Class 12 > Unit # 15:Molecular Theory of Gases > Temperature , Conversion and Triple Point of Water

Q.1 Define temperature and thermal equilibrium. Derive the empirical formulas for conversion between different scales of temperature.

Temperature

Temperature is a macroscopic measure of the average translational kinetic energy of the molecules composing a body. In simpler terms, it is the fundamental property of matter that quantifies the degree of hotness or coldness of an object. The S.I. unit of temperature is the Kelvin (K).

Thermal Equilibrium

When two bodies at different temperatures are brought into physical/thermal contact with each other, heat spontaneously flows from the body at a higher temperature to the body at a lower temperature. This net energy transfer continues until the temperatures of both bodies equalize. At this point, no further net heat flow occurs, and the bodies are said to be in a state of thermal equilibrium.

Scales of Temperature

Thermometers are calibrated using two fixed references: the melting point of pure ice (Lower Fixed Point) and the boiling point of pure water at standard atmospheric pressure (Upper Fixed Point).

1. Celsius Scale (°C)
Designed by Anders Celsius

• Lower Fixed Point = $0^\circ\text{C}$
• Upper Fixed Point = $100^\circ\text{C}$
• Total divisions = $100$ equal parts.

2. Fahrenheit Scale (°F)
Designed by Daniel Fahrenheit

• Lower Fixed Point = $32^\circ\text{F}$
• Upper Fixed Point = $212^\circ\text{F}$
• Total divisions = $180$ equal parts.

3. Kelvin Scale (K)
Designed by Lord Kelvin

• Lower Fixed Point = $273\text{ K}$
• Upper Fixed Point = $373\text{ K}$
• Lowest point = Absolute Zero ($0\text{ K}$)

Mathematical Relation & Derivation

If three calibrated thermometers are placed simultaneously into a uniform liquid bath, the mercury column will reach an identical relative height in each. We derive the relationship based on the principle that the ratio of the reading length to total scale interval length is constant across all linear scales:

$$\frac{\text{Reading} - \text{Lower Fixed Point}}{\text{Upper Fixed Point} - \text{Lower Fixed Point}} = \text{Constant}$$ $$\frac{T_C - 0}{100 - 0} = \frac{T_F - 32}{212 - 32} = \frac{T_K - 273}{373 - 273}$$ $$\frac{T_C}{100} = \frac{T_F - 32}{180} = \frac{T_K - 273}{100}$$

Simplifying the denominators by dividing each term by $20$ yields the core identity line:

$$\frac{T_C}{5} = \frac{T_F - 32}{9} = \frac{T_K - 273}{5}$$

I) Conversion Between Kelvin and Celsius:

Equating the first and third parts of our core master identity:

$$\frac{T_C}{5} = \frac{T_K - 273}{5} \implies T_C = T_K - 273$$ $$T_K = T_C + 273$$

II) Conversion Between Fahrenheit and Celsius:

Equating the first and second parts of our core master identity:

$$\frac{T_C}{5} = \frac{T_F - 32}{9} \implies T_F - 32 = \frac{9}{5}T_C$$ $$T_F = \frac{9}{5}T_C + 32 \quad \text{or} \quad T_F = 1.8T_C + 32$$ $$T_C = \frac{5}{9}(T_F - 32)$$

Q.2 Define the Triple Point of water. Explain its thermodynamic significance.

The Triple Point of Water

The Triple Point of water is the unique, highly specific combination of temperature and pressure at which the solid (ice), liquid (pure water), and gaseous (water vapor) phases of water coexist in perfect thermodynamic equilibrium without any net phase change occurring between them.

Standard International Parameters: By international agreement, these unique equilibrium conditions occur at a precise absolute temperature and vapor pressure:

$$\text{Temperature } (T_3) = 273.16\text{ K} \quad (0.01^\circ\text{C})$$ $$\text{Pressure } (P_3) = 611.73\text{ Pa} \quad (4.58\text{ mm of Hg})$$

Thermodynamic Scale Definition

Unlike the standard melting point of ice (which shifts significantly with ambient atmospheric pressure variations), the Triple Point of water is entirely invariant. For this reason, it serves as the foundational calibration reference point for standard international thermometry. By agreement, the size of a single Kelvin unit is defined precisely as $\frac{1}{273.16}$ of the difference between the Triple Point temperature of water ($T_3$) and absolute zero ($0\text{ K}$).