Heat and Temperature

Heat is energy transferred between bodies without any mechanical work involved.

Key distinctions: • Energy flows from hot body to cold body • No displacements are involved (unlike mechanical work) • Occurs through thermal contact, not through forces causing displacement • Results in internal energy changes

Example: When a hot cup of coffee and cold spoon touch, energy transfers from coffee to spoon without mechanical work.

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.

Mathematical definition: $t = \frac{l-l_0}{l_{100}-l_0} \times 100$ degrees

Where: • $l$ = current length of mercury column • $l_0$ = length at ice point (0°C) • $l_{100}$ = length at steam point (100°C)

Assumptions: • Linear relationship between temperature and mercury expansion • Uniform cross-section of capillary tube • Constant coefficient of expansion across measured temperature range • Negligible expansion of glass container • Stable reference points (ice point at 1 atm, steam point at 1 atm)

This definition forces uniformity on the mercury scale by definition rather than by physical properties.

For an isotropic material: $\gamma = 3\alpha$

Derivation:

1. For linear expansion: $L = L_0(1 + \alpha\Delta T)$
2. For a cube with initial side $L_0$: • Final side length = $L = L_0(1 + \alpha\Delta T)$ • Initial volume = $V_0 = L_0^3$ • Final volume = $V = L^3 = L_0^3(1 + \alpha\Delta T)^3$
3. Expand: $V = L_0^3(1 + 3\alpha\Delta T + 3\alpha^2(\Delta T)^2 + \alpha^3(\Delta T)^3)$
4. For small $\alpha\Delta T$, higher-order terms are negligible: $V \approx L_0^3(1 + 3\alpha\Delta T) = V_0(1 + 3\alpha\Delta T)$
5. Compare with volume expansion formula: $V = V_0(1 + \gamma\Delta T)$
6. Therefore: $\gamma = 3\alpha$

This relationship only holds for isotropic materials (same expansion in all directions) and small temperature changes.

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.

Discrepancy causes: • Real gases deviate from ideal behavior • Intermolecular forces affect pressure-temperature relationship • Different molecular properties (size, mass, interactions) • Varying degrees of adsorption on container walls • Differing Joule-Thomson coefficients • Different virial coefficients in gas equations

Resolution through ideal gas temperature scale: • Temperatures converge as gas pressure approaches zero • Defined mathematically as: $T = \lim_{p_{tr} \to 0} \frac{p}{p_{tr}} \times 273.16\text{ K}$ • Takes limit as gas approaches ideal behavior • Eliminates effects of gas-specific properties • All gases behave identically in zero-pressure limit • Creates universal scale independent of thermometric substance • Mathematically equivalent to thermodynamic temperature

Example: Hydrogen, nitrogen, and oxygen thermometers give slightly different readings at normal pressures, but their extrapolated zero-pressure values converge to the same temperature.

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.

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.

Key design features making Callendar's thermometer superior:

• Symmetrical compensation system: Uses identical bulbs (A and C) and capillaries (cd and ef) to eliminate errors from temperature variations in connecting tubes

• Equal gas distribution: Contains equal amounts of the same ideal gas on both sides of the system, creating a balanced reference

• Pressure equality method: Uses a manometer (M) to detect when pressures equalize, rather than measuring absolute pressure

• Volume adjustment mechanism: Uses graduated mercury bulb (B) with stopcock (S) to remove precise amounts of mercury, converting a constant volume problem into a measurable volume change

• Reference system: Keeps one side at a known temperature (typically ice point) for direct comparison

• Mathematical precision: Provides a simple temperature formula T = (V/(V-v'))T₀ that doesn't depend on gas pressure measurement

This design eliminates the "dead space" error where connecting tubes remain at ambient temperature, which is the main source of inaccuracy in simple gas thermometers.

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.