Temperature

1. Concept Overview

Temperature is a fundamental physical quantity that determines the degree of hotness or coldness of a body.
It is a measure of the average kinetic energy of the particles (atoms or molecules) in a substance.

When two bodies are in thermal contact:

• Heat flows from the body at higher temperature to the one at lower temperature, until both reach the same temperature.
  This condition is known as Thermal Equilibrium.

Thus, temperature acts as an indicator of the direction of heat transfer.

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2. Explanation and Mathematical Representation

Definition of Temperature

Temperature is a measure of the average kinetic energy of the particles of a body.
It does not depend on the total energy or mass, but on the average energy per particle.

Mathematically, for an ideal gas:

[
\langle E_t \rangle = \dfrac{3}{2}kT
]

where:

• [\langle E_t \rangle] = average kinetic energy per molecule
• [k] = Boltzmann constant = [1.38 \times 10^{-23} J/K]
• [T] = absolute temperature (in Kelvin)

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3. Temperature Scales

Temperature can be measured using various scales. The most common ones are:

1. Celsius Scale (°C)
2. Fahrenheit Scale (°F)
3. Réaumur Scale (°R)
4. Kelvin Scale (K) — the absolute scale used in scientific work.

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(a) Celsius Scale (°C)

• Defined between the freezing point of water (0°C) and the boiling point of water (100°C) at standard atmospheric pressure.
• The interval between these two fixed points is divided into 100 equal parts, each representing 1°C.

[
T_C = \text{Temperature in Celsius}
]

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(b) Fahrenheit Scale (°F)

• Defined between freezing point of water (32°F) and boiling point of water (212°F).
• The interval between these two points is divided into 180 equal parts, each representing 1°F.

[
T_F = \text{Temperature in Fahrenheit}
]

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(c) Réaumur Scale (°R)

• Defined between freezing point of water (0°R) and boiling point (80°R).
• Thus, one degree Réaumur is 1/80 of the temperature interval between freezing and boiling points of water.

[
T_R = \text{Temperature in Réaumur}
]

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(d) Kelvin Scale (K)

• The Kelvin scale is the absolute scale of temperature.
• It starts from absolute zero (0 K), the point where molecular motion ceases.
• The relation between Celsius and Kelvin is:

[
T_K = T_C + 273.15
]

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4. Relationship Between Temperature Scales

Let:

• [T_C] = temperature on Celsius scale
• [T_F] = temperature on Fahrenheit scale
• [T_R] = temperature on Réaumur scale

Then, the ratio of temperature intervals between these scales is:

[\dfrac{T_C – 0}{100}] [= \dfrac{T_F – 32}{180}] [= \dfrac{T_R – 0}{80}]

or simply,

[\dfrac{T_C}{100}] [= \dfrac{T_F – 32}{180}] [= \dfrac{T_R}{80}]

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5. Conversion Formulas Between Scales

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6. Dimensions and Units

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7. Key Features

• Temperature measures average kinetic energy of particles.
• Celsius, Fahrenheit, and Réaumur scales are relative, while Kelvin is absolute.
• The ratio of divisions among Celsius : Fahrenheit : Réaumur = 100 : 180 : 80 or 5 : 9 : 4.
• Absolute zero corresponds to [0 , K = -273.15°C = -459.67°F].
• All conversions are linear relationships.

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8. Conceptual Questions with Solutions

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9. FAQ / Common Misconceptions

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10. Practice Questions (With Step-by-Step Solutions)

Q1. Convert 25°C to °F and K.
Solution:
[\begin{align*}
T_F &= \dfrac{9}{5}T_C + 32 = \dfrac{9}{5} \times 25 + 32 = 77°F \
T_K &= T_C + 273.15 = 25 + 273.15 = 298.15,K
\end{align*}]

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Q2. Convert 300 K to °C and °F.
Solution:
[\begin{align*}
T_C &= T_K – 273.15 = 26.85°C \
T_F &= \dfrac{9}{5} \times 26.85 + 32 = 80.33°F
\end{align*}]

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Q3. Find the Celsius temperature equivalent to 68°F.
Solution:
[T_C = \dfrac{5}{9}(68 – 32) = 20°C]

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Q4. Convert 40°C to the Réaumur scale.
Solution:
[T_R = \dfrac{4}{5} \times 40 = 32°R]

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Q5. Convert 60°R to °C and °F.
Solution:
[\begin{align*}
T_C &= \dfrac{5}{4} \times 60 = 75°C \
T_F &= \dfrac{9}{5} \times 75 + 32 = 167°F
\end{align*}]