length and ambient temperature variations.

The resistance-temperature conversion in platinum resistance thermometry is governed by:

t = (Rt - R0)/(R100 - R0) × 100 degrees

Where:

• t = temperature in degrees Celsius
• Rt = resistance of platinum at temperature t
• R0 = resistance at 0°C (ice point)
• R100 = resistance at 100°C (steam point)

Practical application process:

1. Calibrate the thermometer by measuring resistance at known fixed points (0°C and 100°C)
2. Place the platinum sensor in an unknown temperature environment
3. Measure the new resistance (Rt) using the Wheatstone bridge arrangement
4. Apply the formula to calculate the precise temperature

For a sliding contact Wheatstone bridge configuration:

1. The position of point C on a uniform wire AB determines the ratio of resistances
2. When balanced, the position of C relates directly to temperature
3. The scale can be calibrated to read temperature directly

Modern platinum resistance thermometers achieve accuracy of ±0.001°C for scientific applications and are the standard for precise temperature measurement in the range -200°C to 660°C.

Sources of error in constant volume gas thermometers:

1. Capillary tube temperature variation:

   • The gas in capillary BC is often not at the same temperature as the gas in bulb A
   • This creates non-uniform gas temperature distribution
   • Results in inaccurate pressure-temperature relationship

2. Thermal expansion of the glass bulb:

   • The volume of the glass bulb changes slightly with temperature
   • This causes small volume changes in the gas, violating the constant volume assumption
   • Leads to systematic errors in pressure readings

3. Gas-specific properties:

   • Real gases deviate from ideal behavior, especially at high pressures or low temperatures
   • Different gases give slightly different temperature readings
   • This is why the ideal gas temperature scale is defined as the limiting case when pressure approaches zero

4. Measurement precision:

   • Difficulty in precisely maintaining mercury at mark C
   • Reading errors in the height difference h
   • Atmospheric pressure fluctuations

To minimize these errors, scientists use minimal amounts of gas and make corrections for the expansion of the containing vessel.

When different gases (O₂, Air, N₂, He, H₂) are used in gas thermometers:

• Each gas produces slightly different steam point temperature readings at normal pressures
• As gas pressure decreases toward zero, all gases converge to the same temperature value (373.15K)
• Gases like O₂ read higher than 373.15K at higher pressures
• Gases like H₂ read lower than 373.15K at higher pressures

This convergence at low pressures is the foundation for the ideal gas temperature scale, defined as:

$T = \lim_{p_{tr} \to 0} \frac{p}{p_{tr}} \times 273.16 \text{ K}$

This is significant because it provides a gas-independent absolute temperature scale that doesn't depend on the thermometric substance used, unlike mercury or resistance thermometers which can give conflicting readings.

Example: When calibrating precision scientific equipment, using the ideal gas limit eliminates systematic errors that would occur from the specific properties of any single gas.

Limiting behavior in gas thermometry establishes an absolute temperature scale through:

• Extrapolation to zero pressure (p→0) where real gases behave ideally
• At this limit, all gases converge to measure the same temperature regardless of their molecular properties
• This convergence point defines the ideal gas temperature scale

Different gases appear at different positions because:

• Molecular properties affect non-ideal behavior at measurable pressures
• O₂ (diatomic, relatively large) reads higher temperatures at given pressures due to stronger intermolecular forces
• H₂ (diatomic, small) reads lower temperatures due to quantum effects and weaker intermolecular forces
• He (monatomic, inert) shows minimal deviation because of its weak intermolecular interactions
• N₂ and Air show intermediate behavior based on their composition

The formula that utilizes this limiting behavior is: $T = \lim_{p_{tr} \to 0} \frac{p}{p_{tr}} \times 273.16 \text{ K}$

This approach eliminates dependence on any specific substance's properties, creating a truly universal temperature scale.

• Water's maximum density at 4°C creates temperature stratification in lakes and ponds during winter:

• 4°C water sinks to the bottom (being densest)
• Colder water (0-4°C) and ice remain at the top
• This prevents lakes from freezing solid from bottom to top

• Ecological benefits:

• Creates a 4°C refuge at lake bottoms where aquatic organisms can survive
• Allows fish and other aquatic life to remain active below the ice layer
• Preserves underwater ecosystems during harsh winters
• Enables year-round underwater biodiversity in cold climates

Note: Without this property, lakes would freeze completely in winter, making year-round aquatic life impossible in colder regions.