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.

Mathematical comparison: • Mercury: $t = \frac{l-l_0}{l_{100}-l_0} \times 100$ degrees • Platinum: $t = \frac{R_t-R_0}{R_{100}-R_0} \times 100$ degrees

Where: • $R_t$ = resistance at temperature t • $R_0$ = resistance at ice point • $R_{100}$ = resistance at steam point

Key operational differences: • Platinum uses electrical resistance instead of thermal expansion • Uses Wheatstone bridge setup with compensating wires to eliminate connection resistance errors • More precise at extreme temperatures • Less susceptible to environmental factors • Measures temperature indirectly through electrical measurements • Can be designed for remote sensing

Both scales agree at fixed points but diverge at intermediate temperatures due to different material 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.

Calibration: • Measure gas pressure ($p_{tr}$) at triple point of water (273.16 K) • Maintain fixed gas volume by adjusting mercury level to fixed mark

Operation:

1. Place bulb in environment to be measured
2. Allow thermal equilibrium
3. Adjust mercury reservoir to maintain constant gas volume
4. Measure new gas pressure (p)
5. Calculate temperature using formula

Fundamental equation: $T = \frac{p}{p_{tr}} \times 273.16\text{ K}$

For Celsius scale: $t = \frac{p-p_0}{p_{100}-p_0} \times 100\text{°C}$

Where: • $p_0$ = pressure at ice point • $p_{100}$ = pressure at steam point

This directly relates measured pressure ratio to absolute temperature, approaching true thermodynamic temperature as gas pressure approaches zero.

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 constant volume gas thermometer operates on the principle that at constant volume, the pressure of a gas is directly proportional to its absolute temperature.

Key components and operation:

• A gas-filled bulb is connected to a U-shaped mercury manometer
• The mercury level in one arm is maintained at a fixed mark (C) by adjusting reservoir F
• This ensures the gas volume remains constant
• The pressure is calculated as: P = P₀ + hρg, where:
  • P₀ is atmospheric pressure
  • h is the height difference between mercury columns
  • ρ is the density of mercury
  • g is acceleration due to gravity

The temperature is determined using: T = (p/pₜᵣ) × 273.16 K, where pₜᵣ is the pressure at the triple point of water.

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.

• Water's volume DECREASES as temperature rises from 0°C to 4°C • This is anomalous (unusual) behavior, as most substances continuously expand when heated • Water has a negative thermal expansion coefficient in this temperature range • Above 4°C, water behaves normally and expands when heated • At precisely 4°C, water reaches its minimum volume and maximum density (1 g/cm³)

Example: A water bottle filled completely at 4°C will overflow if either cooled to 0°C or heated to 10°C, due to expansion in both directions from the density maximum.