Specific heat capacity Totally Explained

Everything about Specific Heat Capacity totally explained

Specific heat capacity, also known simply as specific heat, is the measure of the heat energy required to increase the temperature of a unit quantity of a substance by a certain temperature interval. The term originated primarily through the work of Scottish physicist Joseph Black who conducted various heat measurements and used the phrase “capacity for heat.” More heat energy is required to increase the temperature of a substance with high specific heat capacity than one with low specific heat capacity. For instance, eight times the heat energy is required to increase the temperature of an ingot of magnesium as is required for a lead ingot of the same mass. The specific heat of virtually any substance can be measured, including chemical elements, compounds, alloys, solutions, and composites.
The symbols for specific heat capacity are either C or c depending on how the quantity of a substance is measured (see Symbols and standards below for usage rules). In the measurement of physical properties, the term “specific” means the measure is a bulk property (an intensive property), wherein the quantity of substance must be specified. For example, the heat energy required to raise water’s temperature one kelvin (equal to one Celsius degree) is 4.184 joules per gram—the gram being the specified quantity. Scientifically, this measure would be expressed as c = 4.184 J g^–1 K^–1.

Basic metrics of specific heat capacity

Unit Quantity

When measuring specific heat capacity in science and engineering, the unit quantity of a substance is often in terms of mass: either the gram or kilogram, both of which are an SI unit. Especially in chemistry though, the unit quantity of specific heat capacity may also be the mole, which is a certain number of molecules or atoms. When the unit quantity is the mole, the term molar heat capacity may also be used to more explicitly describe the measure.

Heat energy

The unit of measure for heat energy is usually the SI unit joule. The calorie however, is still often used in chemistry. For example, ť Problem: How many calories would it be required to raise the temperature of 200g of aluminum 20 degrees Celcius?

ť Solution:

:We note that the specific heat of aluminum, c_Al, is 0.897J g⁻¹ K⁻¹; this is, 0.214cal g⁻¹ °C⁻¹.
ť :Thus the heat energy

Delta Q

required is

ť ::

Delta Q = m c Delta T

Delta Q

= (200g) (0.214 cal g⁻¹ °C⁻¹) (20 °C) ť ::: = 856 calories

Temperature interval

The temperature interval in science, engineering and chemistry is usually one kelvin or degree Celsius (both of which have the same magnitude).

Other units

In the U.S., other units of measure for specific heat capacity are typically used in disciplines such as construction and civil engineering. There, the mass quantity is often the pound-mass, the unit of heat energy is the British thermal unit, and the temperature interval is the degree Fahrenheit.
If temperature is expressed in natural rather than historical terms for example as a rate of energy increase per unit increase in state uncertainty, then heat capacity becomes the number of bits of mutual information between system and surroundings lost per two-fold increase in absolute temperature. Thus for instance, with each 2-fold increase in absolute temperature we lose 3/2 bits of mutual information per atom in a monatomic ideal gas.

Basic equations

The equation relating heat energy to specific heat capacity, where the unit quantity is in terms of mass is: ť : $Delta Q = m c Delta T$

where $Delta Q$ is the heat energy put into or taken out of the substance, $m$ is the mass of the substance, $c$ is the specific heat capacity, and $Delta T$ is the temperature differential.

Where the unit quantity is in terms of moles, the equation relating heat energy to specific heat capacity (also known as molar heat capacity) is ť :

Delta Q = n c Delta T

where

Delta Q

is the heat energy put into or taken out of the substance,

n

is the number of moles,

c

is the specific heat capacity, and

Delta T

is the temperature differential.

Factors that affect specific heat capacity

Degrees of freedom: Molecules are quite different from the monatomic gases like helium and argon. With monatomic gases, heat energy comprises only translational motions. Translational motions are ordinary, whole-body movements in 3D space whereby particles move about and exchange energy in collisions—like rubber balls in a vigorously shaken container (see animation here

). These simple movements in the three X, Y, and Z–axis dimensions of space means monatomic atoms have three translational degrees of freedom. Molecules, however, have various internal vibrational and rotational degrees of freedom because they're complex objects; they're a population of atoms that can move about within a molecule in different ways (see animation at right). Heat energy is stored in these internal motions. For instance, nitrogen, which is a diatomic molecule, has five active degrees of freedom at room temperature: the three comprising translational motion plus two rotational degrees of freedom internally. Not surprisingly, the constant-volume molar heat capacity of nitrogen at this temperature is five-thirds that of monatomic gases. At higher temperatures, nitrogen gains two more degrees of internal freedom as the molecule is excited into higher vibrational modes, and then the heat capacity approaches seven-thirds that of monatomic gases See Thermodynamic temperature for more information on translational motions, kinetic (heat) energy, and their relationship to temperature.

Molar mass: When the specific heat capacity, c, of a material is measured (lowercase c means the unit quantity is in terms of mass), different values arise because different substances have different molar masses (essentially, the weight of the individual atoms or molecules). Heat energy arises, in part, due to the number of atoms or molecules that are vibrating. If a substance has a lighter molar mass, then each gram of it has more atoms or molecules available to store heat energy. This is why hydrogen—the lightest substance there is—has such a high specific heat capacity on a gram basis; one gram of it contains a relatively great many molecules. If specific heat capacity is measured on a molar basis (uppercase C), the differences between substances is less pronounced and hydrogen’s molar heat capacity is quite unremarkable. Conversely, for molecular-based substances (which also absorb heat into their internal degrees of freedom), massive, complex molecules with high atomic count—like gasoline—can store a great deal of energy per mole and yet, be quite unremarkable on a massbasis.

Sincethe bulk density of a solid chemical element is strongly related to its molar mass, generally speaking, there's a strong, inverse correlation between a solid’s density and its c_p (constant-pressure specific heat capacity on a mass basis). Large ingots of low-density solids tend to absorb more heat energy than a small, dense ingot of the same mass because the former usually has proportionally more atoms. Thus, generally speaking, there a close correlation between the size of a solid chemical element and its total heat capacity (see Volumetric heat capacity). There are however, many departures from the general trend. For instance, arsenic, which is only 14.5% less dense than antimony, has nearly 59% more specific heat capacity on a mass basis. In other words; even though an ingot of arsenic is only about 17% larger than an antimony one of the same mass, it absorbs about 59% more heat energy for a given temperature rise.

Hydrogen bonds: Hydrogen-containing polar molecules like ethanol, ammonia, and water have powerful, intermolecular hydrogen bonds when in their liquid phase. These bonds provide yet another place where kinetic (heat) energy is stored.

Impurities: In the case of alloys, there are several conditions in which small impurity concentrations can greatly affect the specific heat. Alloys may exhibit marked difference in behaviour even in the case of small amounts of impurities being one element of the alloy; for example impurities in semiconducting ferromagnetic alloys may lead to quite different specific heat properties as first predicted by White and Hogan.

Symbols and standards

When mass is the unit quantity, the symbol for specific heat capacity is lowercase c. When the mole is the unit quantity, the symbol is uppercase C. Alternatively—especially in chemistry as opposed to engineering—the uppercase version for specific heat, C, may be used in combination with a suffix representing enthalpy (symbol: either H or h); specifically, when the mole is the unit quantity, the enthalpy suffix is uppercase H and when mass is the unit quantity, the suffix is lowercase h.
   The modern SI units for measuring specific heat capacity are either the joule per gram-kelvin (J g^–1 K^–1) or the joule per mole-kelvin (J mol^–1 K^–1). The various SI prefixes can create variations of these units (such as kJ kg^–1 K^–1 and kJ mol^–1 K^–1). Symbols for alternative units are as follows: pounds-mass (symbol: lb) for quantity, calories (symbol: cal) and British thermal units (symbol: BTU) for energy, and degree Fahrenheit (symbol: °F) for the increment of temperature.
   There are two distinctly different experimental conditions under which specific heat capacity is measured and these are denoted with a subscripted suffix modifying the symbols C or c. The specific heat of substances are typically measured under constant pressure (Symbols: C_p or c_p). However, fluids (gases and liquids) are typically also measured at constant volume (Symbols: C_v or c_v). Measurements under constant pressure produces greater values than those at constant volume because work must be performed in the former. This difference is particularly great in gases where values under constant pressure are typically 30% to 66.7% greater than those at constant volume.
   Thus, the symbols for specific heat capacity are as follows:

	Under constant pressure	At constant volume
Unit quantity = mole	C_p or C_pH	C_v or C_vH
Unit quantity = mass	c_p or C_ph	c_v or C_vh

The ratio of the specific heats (or Heat capacity ratio) is usually denoted by

gamma

(gamma). It is often used in equations, such as for calculating speed of sound in an ideal gas.
   The specific heat capacities of substances comprising molecules (distinct from the monatomic gases) are not fixed constants and vary somewhat depending on temperature. Accordingly, the temperature at which the measurement is made is usually also specified. Examples of two common ways to cite the specific heat of a substance are as follows:
Water (liquid): c_p = 4.1855 J g^–1 K^–1 (15 °C), and…
Water (liquid): C_vH = 74.539 J mol^–1 K^–1 (25 °C)
   The pressure at which specific heat capacity is measured is especially important for gases and liquids. The standard pressure was once virtually always “one standard atmosphere” which is defined as the sea level–equivalent value of precisely 101.325 kPa (760 Torr). In the case of water, 101.325 kPa is still typically used due to water’s unique role in temperature and physical standards. However, in 1982, the International Union of Pure and Applied Chemistry (IUPAC) recommended that for the purposes of specifying the physical properties of substances, “the standard pressure” should be defined as precisely 100 kPa (≈750.062 Torr). Besides being a round number, this had a very practical effect: relatively few people live and work at precisely sea level; 100 kPa equates to the mean pressure at an altitude of about 112 meters (which is closer to the 194–meter, world–wide median altitude of human habitation). Accordingly, the pressure at which specific heat capacity is measured should be specified since one can not assume its value. An example of how pressure is specified is as follows:
Water (gas): C_vH = 28.03 J mol^–1 K^–1 (100 °C, 101.325 kPa)
   Note in the above specification that the experimental condition is at constant volume. Still, the pressure within this fixed volume is controlled and specified.

Heat capacity

Heat capacity (symbol: C_p) — as distinct from specific heat capacity — is the measure of the heat energy required to increase the temperature of an object by a certain temperature interval. Heat capacity is an extensive property because its value is proportional to the amount of material in the object; for example, a bathtub of water has a greater heat capacity than a cup of water.
Heat capacity is usually expressed in units of J K^–1 (or J/K), subject to the caveats and exceptions detailed in both Basic metrics of specific heat capacity and Symbols and standards, above. For instance, one could write that the gasoline in a 55-gallon drum has an average heat capacity of 347 kJ/K.
The uncertainty of an object’s measured quantity is rarely better than one percent and this places an upper limit on the accuracy and precision of most stated values of heat capacity. Accordingly, it's usually unnecessary as a practical matter, to specify the defined state at which the measurement was made; for example “(25 °C, 100 kPa).” In most cases, it's assumed that the substance’s specific heat capacity is a published value and the object’s quantity is subject to such a sizable relative uncertainty that it renders this detail moot. An exception would be when an object has an accurately known or precisely defined quantity; for example “The heat capacity of the International Prototype Kilogram is 133 J/K (25 °C).” Another exception would be when the defined state varies significantly from standard conditions.

Table of specific heat capacities

Note that especially high values, as for parafin, water and ammonia, result from calculating specific heats in terms of moles of molecules. If specific heat is expressed per mole of atoms for these substances, few values exceed the theoretical Dulong-Petit limit of 25 J/K/mole = 3 R per mole.

Substance	Phase	C_p J g⁻¹ K⁻¹	C_p,m J mol⁻¹ K⁻¹	C_v,m J mol⁻¹ K⁻¹	Volumetric heat capacity J cm^-3 K^-1
Air (Sea level, dry, 0 °C)	gas	1.0035	29.07	20.7643	0.001297
Air (typical room conditions^A)	gas	1.012	29.19	20.85
Aluminium	solid	0.897	24.2		2.422
Ammonia	liquid	4.700	80.08		3.263
Antimony	solid	0.207	25.2		1.386
Argon	gas	0.5203	20.7862	12.4717
Arsenic	solid	0.328	24.6		1.878
Beryllium	solid	1.82	16.4		3.367
Bismuth	solid	0.123	25.7		1.20
Copper	solid	0.385	24.47		3.45
Carbon dioxide CO₂	gas	0.839*	36.94	28.46
Diamond	solid	0.5091	6.115		1.782
Ethanol	liquid	2.44	112		1.925
Gasoline	liquid	2.22	228		1.64
Glass	solid	0.84
Gold	solid	0.1291	25.42		2.492
Granite	solid	0.790			2.17
Graphite	solid	0.710	8.53		1.534
Helium	gas	5.1932	20.7862	12.4717
Hydrogen	gas	14.30	28.82
Hydrogen sulfide H₂S	gas	1.015*	34.60
Iron	solid	0.450	25.1		3.537
Lead	solid	0.127	26.4		1.44
Lithium	solid	3.58	24.8		1.912
Magnesium	solid	1.02	24.9		1.773
Mercury	liquid	0.1395	27.98		1.888
Nitrogen	gas	1.040	29.12	20.8
Neon	gas	1.0301	20.7862	12.4717
Oxygen	gas	0.918	29.38
Paraffin wax	solid	2.5	900		2.325
Silica (fused)	solid	0.703	42.2		1.547
Silver	solid	0.233	24.9		2.44
Tungsten	solid	0.134	24.8		2.58
Uranium	solid	0.116	27.7		2.216
Water (steam)	gas (100 °C)	2.080	37.47	28.03
Water	liquid (25 °C)	4.1813	75.327	74.53	4.184
Water (ice)	solid (-10 °C)	2.050	38.09		1.938
Zinc	solid	0.387	25.2		2.76
All measurements are at 25 °C unless otherwise noted. Notable minima and maxima are shown in maroon.

^A Assuming an altitude of 194 meters above mean sea level (the world–wide median altitude of human habitation), an indoor temperature of 23 °C, a dewpoint of 9 °C (40.85% relative humidity), and 760 mm–Hg sea level–corrected barometric pressure (molar water vapor content = 1.16%). *Derived data by calculation

Specific heat capacity of building materials

(Usually of interest to builders and solar designers)

Substance	Phase	c_p J g⁻¹ K⁻¹
Asphalt	solid	0.92
Brick	solid	0.84
Concrete	solid	0.88
Glass, silica	solid	0.84
Glass, crown	solid	0.67
Glass, flint	solid	0.503
Glass, pyrex	solid	0.753
Granite	solid	0.790
Gypsum	solid	1.09
Marble, mica	solid	0.880
Sand	solid	0.835
Soil	solid	0.80
Wood	solid	0.42

Derivations of heat capacity and specific heat capacity

Definition of heat capacity

Heat capacity is mathematically defined as the ratio of a small amount of heat δQ added to the body, to the corresponding small increase in its temperature dT:
ť

C = left(frac.

The heat capacity must be zero at zero temperature in order for the above integral not to yield an infinite absolute entropy, thus violating the third law of thermodynamics. One of the strengths of the Debye model is that (unlike the preceding Einstein model) it predicts an approach of heat capacity toward zero as zero temperature is approached, and also predicts the proper mathematical form of this approach.

Further Information

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