Specific heat capacity

What is specific heat capacity?

It is defined as the amount of heat per unit mass absorbed or given off by the substance to change its temperature by one unit.

NCERT Physics Part II (Class 11), Section 10.6, page 208

In plain words: some substances warm up quickly, others slowly. Specific heat capacity measures how much heat it takes to warm one kilogram of a substance by one kelvin. Water needs a lot; metals need little. For equal absorbed power, negligible losses and the same mass, iron’s lower specific heat gives a faster temperature rise than water’s; equal sunshine alone need not mean equal absorbed power.

What the heat needed depends on

NCERT describes a simple experiment with a burner and a stopwatch (page 208):

• heating a quantity of water through 40 °C takes about twice as long as through 20 °C, so the heat needed is proportional to the temperature rise ΔT;
• heating twice the amount of water through 20 °C also takes twice as long, so the heat needed is proportional to the mass m;
• heating the same mass of mustard oil through 20 °C takes less time than water, so the heat needed depends on the nature of the substance.

Together these give the working formula:

With constant specific heat, a heater of constant power P and no losses, the temperature rises at a steady rate ΔT/Δt = P/(ms), so a temperature–time graph is a straight line of slope P/(ms).

The specific heat capacity depends on the substance and, to some extent, on its temperature. For gases the conditions must also be stated: NCERT (page 208) defines the molar specific heat at constant pressure, C_p, and at constant volume, C_v, and leaves the details to the Thermodynamics chapter. C_p is the larger. For the same temperature rise, an ideal gas gains the same internal energy either way, but at constant pressure it also expands and does work PΔV on its surroundings. For one mole, PΔV = RΔT, so C_p − C_v = R (NCERT Eq. 11.8, page 232).

Values, and why water matters

NCERT Table 10.3 (page 209) gives specific heat capacities with its heading referring to room temperature and atmospheric pressure, in J kg^⁻¹ K^⁻¹: water 4186, ice 2060, kerosene 2118, edible oil 1965, aluminium 900, glass 840, carbon 506.5, iron 450, copper 386.4, silver 236.1, mercury 140, tungsten 134.4, lead 127.7. Ice is an exception to that heading: ordinary ice is stable below its melting point at 1 atm, and its quoted value is a near-freezing approximation. Use each value only over a suitable temperature range.

In the laboratory, the specific heat of a solid or another liquid is found in a calorimeter by the method of mixtures, taking water’s value as known. Of the substances in the table, water has the highest specific heat capacity, and that has everyday consequences:

For this reason water is also used as a coolant in automobile radiators, as well as, a heater in hot water bags.

NCERT Physics Part II (Class 11), Section 10.6, page 209

Water also warms up more slowly than land in summer, so a breeze from the sea feels cool (see sea breezes); and in deserts, where there is little water, the ground heats up quickly by day and cools quickly at night (NCERT, page 209).

Gases. NCERT Table 10.4 (page 209) gives molar specific heats in J mol^⁻¹ K^⁻¹: helium C_p = 20.8, C_v = 12.5; hydrogen 28.8, 20.4; nitrogen 29.1, 20.8; oxygen 29.4, 21.1; carbon dioxide 37.0, 28.5. For every one of them C_p − C_v is close to 8.3 J mol^⁻¹ K^⁻¹, the gas constant R (Example 5).

Solved examples

Use the values of NCERT Tables 10.3 and 10.4 given above.

Common mistakes to avoid

Quick revision summary

• Heat capacity S = ΔQ/ΔT; specific heat capacity s = ΔQ/(mΔT), unit J kg^⁻¹ K^⁻¹; molar specific heat C = ΔQ/(μΔT) = Ms, unit J mol^⁻¹ K^⁻¹.
• The heat needed is proportional to mass and temperature rise and depends on the substance: ΔQ = msΔT.
• Water has a very high specific heat (4186 J kg^⁻¹ K^⁻¹): a good coolant, a slow-warming sea, mild coasts.
• For gases, C_p > C_v; C_p − C_v = R for an ideal gas, and close to R for the real gases of NCERT Table 10.4.
• In short: ΔQ = msΔT.

Questions students ask

Why does water have such a high specific heat capacity?

Water has a high specific heat capacity, 4186 J kg^⁻¹ K^⁻¹, mainly because of hydrogen bonding between its molecules: part of the heat supplied goes into stretching and breaking these bonds rather than into speeding up the molecules, so the temperature rises less for each joule. Its molecules are also light, so a kilogram of water holds many more of them than a kilogram of a metal holds atoms (about 3.3 × 10^²⁵, against about 1.1 × 10^²⁵ in a kilogram of iron), and each molecule can store energy by rotating as well as by moving. As a result, a kilogram of water needs over nine times as much heat as a kilogram of iron for the same rise in temperature.

What is the specific heat of water, ice and steam?

The specific heat of water is about 4186 J kg^⁻¹ K^⁻¹ (1 cal g^⁻¹ °C^⁻¹ in cgs units), while ice and steam need only about half as much heat per kilogram per kelvin: about 2100 J kg^⁻¹ K^⁻¹ for ice (NCERT’s table gives 2060) and about 2000 J kg^⁻¹ K^⁻¹ for steam at constant pressure (a handbook value; NCERT gives none), roughly 0.5 cal g^⁻¹ °C^⁻¹ each. So when ice is heated to steam, each state warms with its own specific heat, and on a graph of temperature against heat the sloping part for ice is steeper than the one for water.

Does specific heat capacity change with temperature?

Yes, specific heat capacity changes with temperature, though for most solids and liquids only slowly near room temperature, which is why tables quote it at a stated temperature. Water’s value varies by only about 1% between 0 °C and 100 °C, with a shallow minimum near 35–40 °C, so 4186 J kg^⁻¹ K^⁻¹ can be used across that range. At very low temperatures the specific heats of solids fall steeply towards zero, which is where the Dulong–Petit law fails.

Does water have the highest specific heat capacity of all substances?

No, water does not have the highest specific heat capacity of all substances, though it has the highest in NCERT’s Table 10.3 of common solids and liquids. Hydrogen and helium have more at constant pressure: dividing NCERT’s molar values of C_p by the molar mass gives about 14 300 J kg^⁻¹ K^⁻¹ for hydrogen and 5200 J kg^⁻¹ K^⁻¹ for helium, because a kilogram of these light gases holds very many molecules. These gas examples are enough to disprove a universal “highest specific heat” claim. Water is widely used because its high heat capacity is combined with practical availability and useful operating properties.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. Why is water used in car radiators and in hot-water bags?

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   Water has a very high specific heat capacity, so a given mass of it can carry away (or store) a large amount of heat for a small change in its temperature.

2. Q2. A 1000 W kettle heats 1.0 kg of water from 25 °C to 75 °C. Ignoring losses and the kettle’s own heat capacity, how long does it take? Take s = 4186 J kg^⁻¹ K^⁻¹.

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   ΔQ = 1.0 × 4186 × 50, about 2.09 × 10^⁵ J. At 1000 J per second this takes about 209 s, roughly 3.5 minutes.

3. Q3. On a sunny day, why does dry sand on a beach get much hotter than the sea?

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   Sand has a far smaller specific heat capacity than water, so the same sunlight raises its temperature much more. (Water also spreads the heat by mixing and evaporating.)