Science Wiki

Specific heat capacity (symbol c) is the amount of heat that must be added to one unit of mass of a substance to raise its temperature by one kelvin (equivalently, one degree Celsius). In the SI it has units of joules per kilogram per kelvin (J·kg⁻¹·K⁻¹). It is an intensive property of a material. The related extensive quantity is the heat capacity C of a particular body, which is the product of the mass and the specific heat capacity, and the molar heat capacity gives the same quantity per mole.

Specific heat capacity matters in many fields, including thermodynamics, materials science, engineering, chemistry, and Earth system science. In the Earth system, the large difference between the specific heat capacity of water and that of dry land materials, together with the mixing of the oceans, is one of the main reasons that oceans warm and cool more slowly than continents. It is therefore an important factor in the seasonal cycle, in sea breezes and monsoons, and in the storage of the excess heat that results from global warming. It is not the only factor in these processes, because evaporation, albedo, cloud cover, and the transport of heat by the atmosphere and the oceans also play important roles.

If an amount of heat Q is added to a mass m of a substance and its temperature rises by Δ_T_, the average specific heat capacity over that interval is:

The value generally depends on temperature, pressure, and the phase of the substance, so the exact definition is differential, c = (1/_m_) (δ_Q_ / d_T_), taken along a specified process. The heat required for a given temperature change is then:

For example, heating 2 kg of liquid water (c ≈ 4,184 J·kg⁻¹·K⁻¹) by 10 K needs Q = 2 × 4,184 × 10 ≈ 83,700 J, about 84 kJ. The same heat would raise 2 kg of dry sand (c ≈ 800 J·kg⁻¹·K⁻¹) by about 52 K. Older units include the calorie, which was defined as the heat needed to raise 1 g of water by 1 °C (about 4.184 J for the thermochemical calorie).

Isobaric vs. Isochoric Partial Derivatives

Because the heat absorbed depends on the process, a specific heat capacity must be specified for the conditions of the heating. The two most common are constant pressure and constant volume.

The specific heat capacity at constant pressure, c_p, is the partial derivative of the specific enthalpy h with respect to temperature at constant pressure:

The specific heat capacity at constant volume, c_v, is the partial derivative of the specific internal energy u with respect to temperature at constant volume:

Mayer's Relation and Compressibility

For an ideal gas, the two are related by Mayer's relation, c_p − c_v = R_s, where R_s is the specific gas constant (the universal gas constant divided by the molar mass). The ratio γ = c_p / c_v is the heat capacity ratio (adiabatic index), which is about 1.4 for dry air near room temperature and about 1.67 for monatomic gases. For dry air, c_p is about 1,005 J·kg⁻¹·K⁻¹ and c_v is about 718 J·kg⁻¹·K⁻¹. The heat capacity of a gas is larger at constant pressure because some of the heat does work in expanding the gas.

For a general substance, the difference is:

where v is the specific volume, α_V is the volumetric thermal expansion coefficient, and κ_T is the isothermal compressibility. For liquids and solids, expansion is small, so c_p and c_v are close, differing by a few percent or less. For liquid water at room temperature, the difference is small, and the difference for seawater at depth is also small. The distinction remains important for gases and in atmospheric science, for example in the dry adiabatic lapse rate, Γ_d = g / c_p ≈ 9.8 K per km.

Mixtures and Temperature Dependence

For a mixture or a composite material, the specific heat capacity is the mass-weighted mean of the components when they are in thermal equilibrium:

where w_i is the mass fraction of component i. For example, wet soil with 20 percent water by mass and a dry soil value of 800 J·kg⁻¹·K⁻¹ has c ≈ 0.2 × 4,184 + 0.8 × 800 ≈ 1,480 J·kg⁻¹·K⁻¹. Water content is therefore the main reason that the heat capacity of soil varies.

Specific heat capacity changes with temperature. For solids at low temperature, it falls toward zero, as predicted by quantum theory and described by the Debye model, in which c is proportional to T³ for insulators well below the Debye temperature. At high temperature, the molar heat capacity of many simple solids approaches 3_R_ ≈ 25 J·mol⁻¹·K⁻¹, the Dulong-Petit law. Heat capacity is also large near a phase transition.

Equipartition of Energy and Quantum Modes

Heat added to a substance is stored in the energy of its molecules and lattice. In the classical equipartition theorem, each quadratic degree of freedom of a molecule contributes ½_k__BT of energy per molecule (and ½_R_T per mole) at temperatures that are high enough for the mode to be active. A monatomic ideal gas has three translational degrees of freedom, so the molar c_v is (3/2)_R_ and c_p is (5/2)_R_. A diatomic gas such as N₂ or O₂ has, at room temperature, two additional rotational modes, giving c_v = (5/2)_R_. Its vibrational mode is largely frozen at room temperature because the energy quantum of the vibration is much larger than k_BT, which is a quantum effect and one of the early successes of quantum theory. The heat capacity of a gas therefore increases with temperature as vibrational modes become active.

Temperature is a measure of the mean translational kinetic energy of the molecules, and not of all stored energy. Energy that is stored in rotation, vibration, and intermolecular interactions does not raise the temperature in the same way, and a substance that has more ways to store energy at a given temperature needs more heat for the same temperature change. Specific heat capacity per unit mass also depends on the molar mass, because a lighter molecule means more molecules per kilogram.

Why Water Has a High Heat Capacity

Liquid water has an unusually high specific heat capacity, about 4,184 J·kg⁻¹·K⁻¹ (4.18 kJ·kg⁻¹·K⁻¹) at 15 to 25 °C, higher than almost all common liquids and solids. Its value per unit mass is also high because the molar mass of water is small (18 g/mol). The molar value is about 75.3 J·mol⁻¹·K⁻¹, which is about 9_R_, much more than the roughly 3_R_ that is typical for solids.

The high value is due to the extensive network of hydrogen bonds. As temperature rises, hydrogen bonds are stretched, bent, and broken, and this absorbs energy that does not increase the translational kinetic energy. Water also has large contributions from the librational (hindered rotational) and vibrational modes. Quantum effects of the light hydrogen nuclei contribute to the heat capacity. A detailed account is still the subject of research, but the role of hydrogen bonding is well established. Water has its minimum specific heat capacity near 35 °C, and its value changes by less than 1 percent between 0 and 100 °C. The value for ice is about 2,100 J·kg⁻¹·K⁻¹, and for water vapor at constant pressure about 1,860 J·kg⁻¹·K⁻¹. Seawater has a somewhat lower value than pure water, about 3,990 J·kg⁻¹·K⁻¹ for a salinity of 35 g/kg, and the value changes slowly with temperature and pressure.

Heat capacity can also be compared per unit volume (volumetric heat capacity), which is the product of the density and the specific heat capacity. Water has a volumetric heat capacity of about 4.2 MJ·m⁻³·K⁻¹, whereas dry air is about 1.2 kJ·m⁻³·K⁻¹ at sea level, which is about 3,500 times lower. The entire atmosphere has a heat capacity that is equal to that of only about the upper 2.5 to 3.5 m of the ocean, which is why the ocean can store far more heat than the atmosphere.

Land and Ocean Thermal Contrast

Dry land materials have specific heat capacities of about 800 to 1,000 J·kg⁻¹·K⁻¹, about a fifth to a quarter of that of water. The difference between land and ocean is, however, caused by several effects acting together, and the difference in specific heat capacity is not the only one:

• Mixing and depth. Solar radiation penetrates tens of meters into clear ocean water, and wind and convection mix heat through a surface mixed layer of typically 20 to 100 m (much deeper in winter at high latitudes). On land, heat is carried only by conduction, and the seasonal temperature wave penetrates only a few meters, with a diurnal wave penetrating tens of centimeters. The heat is therefore spread through a much larger mass in the ocean than on land.
• Evaporation. A large part of the energy that the ocean absorbs is used for evaporation (latent heat flux), which lowers the surface temperature. Over dry land there is less evaporation, so more of the energy goes into raising the temperature.
• Thermal conductivity and transparency. Water is partly transparent to sunlight, whereas soil is opaque, so land absorbs energy at the surface.

The effective heat capacity of the top 3 m of the ocean is about equal to that of the whole atmosphere. As an example of the time scale, the seasonal heating of a surface mixed layer of 50 m with an energy input of 100 W·m⁻² for 90 days (7.8 × 10⁶ s) is 7.8 × 10⁸ J·m⁻². The heat capacity of the layer per unit area is ρ c h ≈ 1,025 × 3,990 × 50 ≈ 2.0 × 10⁸ J·m⁻²·K⁻¹, so the warming would be about 3.8 K, if all the heat stayed in the layer. For a dry soil layer of 0.5 m with ρ = 1,500 kg·m⁻³ and c = 800 J·kg⁻¹·K⁻¹, the heat capacity is 6 × 10⁵ J·m⁻²·K⁻¹, about 300 times smaller, so the same input would give a very large temperature change, and in practice heat is lost by other routes, so the real change is smaller. This illustrates why land heats and cools much more rapidly than the ocean.

The differential heating of land and ocean creates horizontal temperature and pressure gradients, which drive local sea and land breezes on a daily cycle and the seasonal reversal of winds in monsoon systems. It contributes to the maritime and continental difference in climate, in which coastal locations have a smaller annual temperature range than inland locations at the same latitude. It also leads to the faster warming of land than ocean in global warming, with the land warming about 1.5 times as much as the ocean on average. The difference is not caused by heat capacity alone, but also by the atmospheric adjustment and the supply of moisture. The Hadley circulation and the mid-latitude storm tracks are driven mainly by the difference in the heating between the equator and the poles, and the land-sea contrast modifies them, so the statement that the land-sea contrast alone is "the engine" of global meteorology is not accurate.

Response Time and Thermal Inertia

For a system with heat capacity per unit area C that responds to a change in forcing, with a climate feedback parameter λ (negative for a stable climate), the time constant of the response is:

For a mixed layer of 70 m, C ≈ ρ c h ≈ 2.9 × 10⁸ J·m⁻²·K⁻¹, and for |_λ_| ≈ 1.2 W·m⁻²·K⁻¹, τ ≈ 2.4 × 10⁸ s, or about 8 years. The transport of heat into the deep ocean provides a second, much longer time scale of centuries, so the climate system has a fast and a slow response. The ocean's large heat capacity delays and moderates the surface warming, so that for a given forcing, the warming at the time of the forcing is smaller than the equilibrium warming.

Ocean Heat Uptake and Committed Warming

The imbalance in Earth's energy budget caused by greenhouse gases is mainly stored in the ocean. The Intergovernmental Panel on Climate Change assessed that the oceans took up about 91 percent of the heat gained by the climate system between 1971 and 2018, with the land taking about 5 percent, melting of ice about 3 percent, and the atmosphere about 1 percent. The atmosphere is a small part of the total because of its small heat capacity. The ocean covers about 71 percent of the surface of the Earth. The accumulation is measured as ocean heat content (OHC), which for a volume is:

Changes are often given relative to a reference. A change of 1 zettajoule (10²¹ J) of OHC in the upper 2,000 m corresponds to a mean warming of about 0.0025 K, because the heat capacity of the upper 2,000 m is about 2.5 × 10²⁴ J·K⁻¹ (mass of about 7 × 10²⁰ kg times about 3,990 J·kg⁻¹·K⁻¹ gives 2.8 × 10²⁴ J·K⁻¹). In recent decades, the OHC has increased by roughly 10 to 15 zettajoules per year in the upper 2,000 m, so that the warming of the upper ocean has been much smaller than that of the surface air.

The statement that the ocean would continue to "radiate" heat to the atmosphere for centuries even if emissions stopped is not accurate. The ocean's thermal inertia delays the response of the surface temperature, which means that some warming is "in the pipeline" because the climate has not reached equilibrium with the present forcing, and the surface temperature would continue to increase for a time. When emissions of carbon dioxide stop, however, the atmospheric concentration declines because the ocean and land take up carbon, and the decline in the radiative forcing approximately offsets the continuing warming from the thermal lag, so that the global temperature remains roughly constant for decades to centuries in many model simulations. This is the concept of the zero emissions commitment, which is close to zero in the best estimates, with an uncertainty range that includes both small warming and small cooling. The ocean's thermal inertia therefore causes a delay, and it is not the same as irreversible additional warming.

• Calorimetry. Specific heat capacity is measured by calorimetry, in which a known amount of heat is supplied to a sample (for example by an electrical heater) or exchanged with a known mass of water, and the temperature change is measured. Differential scanning calorimetry is used for small samples and for the study of phase transitions.
• Heat storage and cooling. Water is a common coolant and a medium for sensible heat storage (in hot water tanks and in district heating) because of its high specific heat capacity, low cost, and availability. Other materials with high volumetric heat capacity and phase change materials are used for thermal storage.
• Climate and living things. The high heat capacity of water, which makes up the majority of the mass of most organisms, helps to stabilize body temperatures against changes in the environment, and the high heat capacity of water bodies moderates the climate of coastal and lake regions and allows aquatic organisms to live in a more stable thermal environment than terrestrial organisms.
• Planetary science. The thermal inertia of a planetary surface, defined as I = √(k ρ c), where k is the thermal conductivity, ρ is the density, and c is the specific heat capacity, determines how fast a surface heats and cools and is measured from orbit to identify surface materials on Mars and the Moon.

Why does liquid water have twice the specific heat capacity of solid ice?

In crystalline hexagonal ice (I_h) at −10 °C (c_p ≈ 2,100 J·kg⁻¹·K⁻¹), water molecules are locked into a rigid tetrahedral lattice where thermal energy primarily excites low-energy lattice vibrations (phonons). In liquid water (c_p ≈ 4,184 J·kg⁻¹·K⁻¹), rising temperature continuously stretches, bends, and breaks transient intermolecular hydrogen bonds, absorbing substantial potential energy without increasing translational kinetic energy.

Why do diverse solid metals have nearly identical molar heat capacities near room temperature?

According to the Dulong-Petit law, the molar heat capacity of most crystalline solids approaches 3_R_ ≈ 24.94 J·mol⁻¹·K⁻¹ above their Debye temperature because each atom possesses three translational and three vibrational quadratic degrees of freedom in a 3D crystal lattice (6 × ½_R_). However, when expressed per kilogram, heavy elements like lead (207.2 g·mol⁻¹, c ≈ 130 J·kg⁻¹·K⁻¹) contain far fewer atoms per kilogram than aluminum (27.0 g·mol⁻¹, c ≈ 900 J·kg⁻¹·K⁻¹).

How does specific heat capacity govern the dry adiabatic lapse rate of Earth's atmosphere?

When an unsaturated parcel of dry air rises through the troposphere, it expands against lower ambient pressure and cools adiabatically without exchanging heat with its surroundings. By combining the first law of thermodynamics (d_h_ = c_pd_T_ = _v_d_p_) with hydrostatic equilibrium (d_p_ = −_ρ__g_d_z_), the dry adiabatic cooling rate simplifies to Γ_d = g / c_p = 9.81 m·s⁻² / 1,005 J·kg⁻¹·K⁻¹ ≈ 9.76 K·km⁻¹.

• Heat capacity
• Molar heat capacity
• Ocean heat content
• Earth's energy budget
• Thermal inertia
• Enthalpy
• Calorimetry