Specific heat capacity

In thermodynamics, the specific heat capacity of a substance is the amount of heat that must be added to one unit of mass of the substance in order to cause an increase of one unit in temperature. It is also referred to as massic heat capacity or as the specific heat. More formally it is the heat capacity of a sample of the substance divided by the mass of the sample. The SI unit of specific heat capacity is joule per kelvin per kilogram, J⋅kg⁻¹⋅K⁻¹. For example, the heat required to raise the temperature of of water by is, so the specific heat capacity of water is.
Specific heat capacity often varies with temperature, and is different for each state of matter. Liquid water has one of the highest specific heat capacities among common substances, about at 20 °C, but that of ice, just below 0 °C, is only. The specific heat capacities of iron, granite, and hydrogen gas are about 449 J⋅kg⁻¹⋅K⁻¹, 790 J⋅kg⁻¹⋅K⁻¹, and 14300 J⋅kg⁻¹⋅K⁻¹, respectively. While the substance is undergoing a phase transition, such as melting or boiling, its specific heat capacity is technically undefined, because the heat goes into changing its state rather than raising its temperature.
The specific heat capacity of a substance, especially a gas, may be significantly higher when it is allowed to expand as it is heated than when it is heated in a closed vessel that prevents expansion. These two values are usually denoted by and, respectively; their quotient is the heat capacity ratio.
The term specific heat may also refer to the ratio between the specific heat capacities of a substance at a given temperature and of a reference substance at a reference temperature, such as water at 15 °C; much in the fashion of specific gravity. Specific heat capacity is also related to other intensive measures of heat capacity with other denominators. If the amount of substance is measured as a number of moles, one gets the molar heat capacity instead, whose SI unit is joule per kelvin per mole, J⋅mol⁻¹⋅K⁻¹. If the amount is taken to be the volume of the sample, one gets the volumetric heat capacity, whose SI unit is joule per kelvin per cubic meter, J⋅m⁻³⋅K⁻¹.

History

Discovery of specific heat

One of the first scientists to use the concept was Joseph Black, an 18th-century medical doctor and professor of medicine at Glasgow University. He measured the specific heat capacities of many substances, using the term capacity for heat. In 1756 or soon thereafter, Black began an extensive study of heat. In 1760 he realized that when two different substances of equal mass but different temperatures are mixed, the changes in number of degrees in the two substances differ, though the heat gained by the cooler substance and lost by the hotter is the same. Black related an experiment conducted by Daniel Gabriel Fahrenheit on behalf of Dutch physician Herman Boerhaave. For clarity, he then described a hypothetical, but realistic variant of the experiment: If equal masses of 100 °F water and 150 °F mercury are mixed, the water temperature increases by 20 ° and the mercury temperature decreases by 30 °, even though the heat gained by the water and lost by the mercury is the same. This clarified the distinction between heat and temperature. It also introduced the concept of specific heat capacity, being different for different substances. Black wrote: "Quicksilver ... has less capacity for the matter of heat than water."

Definition

The specific heat capacity of a substance, usually denoted by or, is the heat capacity of a sample of the substance, divided by the mass of the sample:
where represents the amount of heat needed to uniformly raise the temperature of the sample by a small increment.
Like the heat capacity of an object, the specific heat capacity of a substance may vary, sometimes substantially, depending on the starting temperature of the sample and the pressure applied to it. Therefore, it should be considered a function of those two variables.
These parameters are usually specified when giving the specific heat capacity of a substance. For example, "Water : = 4187 J⋅kg⁻¹⋅K⁻¹." When not specified, published values of the specific heat capacity generally are valid for some standard conditions for temperature and pressure.
However, the dependency of on starting temperature and pressure can often be ignored in practical contexts, e.g. when working in narrow ranges of those variables. In those contexts one usually omits the qualifier and approximates the specific heat capacity by a constant suitable for those ranges.
Specific heat capacity is an intensive property of a substance, an intrinsic characteristic that does not depend on the size or shape of the amount in consideration.

Variations

The injection of heat energy into a substance, besides raising its temperature, usually causes an increase in its volume and/or its pressure, depending on how the sample is confined. The choice made about the latter affects the measured specific heat capacity, even for the same starting pressure and starting temperature. Two particular choices are widely used:

If the pressure is kept constant, and the sample is allowed to expand, the expansion generates work, as the force from the pressure displaces the enclosure or the surrounding fluid. That work must come from the heat energy provided. The specific heat capacity thus obtained is said to be measured at constant pressure and is often denoted
On the other hand, if the expansion is prevented for example, by a sufficiently rigid enclosure or by increasing the external pressure to counteract the internal one no work is generated, and the heat energy that would have gone into it must instead contribute to the internal energy of the sample, including raising its temperature by an extra amount. The specific heat capacity obtained this way is said to be measured at constant volume and denoted

The value of is always less than the value of for all fluids. This difference is particularly notable in gases where values under constant pressure are typically 30% to 66.7% greater than those at constant volume. Hence the heat capacity ratio of gases is typically between 1.3 and 1.67.

Applicability

The specific heat capacity can be defined and measured for gases, liquids, and solids of fairly general composition and molecular structure. These include gas mixtures, solutions and alloys, or heterogenous materials such as milk, sand, granite, and concrete, if considered at a sufficiently large scale.
The specific heat capacity can be defined also for materials that change state or composition as the temperature and pressure change, as long as the changes are reversible and gradual. Thus, for example, the concepts are definable for a gas or liquid that dissociates as the temperature increases, as long as the products of the dissociation promptly and completely recombine when it drops.
The specific heat capacity is not meaningful if the substance undergoes irreversible chemical changes, or if there is a phase change, such as melting or boiling, at a sharp temperature within the range of temperatures spanned by the measurement.

Measurement

The specific heat capacity of a substance is typically determined according to the definition; namely, by measuring the heat capacity of a sample of the substance, usually with a calorimeter, and dividing by the sample's mass. Several techniques can be applied for estimating the heat capacity of a substance, such as differential scanning calorimetry.
The specific heat capacities of gases can be measured at constant volume, by enclosing the sample in a rigid container. On the other hand, measuring the specific heat capacity at constant volume can be prohibitively difficult for liquids and solids, since one often would need impractical pressures in order to prevent the expansion that would be caused by even small increases in temperature. Instead, the common practice is to measure the specific heat capacity at constant pressure, determine separately the coefficient of thermal expansion and the compressibility of the material, and compute the specific heat capacity at constant volume from these data according to the laws of thermodynamics.

Units

International system

The SI unit for specific heat capacity is joule per kelvin per kilogram, J⋅K⁻¹⋅kg⁻¹. Since an increment of temperature of one degree Celsius is the same as an increment of one kelvin, that is the same as joule per degree Celsius per kilogram: J/. Sometimes the gram is used instead of kilogram for the unit of mass: 1 J⋅g⁻¹⋅K⁻¹ = 1000 J⋅kg⁻¹⋅K⁻¹.
The specific heat capacity of a substance has dimension L²⋅Θ⁻¹⋅T⁻², or ²/Θ. Therefore, the SI unit J⋅kg⁻¹⋅K⁻¹ is equivalent to metre squared per second squared per kelvin.

Imperial engineering units

Professionals in construction, civil engineering, chemical engineering, and other technical disciplines, especially in the United States, may use English Engineering units including the pound as the unit of mass, the degree Fahrenheit or Rankine as the unit of temperature increment, and the British thermal unit, as the unit of heat.
In those contexts, the unit of specific heat capacity is BTU/lb⋅°R, or 1 = 4186.68. The BTU was originally defined so that the average specific heat capacity of water would be 1 BTU/lb⋅°F. Note the value's similarity to that of the calorie - 4187 J/kg⋅°C ≈ 4184 J/kg⋅°C - as they are essentially measuring the same energy, using water as a basis reference, scaled to their systems' respective lbs and °F, or kg and °C.