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Physics · Thermal Properties of Matter

Specific Heat Capacity

Equal masses of different substances, given equal heat, end up at different temperatures. NCERT §10.6 turns that observation into three precise quantities — heat capacity $S$, specific heat capacity $s$, and molar specific heat capacity $C$ — joined by the working formula $\Delta Q = m\,s\,\Delta T$. This deep-dive separates the extensive quantity from the intensive ones, sets out the gas pair $C_p$ and $C_v$ with Mayer's relation, explains why water's specific heat is anomalously high, and drills the NEET trap that turns the copper-sphere PYQ into a quick win.

Heat capacity of a body

The quantity of heat needed to warm a substance depends on three things: its mass $m$, the temperature change $\Delta T$, and the nature of the substance. The first quantity that bundles these is the heat capacity of a body. If $\Delta Q$ is the heat that changes a body's temperature from $T$ to $T+\Delta T$, the heat capacity is

$$ S = \frac{\Delta Q}{\Delta T} $$

Heat capacity carries the SI unit $\text{J K}^{-1}$. It is a property of the whole body, not of the material alone — a swimming pool and a teacup of the same water have very different heat capacities because they hold different masses. For this reason heat capacity is an extensive quantity: scale up the amount of substance and you scale up $S$ in direct proportion.

Specific heat capacity

To get a number that describes the material itself, NCERT divides out the mass. The specific heat capacity $s$ is the heat absorbed or given off per unit mass to change the temperature by one unit:

$$ s = \frac{S}{m} = \frac{\Delta Q}{m\,\Delta T} $$

Its SI unit is $\text{J kg}^{-1}\,\text{K}^{-1}$. Because the mass has been divided away, specific heat capacity is an intensive quantity: it depends only on the nature of the substance and on its temperature, not on how much of it you have. A drop of water and an ocean of water share the same specific heat capacity, $4186~\text{J kg}^{-1}\text{K}^{-1}$, even though their heat capacities differ enormously. NCERT is explicit that $s$ describes the temperature change of a substance undergoing no phase change; the moment ice begins to melt, the formula no longer applies and latent heat takes over.

Figure 1 · Heat budget Heat in ΔQ Body mass m specific heat s Temp rise ΔT ΔQ = m s ΔT
Heat supplied to a single-phase body raises its temperature in proportion to mass and specific heat. Larger $m$ or larger $s$ means more heat for the same $\Delta T$.

Molar specific heat capacity

When the amount of substance is counted in moles $\mu$ rather than kilograms, the heat capacity per mole is the molar specific heat capacity $C$:

$$ C = \frac{S}{\mu} = \frac{\Delta Q}{\mu\,\Delta T} $$

Its SI unit is $\text{J mol}^{-1}\,\text{K}^{-1}$. Like $s$, the molar specific heat capacity depends on the nature of the substance and its temperature. It is the natural measure for gases, where counting particles in moles is more useful than weighing them. NCERT flags one subtlety here: for a gas, heat can be supplied while holding either the pressure or the volume fixed, and these two routes need two different molar specific heats. That distinction is the subject of the $C_p$–$C_v$ section below.

The three quantities side by side

The three definitions differ only in what is divided out. Hold this table in mind and most NEET wording traps dissolve.

QuantitySymbol & definitionSI unitNature
Heat capacity$S = \dfrac{\Delta Q}{\Delta T}$J K⁻¹Extensive — depends on amount of substance
Specific heat capacity$s = \dfrac{\Delta Q}{m\,\Delta T}$J kg⁻¹ K⁻¹Intensive — per unit mass
Molar specific heat capacity$C = \dfrac{\Delta Q}{\mu\,\Delta T}$J mol⁻¹ K⁻¹Intensive — per mole

The links between them follow at once: $S = m\,s = \mu\,C$. If the molar mass is $M$ (in kg per mole), then $C = M\,s$, because one mole has mass $M$ and $C$ is just $s$ scaled up to a mole's worth of substance.

Using ΔQ = m s ΔT

Rearranging the definition of specific heat capacity gives the single most-used formula in calorimetry:

$$ \Delta Q = m\,s\,\Delta T $$

This is the heat exchanged when a mass $m$ of a substance of specific heat $s$ changes temperature by $\Delta T$, with no change of state. A positive $\Delta T$ (warming) means heat absorbed; a negative $\Delta T$ (cooling) means heat released. The same expression with $C$ in place of $s$ and moles in place of mass reads $\Delta Q = \mu\,C\,\Delta T$.

Worked example

How much heat is needed to raise the temperature of 2.0 kg of water from 25 °C to 75 °C? Take $s_{\text{water}} = 4186~\text{J kg}^{-1}\text{K}^{-1}$.

A temperature change of 50 °C equals a change of 50 K, because both scales share the same degree size. Then $\Delta Q = m\,s\,\Delta T = 2.0 \times 4186 \times 50 = 4.186 \times 10^{5}~\text{J}$, that is about 419 kJ. The large figure is a direct consequence of water's high specific heat — the same calculation for 2.0 kg of copper would need less than a tenth of this heat.

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Related drill

The "heat lost equals heat gained" balance built on $\Delta Q = ms\Delta T$ is worked out fully in calorimetry.

Cp and Cv of a gas

For a gas, the molar specific heat depends on what is held fixed during heating. If pressure is held constant, the molar specific heat is $C_p$, the molar specific heat at constant pressure. If volume is held constant, it is $C_v$, the molar specific heat at constant volume. Across every gas in the NCERT table, $C_p$ exceeds $C_v$.

The reason is bookkeeping of energy. At constant volume the gas cannot expand, so all the supplied heat goes into raising the internal energy — and hence the temperature. At constant pressure the gas expands as it warms and does work pushing back its surroundings, so part of the heat is spent on that work and extra heat is needed to achieve the same temperature rise. The excess is exactly the universal gas constant, a result known as Mayer's relation:

$$ C_p - C_v = R, \qquad R = 8.31~\text{J mol}^{-1}\text{K}^{-1} $$
Gas$C_p$ (J mol⁻¹ K⁻¹)$C_v$ (J mol⁻¹ K⁻¹)$C_p - C_v$ (J mol⁻¹ K⁻¹)
He20.812.58.3
H₂28.820.48.4
N₂29.120.88.3
O₂29.421.18.3
CO₂37.028.58.5

The final column hovers near $8.3~\text{J mol}^{-1}\text{K}^{-1}$ for every gas — a clean experimental confirmation of $C_p - C_v = R$. NCERT defers the derivation to the chapter on thermodynamics; for this topic the relation and its direction $C_p > C_v$ are what NEET tests.

Water's anomalous specific heat

Among the substances NCERT tabulates, water stands out: its specific heat capacity of $4186~\text{J kg}^{-1}\text{K}^{-1}$ is the highest of the common materials, several times that of metals. So much heat is absorbed for so little temperature rise because a large share of the energy goes into rearranging the network of hydrogen bonds between water molecules rather than into raising their kinetic energy directly.

This single number explains a string of NCERT applications. Water is used as a coolant in automobile radiators because it can soak up a great deal of engine heat for a modest temperature rise. It is used as a heater in hot-water bags because, having stored that heat, it releases it slowly. On a planetary scale, water warms up more slowly than land during summer, so a sea breeze has a cooling effect; in desert regions, by contrast, the dry earth heats quickly by day and cools quickly by night. The high specific heat of water is the quiet regulator of coastal climate.

Figure 2 · Specific-heat comparison Lead 127.7 Tungsten 134.4 Silver 236.1 Copper 386.4 Carbon 506.5 Aluminium 900.0 Ice 2060 Water 4186.0 specific heat capacity / J kg⁻¹ K⁻¹
Water (highlighted) tops the NCERT list at $4186~\text{J kg}^{-1}\text{K}^{-1}$ — over four times copper and more than ten times lead. This anomaly makes water the standard coolant and a stabiliser of climate.

NCERT specific-heat values

These are the room-temperature, atmospheric-pressure values NCERT lists in Table 10.3. Use them directly in NEET numericals; do not round mid-calculation.

SubstanceSpecific heat capacity (J kg⁻¹ K⁻¹)
Water4186.0
Kerosene2118
Ice2060
Edible oil1965
Aluminium900.0
Carbon506.5
Copper386.4
Silver236.1
Tungsten134.4
Lead127.7

Two patterns are worth committing to memory. Liquids and ice cluster high (water far above the rest); dense metals such as lead and tungsten sit low. A low specific heat means a substance heats up and cools down with very little heat exchanged — the opposite of water's behaviour.

Quick recap

Specific heat capacity in one breath

  • Heat capacity $S = \Delta Q/\Delta T$, unit $\text{J K}^{-1}$ — extensive, depends on amount of substance.
  • Specific heat capacity $s = \Delta Q/(m\,\Delta T)$, unit $\text{J kg}^{-1}\text{K}^{-1}$ — intensive, per unit mass.
  • Molar specific heat $C = \Delta Q/(\mu\,\Delta T)$, unit $\text{J mol}^{-1}\text{K}^{-1}$; links: $S = m\,s = \mu\,C$ and $C = M\,s$.
  • Working formula (no phase change): $\Delta Q = m\,s\,\Delta T$.
  • For a gas $C_p > C_v$; Mayer's relation $C_p - C_v = R$, with $R = 8.31~\text{J mol}^{-1}\text{K}^{-1}$.
  • Water has the highest specific heat among common substances, $4186~\text{J kg}^{-1}\text{K}^{-1}$ — coolant, heater, climate moderator.

NEET PYQ Snapshot — Specific Heat Capacity

Two PYQs that turn on $\Delta Q = m s \Delta T$. Same idea each time: same material means same specific heat, so heat tracks mass.

NEET 2020

The quantities of heat required to raise the temperature of two solid copper spheres of radii $r_1$ and $r_2$ ($r_1 = 1.5\,r_2$) through 1 K are in the ratio:

  1. $9/4$
  2. $3/2$
  3. $5/3$
  4. $27/8$
Answer: (4) 27/8

Specific-heat driven. $\Delta Q = m\,s\,\Delta T$. Both spheres are copper, so $s$ is the same, and $\Delta T = 1$ K for both. Hence $\Delta Q \propto m \propto$ volume $\propto r^3$. Therefore $\dfrac{\Delta Q_1}{\Delta Q_2} = \left(\dfrac{r_1}{r_2}\right)^3 = (1.5)^3 = \dfrac{27}{8}$.

NEET 2016

A piece of ice falls from a height $h$ so that it melts completely. Only one-quarter of the heat produced is absorbed by the ice and all energy of ice gets converted into heat during its fall. The value of $h$ is (latent heat of ice $= 3.4 \times 10^{5}~\text{J kg}^{-1}$, $g = 10~\text{N kg}^{-1}$):

  1. 544 km
  2. 136 km
  3. 68 km
  4. 34 km
Answer: (2) 136 km

Heat balance. Gravitational energy $mgh$ becomes heat; only a quarter is absorbed by the ice, and that quarter must equal the heat of melting $m L_f$. So $\tfrac{1}{4} mgh = m L_f$, giving $h = \dfrac{4 L_f}{g} = \dfrac{4 \times 3.4 \times 10^{5}}{10} = 1.36 \times 10^{5}~\text{m} = 136$ km. Note that melting uses latent heat $L_f$, not $s\,\Delta T$ — temperature stays at the melting point throughout the change of state.

FAQs — Specific Heat Capacity

Short answers to the questions NEET aspirants get wrong most often on this topic.

What is the difference between heat capacity and specific heat capacity?
Heat capacity $S = \Delta Q/\Delta T$ is the heat needed to raise the temperature of a whole sample by one unit; it is an extensive quantity that depends on how much material you have. Specific heat capacity $s = \Delta Q/(m\,\Delta T)$ is the heat needed per unit mass, an intensive quantity that depends only on the substance and its temperature. Numerically $S = m\,s$, so doubling the mass doubles the heat capacity but leaves the specific heat unchanged.
What is the SI unit of specific heat capacity and of molar specific heat capacity?
Specific heat capacity $s$ is measured in $\text{J kg}^{-1}\text{K}^{-1}$ because it is heat per unit mass per unit temperature rise. Molar specific heat capacity $C$ is measured in $\text{J mol}^{-1}\text{K}^{-1}$ because the amount of substance is counted in moles instead of kilograms. Heat capacity $S$ itself carries the unit $\text{J K}^{-1}$.
Why is the specific heat capacity of water so high?
Water has the highest specific heat capacity among common substances, $4186~\text{J kg}^{-1}\text{K}^{-1}$ in the NCERT table. A large fraction of any heat supplied goes into breaking and rearranging the hydrogen-bond network between molecules rather than into raising the average kinetic energy, so the temperature climbs slowly. This is why water is used as a coolant in radiators and as a heater in hot-water bags, and why coastal climates are milder than interior deserts.
Why is Cp greater than Cv for a gas?
At constant volume all the heat supplied raises the internal energy, so $C_v$ accounts only for the temperature rise. At constant pressure the gas also expands and does work on its surroundings, so extra heat is needed for the same temperature rise. The difference is exactly $C_p - C_v = R$, Mayer's relation, where $R$ is the universal gas constant. Hence $C_p$ is always larger than $C_v$ by $R$.
Does specific heat capacity depend on temperature?
Yes. Specific heat capacity depends on the nature of the substance and on its temperature, as NCERT states. The values listed in tables are quoted at room temperature and atmospheric pressure. For NEET numerical problems these tabulated values are treated as constant over the temperature range of the question unless a variation is explicitly given.
How is the heat ΔQ calculated when there is no change of state?
When a substance only warms or cools without melting, boiling or freezing, the heat exchanged is $\Delta Q = m\,s\,\Delta T$, where $m$ is the mass, $s$ the specific heat capacity and $\Delta T$ the temperature change. If a phase change occurs, this formula no longer applies during the change because the temperature stays constant; latent heat must be used instead. The $ms\Delta T$ formula is valid strictly for single-phase heating or cooling.
Go deeper
Thermal Properties of Matter — Parent chapter Full chapter map: temperature, expansion, heat transfer, calorimetry. Temperature & Heat Heat as energy in transit; the meaning of thermal equilibrium. Calorimetry Heat lost = heat gained; finding unknown specific heats. Change of State & Latent Heat Where ΔQ = msΔT stops and latent heat takes over. Ideal Gas Equation & Absolute Temperature The gas constant R behind Mayer's relation Cp − Cv = R. Heat Transfer Modes Conduction, convection, radiation — how the heat arrives.