Heat and Temperature

Heat and temperature are fundamental concepts in science, especially in physics and thermodynamics. While often used interchangeably in everyday language, they refer to different things. Temperature measures the average kinetic energy of particles in a substance, indicating how hot or cold it is. Heat, on the other hand, is a form of energy transferred between objects due to a temperature difference. Understanding the difference between heat and temperature is essential in fields like engineering, climate science, and everyday applications like cooking or heating systems.

1.0Heat

Heat is energy transferred between two bodies (or within parts of a body) due to a temperature difference. It exists only during transfer; once absorbed, it becomes internal energy.
Phrases like “heat in a body” or “heat of a body” are incorrect. Heat is not stored—it is transferred.

Effect of Heating: When a body is heated, its molecules move faster, increasing their kinetic energy.

Units of Heat:

SI unit: Joule (J)
Common unit: Calorie (cal)
1 calorie = 4.18 joules

Definition of 1 Calorie: The heat needed to raise the temperature of 1 gram of water from 14.5°C to 15.5°C at 1 atm pressure.

Mechanical Equivalent of Heat

Heat was once seen as different from mechanical energy. Later experiments proved they are equivalent.

Mechanical Work and Heat: Mechanical work can raise temperature just like heat. The relationship is: $W = J \times H$

W: mechanical work ,H: heat supplied ,J: mechanical equivalent of heat.

Value of J (Joule/Calorie): J = 4.18 J/cal

Note: This means 4.18 joules of mechanical work produces the same effect as 1 calorie of heat.

Specific Heat

Specific heat (s) is the amount of heat required to raise the temperature of 1 unit mass of a substance by 1°C (or 1 K).

Heat Exchange Formula:

Heat gained or lost (ΔQ) is proportional to:

Mass (m) of the substance
Change in temperature (ΔT)

$Δ Q \propto m Δ T \Rightarrow Δ Q = m s Δ T$

Differential Form:dQ=ms dT

If specific heat depends on temperature: $Q = \int m s d T$

$s = \frac{Q}{m Δ T}$

Units of Specific Heat: SI Unit: J/kg·K; CGS Unit: cal/g·°C

Specific heat of water: 4200 J/kg·°C=1000 cal/kg·°C=1 kcal/kg·°C=1 cal/g·°C

Specific heat of steam = half of specific heat of water = specific heat of ice

Heat Capacity or Thermal Capacity

Heat capacity of a body is defined as the amount of heat required to raise the temperature of that body by1°C . If 'm' is the mass and 's' the specific heat of the body, then Heat capacity = ms.

Units of heat capacity in: CGS system is, cal $° C^{- 1}$ ; SI unit is, $J K^{- 1}$

Key Points

Specific Heat During Phase Change: During a change of state (like melting or boiling), the temperature remains constant → ΔT = 0.Heat is still absorbed or released $(Q \neq = 0)$

$s = \frac{Q}{m Δ T} \Rightarrow s = \frac{Q}{0} = \infty$
Specific heat becomes infinite during phase changes

Specific Heat When Q = 0 :If no heat is transferred (Q = 0) but temperature changes, then:

$s = \frac{Q}{m Δ T} \Rightarrow s = \frac{0}{m Δ T} = 0$
Example: Shaking a liquid in a thermos increases its temperature due to mechanical energy, not heat. Specific heat is zero in such cases.

Specific Heat of Saturated Water Vapour: For saturated water vapour, increasing temperature requires removal of heat (Q < 0, ΔT > 0).

$s = \frac{Q}{m Δ T} < 0$ . Specific heat is negative

Specific Heat of Water: Specific heat of water shows a slight variation with temperature (0°C to 100°C) at 1 atm pressure .The variation is less than 1%. It is usually treated as a constant:

Relation between Specific Heat and Water Equivalent

It is the amount of water which requires the same amount of heat for the same temperature rise as that of the object (mass m and specific heat s).

$m s Δ T = m_{w} s_{w} Δ T \Rightarrow m_{w} = \frac{m s}{s _{w}}$

In calorie $s_{w} = 1 m_{w} = m_{s}$

$m_{w}$ is also represented by W

W=ms

Phase Change: Heat required for the change of phase or state, Q = mL, L = latent heat.
Latent heat (L): The heat supplied to a substance of unit mass which changes its state at constant temperature is called latent heat of the body.
Latent heat of Fusion ( $L_{f}$ ):The heat supplied to a substance which changes it from solid to liquid state at its melting point and1 atm pressure is called latent heat of fusion. Latent heat of fusion of ice is 80 kcal/kg
Latent heat of vaporization ( $L_{v}$ ): The heat supplied to a substance which changes it from liquid to vapour state at its boiling point and 1 atm pressure is called latent heat of vaporization. Latent heat of vaporization of water is 540 kcal kg-1
Latent heat of ice : L = 80 cal/gm = 80 Kcal/kg = 4200 × 80 J/kg
Latent heat of steam : L = 540 cal/gm = 540 Kcal/kg = 4200 × 540 J/kg

Heating Curve

If to a given mass (m) of a solid, heat is supplied at constant rate and a graph is plotted between temperature and time as shown in figure, then the graph is called heating curve.

Region OA (Solid heating)

Heat supplied at constant rate P
Temperature of solid rises; $Q = m c_{s} Δ T = P Δ T$
Slope of temperature-time curve: $\frac{Δ T}{Δ t} = \frac{P}{m c _{s}}$
Specific heat $c_{s}$ is inversely proportional to the slope of OA.

Region AB (Melting at constant temperature $T_{1}$ )

Temperature remains constant → phase change (solid → liquid).
Heat supplied is used for melting: $Q = m L_{f} = P (t_{2} - t_{1})$
Latent heat of fusion Lf ∝ length of line AB (zero slope).
Specific heat in this region is infinite (since temperature doesn’t change).

Region BC (Liquid heating)

The temperature of the liquid rises.
Slope of BC: $\frac{Δ T}{Δ t} = \frac{P}{m c _{l}}$
Specific heat of liquid $c_{l}$ inversely proportional to the slope of BC.

Region CD (Boiling at constant temperature T2)

Temperature constant → phase change (liquid → vapor).
Heat supplied used for vaporization: $Q = m L_{v} = P (t_{4} - t_{3})$
Latent heat of vaporization Lv ∝ length of line CD (zero slope).
Specific heat is infinite during boiling.

Region DE (Vapour heating)

The temperature of vapor increases linearly.
Slope of DE: $\frac{Δ T}{Δ t} = \frac{P}{m c _{v}}$
Specific heat of vapor cv inversely proportional to slope of DE.

2.0Temperature

The branch of thermodynamics which deals with the measurement of temperature is called thermometry. A thermometer is a device used to measure the temperature of a body. The substances like liquids and gases which are used in the thermometer are called thermometric substances.

Different Scales of Temperature

A thermometer can be graduated into following scales.

(a) The Centigrade or Celsius scale (°C)

(b) The Fahrenheit scale (°F)

(c) Kelvin scale of temperature

Comparison between Different Temperature Scales

The formula for the conversion between different temperature scales is:

$\frac{K - 273}{100} = \frac{C}{100} = \frac{F - 32}{180}$

General formula for the conversion of temperature from one scale to another:

$\frac{Temp. on one scale ( S _{1} ) - Lower fixed point ( S _{1} )}{Upper fixed point ( S _{1} ) - Lower fixed point ( S _{1} )} = \frac{Temp. on one scale ( S _{2} ) - Lower fixed point ( S _{2} )}{Upper fixed point ( S _{2} ) - Lower fixed point ( S _{2} )}$

Thermometers

Thermometers are devices that are used to measure temperatures. All thermometers are based on the principle that some physical property of a system changes as the system temperature changes.

Required properties of a good thermometric substance.

(1) Non-sticky (absence of adhesive force)

(2) Low melting point (in comparison with room temperature)

(3) High boiling temperature

(4) Coefficient of volumetric expansion should be high (to increase accuracy in measurement).

(5) Heat capacity should be low.

(6) Conductivity should be high

Note: Mercury (Hg)suitably exhibits above properties.

3.0Heat Transfer

Heat is a form of energy which transfers from a body at higher temperature to a body at lower temperature. The transfer of heat from one body to another may take place by any of the following modes :

1. Conduction: The process in which the material takes an active part by molecular action and energy is passed from one particle to another is called conduction. It is predominant in solids.

2. Convection: The transfer of energy by actual motion of particles of medium from one place to another is called convection. It is predominant in fluids (liquids and gases).

3. Radiation: The quickest way of transmission of heat is known as radiation. In this mode of energy transmission, heat is transferred from one place to another without heating the inter–vening medium.

Conduction	Convection	Radiation
Due to free electron or vibration motion of molecules	Actual motion of particles	Electromagnetic radiation
Heat transfer in solid body (in mercury also)	Heat transfer in fluids (Liquid + gas)	All
Slow process	Slow process	Fast process $(3 \times 1 0^{8} m / sec)$

Process	Medium
Conduction	Required
Convection	Required
Radiation	No Required

Steady and Variable State

A metal rod AB has end A placed inside a heater and end B exposed to the surroundings. The rod is insulated on the sides with a poor conductor like cotton. Three thermometers ( $T_{1}, T_{2}, T_{3}$ )are placed along the rod at points 1, 2, and 3 from A to B.
Initially, the entire system is at room temperature, and all thermometers show the same reading. When the heater is switched on, end A heats up first, and heat begins to conduct along the rod. This causes $T_{1} > T_{2} > T_{3}$ with each section's temperature rising over time. This phase is called the variable state, where heat is absorbed by the rod.
Eventually, when the temperature at end B stabilizes and matches the surroundings, all temperatures along the rod become constant. This is the steady state, where heat flows through the rod without further temperature change at any point.

Conduction

Conduction is the transfer of heat through a material without movement of the particles themselves—heat is passed from one particle to the next.
Example: Holding an iron rod in fire—heat travels from the hot end to the handle, making it warm over time.

Mechanism:

At the hot end, atoms and electrons vibrate more due to higher temperature.
These vibrations are passed on to neighboring particles through collisions.
This creates a temperature gradient along the rod.

Heat Transfer Through a Slab:

Consider a slab of area A and thickness L.
One face is at temperature $T_{H}$ , the other at $T_{C} (T_{H} > T_{C}) .$
Take two points A and B in the slab separated by a small distance dx.
Let temperature at A = T, and at B = T + dT.

The rate of heat flow Q through area A in time t is given by:

$\frac{Q}{t} = - K A \frac{d T}{d x}$

Thermal Conductivity (K):A constant specific to the material, indicating how well it conducts heat.

Temperature Gradient( $\frac{d T}{d x}$ ) : The rate of change of temperature with distance in the slab.

Negative Sign (–):Indicates heat flows from higher to lower temperature since $\frac{d T}{d x}$ is negative.

SI Unit of Thermal Conductivity $J s^{- 1} m^{- 1} K^{- 1}$
Dimensions of $K = [M^{1} L^{1} T^{- 3} Θ^{- 1}]$
Perfect conductor: $K = \infty$
Perfect insulator: K=0
For cooking the food, low specific heat and high conductivity utensils are most suitable.

Steady State in Heat Conduction

When temperature at any cross-section x of the slab remains constant over time (but varies with position), the slab is in steady state.
In steady state, temperature varies along the slab; in thermal equilibrium, temperature is uniform everywhere.
No heat is absorbed or emitted at any cross-section (temperature is time-constant).
The two faces of the slab are kept at constant temperatures $T_{H}$ (hot) and $T_{C}$ (cold).
All other faces are insulated (adiabatic) to prevent heat loss.

Heat flow is constant across all cross-sections

$Q_{1} = Q_{2} = Q_{3}$

Consequently, the temperature gradient is constant throughout the slab.

$\frac{d T}{d x} = \frac{Δ T}{L} = \frac{T _{H} - T _{C}}{L} = \frac{T _{C} - T _{H}}{L}$

$\frac{Q}{t} = - K A \frac{Δ T}{L}$

$\frac{Q}{t} = K A (\frac{T _{H} - T _{C}}{L})$

Here Q is the amount of heat flowing through a cross-section of slab at any position in a time interval of t.

Heat Flow from a Uniform Rod at Steady State

In steady state temperature of each element of the rod becomes constant w.r.t. to time i.e. rate of heat flow at energy cross-section of the rod will be the same.

In steady state $\frac{d Q}{d t}$ is same for each element

$\frac{d Q}{d t} = K A \frac{d T}{d x}$

For uniform rod K and A same for each element

$\frac{d T}{d x} is same \Rightarrow \frac{d T}{d x} = \frac{Δ T}{Δ x}$

$\Rightarrow \frac{d Q}{d t} = K A (\frac{Δ T}{Δ x})$

Thermal Resistance to Conduction

Thermal resistance R helps quantify how well a material resists heat flow—important for insulation (e.g., houses, tiffin boxes).

$R = \frac{L}{K A}$

L = thickness, A = cross-sectional area, K = thermal conductivity.

Heat Flow in Terms of R:

$\frac{Q}{t} = \frac{T _{H} - T _{C}}{R}$

Thermal Current ( $i_{T}$ )

$i_{T} = \frac{Q}{t} = \frac{T _{H} - T _{C}}{R}$

Temperature difference acts like voltage (electric potential).
Thermal current is analogous to electric current.
Thermal resistance corresponds to electrical resistance.
Thermal current $i_{T} $ is constant across cross-sections in steady state, similar to Kirchhoff’s current law in electrical circuits.

Heat Flow Through Slabs in Series

$R_{eq} = R_{1} + R_{2}$

$\frac{L _{1} + L _{2}}{K _{eq} A} = \frac{L _{1}}{K _{1} A} + \frac{L _{2}}{K _{2} A}$

Equivalent thermal conductivity of the system is

$K_{eq} = \frac{L _{1} + L _{2}}{\frac{L _{1}}{K _{1}} + \frac{L _{2}}{K _{2}}} = \frac{\sum L}{\sum \frac{L _{i}}{K _{i}}}$

Heat Flow Through Slabs in Parallel

$\frac{1}{R _{eq}} = \frac{1}{R _{1}} + \frac{1}{R _{2}}, R = \frac{L}{K A}$

$\frac{K _{eq}}{L} (A_{1} + A_{2}) = \frac{K _{1} A _{1}}{L} + \frac{K _{2} A _{2}}{L}$

Equivalent thermal conductivity

$K_{eq} = \frac{K _{1} A _{1} + K _{2} A _{2}}{A _{1} + A _{2}} = \frac{\sum K _{i} A _{i}}{\sum A _{i}}$

Convection

Convection is the mode of heat transfer in fluids (liquids and gases) where the particles themselves move, carrying heat energy with them.
It occurs due to the difference in temperature and density within the fluid. Hotter, lighter fluid rises, and cooler, denser fluid sinks, forming a convection current.

Example:

Boiling water in a pot: Water at the bottom gets heated, becomes lighter, and rises. Cooler water at the top sinks to take its place, forming a circular motion of fluid – this is convection.

Thermal Radiation

Transfer of heat without a material medium, through electromagnetic waves.
Example: The Sun’s heat reaching Earth is due to radiation.
Medium Not Required: Unlike conduction and convection, radiation works in a vacuum.

Properties of Thermal Radiation:

Travels in straight lines – forms shadows when blocked.
Can pass through vacuum – no medium is needed.
Follows inverse square law – $in t e n s i t y \propto \frac{1}{r ^{2}}$
Can be polarised – similar to light (e.g., using Nicol prism).
Electromagnetic in nature – behaves like light but has a longer wavelength than visible light.

Radiation

Radiation is the transfer of energy via electromagnetic waves, requiring no medium.
Even cold objects like ice cubes emit radiation (mostly not visible to the eye).
The amount of energy absorbed or emitted depends on the surface texture and color.

Example – Two Blocks in Sunlight:

Lampblack-coated block:

Rough black surface
Absorbs ~97% of radiation
Heats up faster

Blackbody Concept:

A blackbody absorbs all electromagnetic radiation falling on it.
It appears black because it reflects almost no light.

4.0Sample Questions on Heat And Temperature

Q-1.Calculate the quantity of heat conducted through $2 m^{2}$ of a brick wall 12 cm thick in 1 hour if the temperature on one side is 8 ℃ and on the other side is 28 ℃. $(Thermal conductivity of brick = 0.13 W m^{- 1} K^{- 1})$

Solution:

$Temperature gradient = \frac{28 - 8}{12 \times 1 0 ^{- 2}} K m^{- 1} and t = 3600 s$

$Q = k A t ✕ t e m p er a t u re g r a d i e n t = 0.13 \times 2 \times 3600 \times \frac{28 - 8}{12 \times 1 0 ^{- 2}} = 156000 J$

Q-2.Thick spherical shell. Find rate of heat flow ?

Solution:

$\frac{d Q}{d t} = P = - K 4 π x^{2} \frac{d T}{d x}$

$\int_{r_{1}}^{r_{2}} \frac{d x}{x ^{2}} = \frac{K 4 π}{P} \int_{T_{1}}^{T_{2}} d T$

$[\frac{1}{r _{1}} - \frac{1}{r _{2}}]$ $= \frac{K 4 π}{P} (T_{1} - T_{2})$

$P = K 4 π \frac{r _{1} r _{2}}{r _{2} - r _{1}} (T_{1} - T_{2})$

Sample Solutions on Heat and temperature

Q-3.The readings of a thermometer at 0°C and 100°C are 50 cm and 75 cm of mercury columns respectively. Find the temperature at which its reading is 80 cm of the mercury column?

Solution: Let reference temperature at 0°C

75=501+ ✕ 100…..(1)

80=501+ ✕ T……..(2)

From equation (1) and (2)

$\frac{80 - 50}{75 - 50} = \frac{T - 0}{100 - 0} \Rightarrow T = 120° C$

Q-4.Specific heat of a substance depends on temperature as $s = 3 T^{2}$ , where temperature is in kelvin and the constant 3 is in SI units. Find heat required to raise the temperature of unit mass from 2K to 5K ?

Solution:

$\int d Q = \int m s d T$

$Δ Q = 1 \int_{2}^{5} 3 T^{2} d T$

$Δ Q = [5^{3} - 2^{3}] = 117 J$

Heat and Temperature

1.0Heat

Mechanical Equivalent of Heat

Specific Heat

Heat Capacity or Thermal Capacity

Heating Curve

2.0Temperature

Different Scales of Temperature

Comparison between Different Temperature Scales

Thermometers

3.0Heat Transfer

Conduction

Convection

Radiation

4.0Sample Questions on Heat And Temperature

Table of Contents

Frequently Asked Questions

1

What is heat?

2

What are Thermometers?

3

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