Heat capacity

Depending on their internal mobility, molecules can accumulate agitation energy according to their degrees of internal freedomwhich contributes to its thermal capacity.

The heat capacity or thermal capacity of a body is the ratio between the amount of heat energy transferred to a body or system in any process and the change in temperature that experiences. In a more rigorous form, it is the energy necessary to increase the temperature of a certain substance by one unit of temperature. It indicates the greater or lesser difficulty that said body presents to experience temperature changes under the supply of heat. It can be interpreted as a measure of thermal inertia. It is an extensive property, since its magnitude depends not only on the substance but also on the amount of matter in the body or system; therefore, it is characteristic of a particular body or system. For example, the heat capacity of the water in an Olympic swimming pool will be greater than that of the water in a glass. In general, the heat capacity also depends on temperature and pressure.

Heat capacity (thermal capacity) should not be confused with specific heat capacity (specific thermal capacity) or specific heat, which is the intensive property that refers to a body's ability "to store heat", y is the ratio between the heat capacity and the mass of the object. Specific heat is a characteristic property of substances and depends on the same variables as heat capacity.

Background

Before the development of modern thermodynamics, heat was thought to be an invisible fluid, known as caloric. Bodies were capable of storing a certain amount of that fluid, hence the term heat capacity, named and first investigated by the Scottish chemist Joseph Black in the 1750s. Today, the notion of caloric has been replaced by the notion of the internal energy of a system. That is, heat is no longer considered a fluid but a disordered energy transfer. However, in many languages, the expression caloric capacity survives although in others thermal capacity is used.

Measurement of heat capacity

To measure the heat capacity under certain conditions it is necessary to compare the heat absorbed by a substance (or a system) with the resulting increase in temperature. The heat capacity is given by:

C=limΔ Δ T→ → 0QΔ Δ T{displaystyle C=lim _{Delta Tto 0}{frac {Q}{Delta T}}}}}{displaystyle C=lim

Symbol	Name	Unit	Description
C{displaystyle C}	Heat capacity	(J/K) or (cal/°C)	In general it will be function of the state variables
Q{displaystyle Q}	Heat absorbed by the system	J or lime
Δ Δ T{displaystyle {Delta T}	Temperature variation	K o °C

The heat capacity (C) of a physical system depends on the amount of substance or mass in the system. For a system made up of a single homogeneous substance, the specific heat or specific heat capacity (c) is also defined from the relationship:

C=QΔ Δ T=c⋅ ⋅ m{displaystyle C={frac {Q}{Delta T}}}=ccdot m}

Symbol	Name	Unit
C{displaystyle C}	Heat capacity of the body or system
c{displaystyle c}	Specific heat (specific heat capacity)
m{displaystyle m}	Mass of substance considered

From the above relationships it is easy to infer that by increasing the mass of a substance, its heat capacity increases since thermal inertia increases, and with it the difficulty of the substance to vary its temperature increases. An example of this can be seen in coastal cities where the sea acts as a great thermostat, regulating temperature variations.

Formal approach to heat capacity

Be a thermodynamic system in the state A{displaystyle A}. The heat capacity is defined Cc{displaystyle C_{c}} associated with an elementary quasistatic process c{displaystyle c} that part of A{displaystyle A} and ends in the state B{displaystyle B} like the limit of the quotient between the amount of heat Q{displaystyle Q} absorbed by the system and the temperature increase Δ Δ T{displaystyle Delta T} experienced when the final state B{displaystyle B} tends to be confused with the initial A{displaystyle A}.

Cc=limA→ → B(QΔ Δ T)c=␡ ␡ iqi(dcidT)=(d! ! QdT)c{displaystyle C_{c}=lim _{Ato B}left({Q over Delta} Tright)_{c}=sum _{i}q_{i}left({dc_{i} over dT}right)=left({{{bar {d}}Q over dT}right)}

Where (c1(T),...... ,cn(T)){displaystyle scriptstyle (c_{1}(T),dotsc_{n}(T))}}, is a parameterized curve by temperature, which represents the path followed in the phase space during the process c. The heat capacity is, thus, a thermodynamic variable and is perfectly defined in each state of system balance (the sign d! ! {displaystyle {bar {d}}} indicates that not a Q function whose differential is precisely d! ! Q{displaystyle {bar {d}Q}, that is, it is 1-form not exact).

Heat capacities of solids and gases

The heat capacity of solids and gases depends, according to the equipartition of energy theorem, on the number of degrees of freedom that a molecule has, as will be explained below.

Specific heat capacity

It is the amount of energy, in the form of heat, that a system gains or loses per unit of mass, so that a temperature change of one degree Celsius occurs in it, without there being a change of state.

Cp=Q/(mΔ Δ T){displaystyle Cp=Q/(mDelta T)}

Symbol	Name	Unit
Cp{displaystyle Cp}	Consistent pressure specific heat	kJ/(kg °C)
Q{displaystyle Q}	Heat earned or lost	kJ
m{displaystyle m}	Masa	kg
Δ Δ T{displaystyle Delta T}	Change in temperature	K o °C

In practice, it is only when working with gases that it is necessary to distinguish between specific heat at constant pressure and specific heat at constant volume Cv.

Sensible heat

It is the caloric content or energy level of a material, referring to the one it has at an arbitrary temperature in which it assigns a zero level (generally –40 °C for frozen products or 0 °C for other systems). This concept is widely used to study the thermal phenomena of pure substances or gases such as steam and air; in the case of food, it has its greatest applicability for frozen products. Its units in the SI system are J/kg.

The amount of heat to heat or cool a material from a temperature T₁ to T₂ is

Q=m(H2− − H1){displaystyle Q=m(H_{2}-H_{1})}

where m is the mass of the material; H₂ and H₁ the enthalpies at temperatures T₂ and T₁ respectively

Thermal conductivity

The thermal conductivity q is given by:

q=− − kA► ► T{displaystyle mathbf {q} =-kA{boldsymbol {nabla }}T}

Symbol	Name	Unit
q{displaystyle mathbf {q} }	Thermal conductivity
k{displaystyle k}	Thermal conductivity	W/(m K)
► ► T{displaystyle {boldsymbol {nabla }T}	Temperature gradient

The orders of magnitude of the thermal conductivity, according to the different types of materials, can be appreciated between the following values:

Monatomic Gas

The energy of a monatomic gas and therefore the heat capacity at constant volume are given by:

U=32NkT=32nRTCV=(▪ ▪ U▪ ▪ T)V=32Nk=32nR{displaystyle U={frac {3}{2}}NkT={frac {3{2}}}}nRTqquad C_{V}=left({frac {partial U}{partial T}}}{right)_{V}={frac {3}{2}{frac}}{3}n}

Symbol	Name	Unit
N{displaystyle N}	Number of gas atoms within the studied system
n{displaystyle n}	Number of moles
k{displaystyle k}	Boltzmann Constant
R{displaystyle R}	Universal consistency of ideal gases
T{displaystyle T}	Absolute temperature

Thus the molar specific heat of a monatomic ideal gas is simply c_v = 3R/2 or c_p = 5R/2. The real monatomic gases also fulfill the previous equalities, although in an approximate way.

Diatomic Gas

In a diatomic gas, the total energy can be found in the form of translational kinetic energy and also in the form of rotational kinetic energy, which means that diatomic gases can store more energy at a given temperature. At temperatures close to room temperature, the internal energy and heat capacity are given by:

U=52NkT=52nRTCV=(▪ ▪ U▪ ▪ T)V=52Nk=52nR{displaystyle U={frac {5}{2}}NkT={frac {5{2}}}}nRTqquad C_{V}=left({frac {partial U}{partial T}}}{right)_{V}={frac {5}{2}{frac}}{1⁄2}n}

For extremely high temperatures, the vibration energy of the bonds becomes important and diatomic gases deviate somewhat from the previous conditions. At even higher temperatures the contribution of the thermal motion of the electrons produces additional deflections. However, all real gases such as hydrogen (H₂), oxygen (O₂), nitrogen (N₂) or carbon monoxide (CO), meet the above relationships at moderate ambient temperatures. Therefore these gases have specific heats or molar heat capacities close to c_v = 5R/2.

Polyatomic Gases

The equipartition theorem for polyatomic gases suggests that polyatomic gases that have "soft" or flexible and vibrating easily with q frequencies, they should have a molar heat capacity given by:

Cv=(32+r2+q)R{displaystyle C_{v}=left({frac {3}{2}}}}} +{frac {r}{2}} +qright)R}

Where r measures the rotational degrees of freedom (r = 1 for linear molecules, r = 2 for planar molecules and r = 3 for three-dimensional molecules). However, these predictions do not hold at room temperature. The molar heat capacity increases moderately as the temperature increases. That is due to quantum effects that make the vibration modes quantized and only accessible as the temperature increases, and the expression (*) can only be a limit at very high temperatures. However, before reaching temperatures where this expression is a reasonable limit, many molecules break due to the effect of temperature, never reaching the previous limit. A rigorous treatment of heat capacity therefore requires the use of quantum mechanics, in particular quantum-type statistical mechanics.

Crystalline solids

Representation of three-dimensional heat capacity, depending on the temperature, according to Debye's model and Einstein's first model. The axis of abscises corresponds to the temperature divided by Debye's temperature. Note that the dimensional heat capacity is zero at absolute temperature zero and increases to value 3 when the temperature increases well above Debye's temperature. The red line represents the classic limit given by the law of Dulong-Petit.

It is a known experimental fact that non-metallic crystalline solids at room temperature have a heat capacity c_v more or less constant and equal to 3R (while the heat capacity at constant pressure continues to increase). This empirical verification is called the Dulong and Petit rule, although the Dulong and Petit rule fits the predictions of the equipartition theorem, at low temperatures this rule fails miserably. In fact for solids and liquids at low temperatures, and in some cases at room temperature, the expression (*) given by the energy equipartition theorem gives even worse results than for complicated polyatomic gases. Thus it is necessary to abandon classical statistical mechanics and study the problem from the quantum point of view.

Einstein was the first to propose a theory that reasonably predicted the evolution of the heat capacity of solids over a wide range of temperatures, which was qualitatively correct. Debye later proposed an improvement that made the theory quantitatively correct., and later this theory was improved by Blackman and others. Einstein's theory predicts that the molar heat capacity of a solid should vary according to the expression:

cv3R=(θ θ ET)2eθ θ ET(eθ θ ET− − 1)2{displaystyle {frac {c_{v}}{3R}}=left({frac {theta _{E}}{T}}{T}}{2}{frac {e^{frac}{frac {theta}{{E}}}{T}}}{left(eFF}{frac {theta _{E}{ right}}{

Where θ_E is a parameter proper to each solid called the Einstein characteristic temperature of the solid. This equation predicted the correct behavior at high temperatures:

limT→ → ∞ ∞ cv3R→ → 1{displaystyle lim _{Tto infty }{cfrac {c_{v}{3R}}}{1}}{1}

The Debye correction took into account, in addition to the quantum effects, the frequency distribution of the various modes of vibration (Einstein had assumed to simplify that all molecules were vibrating around the same frequency fundamental) with that innovation, Debye arrived at the somewhat more complicated expression:

cv3R=∫ ∫ 0θ θ DTx4ex(ex− − 1)2dx{displaystyle {frac {c_{v}}{3R}}=int _{0}{frac {frac {theta _{D}{T}}{frac {x^{4}e^{x}{x}}{(e^{x}{x}}{2}}}}{x}}

This expression coincides with Einstein's and the Dulong and Petit rule at high temperatures and at low temperatures explains the proportional behavior T³ observed:

limT→ → 0cv3R≈ ≈ 4π π 45(Tθ θ D)3{displaystyle lim _{Tto 0}{cfrac {c_{v}}}{3R}approx {frac {4pi ^{4}{5}}}}{left({frac {T}{theta _{D}}}}{right)^{3}

This last expression is called Debye's law T³.

Specific heat and heat capacity of some materials

In the table it can be seen that of the common materials, water, walls of water, earth or dry compacted soil (adobe, wall), and dense stones such as granite together with metals such as steel have a high heat capacity.. These are between 500 and 1000 kcal/m³ °C.

Then there is another group that goes from 300 to 500 kcal/m³ °C, among which most of the usual materials in current construction are located, such as brick, concrete, wood, rock plaster boards and the sandstones.

A final group includes (3 to 35 kcal/m³ °C), mass thermal insulators such as glass wool, mineral wool, expanded polystyrene and expanded polyurethane which, due to their low density" Because they contain a lot of air, they have a very low heat capacity but serve as thermal insulators.

A special case is air (0.29 kcal/m³ K; 1.214 J/m³ K), which serves as a medium to transport heat in passive systems but not to store heat inside.

Heat capacity of chemical elements

The following table shows the heat capacity of the individual pure components taken under laboratory conditions (at a temperature of 25 °C and a pressure of 100 kPa). ("H, N, O, F, Cl, Br, I" are respectively H
2 , N
2, O
2, F
2, Cl
2, Br
2 and I
2)

These values, expressed in J⋅mol^-1⋅K^-1, result:

• Maximum value = 37,030 J⋅mol^-1⋅K^-1 for gadolinium
• Minimum value = 8.517 J⋅mol^-1⋅K^-1 for carbon

The same values, converted to J⋅g^-1⋅K^-1, result in:

• Maximum value = 14,304 J⋅g^-1⋅K^-1 for hydrogen
• Minimum value = 0.094 J⋅g^-1⋅K^-1 for radio