Introduction

Heat capacity is calculated of the heat energy needed to amplify the temperature of an object via an indeed temperature interval. Heat capacity is a wide property since its value is proportional to the amount of material in the object; for instance, a bathtub of water has a greater heat capacity than a cup of water. Heat capacity is generally conveyed in units of J/K (or J K^-1). For example, one could write that the gasoline in a 55-gallon drum has an standard heat capacity of 347 kJ/K.

Definition of Heat Capacity

The heat capacity of a material, C, is amplify in internal energy of a substance U per unit raise in temperature T:

C = (∂U/∂T)

Heat capacity is mathematically described as the ratio of a tiny amount of heat δQ added to the body, to the equivalent tiny enhance in its temperature dT:

For thermodynamic systems through more than one physical dimension, the above explanation  doesn't give a single, unique amount unless a meticulous infinitesimal path through the system's phase space has been described (this means that one requires to know at all times where all parts of the system are, how much mass they have, and how fast they are moving). This information is utilized to account for different ways that heat can be stored as kinetic energy (energy of motion) and potential energy (energy stored in force fields), as an object increases or contracts. For all real systems, the path through such modifies must be explicitly explained, because the value of heat capacity based on that path from one temperature to an additional, is chosen. Of particular helpfulness in this context are the values of heat capacity for steady volume, CV, and stable pressure, CP. Such will be described below.

Heat Capacity of Compressible Bodies

The state of an easy compressible body through fixed mass is described via 2 thermodynamic parameters such as temperature T and pressure p. Consequently as revealed above, one might distinguish between heat capacity at constant volume, CV, and heat capacity at steady pressure, Cp:

Where

C_v = ( Δq/ δT)_v  = T( δS/ δT)_v

C_p = ( δQ/ Δt)_p = T( Δs/ δT)_p

δQ is the infinitesimal amount of heat added,

dT is the subsequent rise in temperature.

The increment of internal energy is the heat added and the work added:

dU = TdS - pdV

So the heat capacity at constant volume is

C_v = (δU/δT)_v

The enthalpy is defined by H = U + PV. The increment of enthalpy is

dH = dU + (pdV + V dp)

which, after replacing dU with the equation above and cancelling the PdV terms reduces to?

dH = TdS + Vdp

So the heat capacity at constant pressure is

C_p = ( δH/ δT)_p

As we know that this last 'definition' is a bit circular, because the idea of "enthalpy" itself was discovered to be a estimate of heat absorbed or generated at constant pressures (the situations in that chemists generally work). As such, enthalpy merely accounts for the extra heat that is generated or absorbed via pressure-volume work at steady pressure.

Therefore, it isn't surprising that constant-pressure heat capacities might be described in terms of  enthalpy, because "enthalpy" was described in the 1^st place to build this so. Analogous expressions for precise heat capacities at steady volume and at stable pressure are:

C_v = ( δu/ δT)_v

C_p = ( δh/ δT)_p

Where u = specific internal energy and h = specific enthalpy. Such intensive quantities can be expressed on a per mole basis or on a per mass basis, as long as constancy is preserved all through the equation.

Relation between Specific Heat

Measuring the heat capacity at steady volume can be prohibitively hard for liquids and solids. That is, small temperature transforms classically need huge pressures to preserve a liquid or solid at constant volume implying the enclosing vessel must be almost rigid or at least extremely strong. In its place it is easier to calculate the heat capacity at steady pressure (allowing the material to enlarge or contract as it wishes) and solve for the heat capacity at constant volume using mathematical associations derived from the basic thermodynamic laws. Starting from the fundamental Thermodynamic relation one can illustrate,

C_p - C_v = α²T/αβT

where,

α is the coefficient of thermal expansion,

β_T is the isothermal compressibility, and

ρ is density.

For an  ideal  gas,  if  ρ  is  expressed  as  molar  density  in  the  above equation, this equation reduces simply to Mayer's relation,

C_p,m - C_v,m = R

Where C_p,m and C_v,m are intensive property heat capacities conveyed on a per mole basis at steady pressure and stable volume, correspondingly.

Specific heat capacity

The specific heat capacity of a substance on a per mass basis is

C= δC/δm'

Which in the absence of phase transitions is equivalent to

c = E_m = C/m = C/ρV'

Where C is the heat capacity of a body made of the material in question,

m is the mass of the body,

V is the volume of the body, and

Ρ = m/V

is the density of the material.

For gases, and as well for other substances under high pressures, there is require to differentiate   between dissimilar boundary conditions for the processes under consideration (because values differ significantly between different conditions). Typical procedures for that heat capacity may be described contain isobaric (constant pressure, dp = 0) or isochoric (constant volume, dV = 0) processes. The equivalent specific heat capacities are expressed as

Cp = (δc/ δm)_p

Cv = (δC/δm)_v

A related parameter to c is Cv^-1, the volumetric heat capacity. In engineering practice, C_v for solids or liquids frequently signifies a volumetric heat capacity, rather than a constant-volume one.  In these cases, the mass-specific heat capacity (exact heat) is frequently explicitly written through the subscript m, as Cm. Of course, from the above associations, for solids one writes

C m = C/m = cV/ Ρ.

For pure homogeneous chemical compounds by established molecular or molar mass or a  molar quantity is established, heat capacity as an intensive property can be expressed on a per mole basis instead of a per mass  basis  via  the  subsequent  equations  equivalent  to  the  per  mass equations:

  C_p,m  = (δC/δn)_p= molar heat capacity at constant pressure

   C_v,m = (δC/δn)_v= molar heat capacity at constant volume

Where n = number of moles in the body or thermodynamic system. One may term to such a 'per mole' quantity as molar heat capacity to differentiate it from precise heat capacity on a per mass basis.

Dimensionless heat capacity

The dimensionless heat capacity of a material is

C* = C/nR = C/NK

Where

C is the heat capacity of a body made of the material in question (J/K)

n is the amount of matter in the body (mol)

R is the gas constant (J/(K·mol) N is the number of molecules in the body. (Dimensionless)

k is Boltzmann's constant (J/(K·molecule)

Solid phase

Fig: The Dimensionless Heat Capacity Divided via three, as a Function of Temperature as Predicted by the Debye model and by Einstein's Earlier Model

The horizontal axis is the temperature separated via the Debye temperature. As we know that, as supposed, the dimensionless heat capacity is zero at absolute zero, and increases to a value of 3 as the temperature becomes much larger than the Debye temperature. The horizontal line above the plots corresponds to the classical limit of the Dulong-Petit law.

For issue in a crystalline solid phase, the Dulong-Petit law, that was computed empirically, states that the mole-specific heat capacity supposes the value 3 R. Indeed, for solid metallic chemical components at room temperature, molar heat capacities range from about 2.8 R to 3.4 R (beryllium being a notable exception at 2.0 R).

The theoretical maximum heat capacity for superior multi-atomic gases at higher temperatures as well approaches the Dulong-Petit limit of 3 R, so long as this is computed per mole of atoms, not molecules. The cause is that gases through extremely huge molecules, in theory have approximately the similar high-temperature heat capacity as solids, lacking only the (small) heat capacity contribution that comes from potential energy that can't be stored between divide molecules in a gas. The Dulong-Petit "limit" consequences from the equipartition theorem, and as these is only valid in the standard limit of a microstate continuum that is an elevated temperature limit. For light and non-metallic elements, in addition to most of the ordinary molecular solids depend on carbon compounds at standard ambient temperature, quantum results might as well play a significant role, as they do in multi-atomic gases. Such outcomes generally join to provide heat capacities lower than 3 R per mole of atoms in the solid, even though heat capacities computed per mole of molecules in molecular solids might be more than 3R. For instance, the heat capacity of water ice at the melting point is about 4.6 R per mole of molecules, but only 1.5 R per mole of atoms. The lower number consequences from the 'freezing out' of possible vibration modes for light atoms at correctly low temperatures, just as in many gases. Such results are seen in solids more often than liquids: for instance the heat capacity of liquid water is once more close to the theoretical 3 R per mole of atoms of the Dulong-Petit academic maximum.

For a more modern and exact analysis of the heat capacities of solids, particularly at low temperatures, it is helpful to employ the thought of phonons.

A Classical Approach to Heat Capacity

A good deal of insight might be attained from an extremely easy model of a solid. Consider the solid to be a 3 dimensional lattice of atoms in that the atoms are held near equilibrium positions via forces. If the force on an atom is proportional to its digression from its equilibrium position then it is termed a harmonic oscillator. For zero deviation the force is zero so the force for tiny deviations is proportional to the deviation. Therefore any these solid can be believed to be composed of harmonic oscillators. In a cubic lattice the atoms can oscillate in three directions.

The standard energy E of a harmonic oscillator in one dimension is kT, where k is Boltzmann's constant. In 3 dimensions the average energy is 3kT. If there are N atoms in the lattice then the internal energy is U= N(3kT). Let A be Avogadro's number (6.025×1023). Then separating and multiplying the equation of U by A gives

U = (N/A)(3AkT)

The ratio (N/A) is the number of moles of the substance, n, and Ak is denoted as R. Thus

U = n(3RT)

so

U_M = (U/n) = 3RT

The heat capacity per unit mole of a substance at constant pressure is then defined as

C_p = (∂UM/∂T)

and thus from the above C_p = 3R

The value for Cp of 3R is about 6 calories per degree Kelvin. This is identified as the Dulong and Petit value. It is a good approximation for the calculated values for solids at room temperatures (300°K). At low temperatures the Dulong and Petit value isn't a good approximation. Below is exposed the heat capacity of metallic silver as a function of temperature.

Fig: Heat Capacity for Silver at Different Temperatures

The shape of the curve for T near zero is of interest. It appears to be proportional to a power of T, say T² or T³.

A Quantum Mechanical Derivation of Heat Capacity

According to Planck's law for the distribution of energy for an ensemble (collection) of harmonic oscillators the average energy E is given by

E = hω/[exp(hω/kT) - 1]

Where h is Planck's constant h divided by 2*pi; and ω is the characteristic frequency (circular) of the oscillators. This frequency is in the nature of a parameter for the substance and has to be determined empirically.

For temperature such that kT is much larger than hω the denominator becomes approximately hω/kT so E is approximately kT.

In general though

U = NE = N3hω/[exp(hω/kT) - 1]

UM = 3Ahω/[exp(hω/kT) - 1]

and therefore

C_p = -3Ahω/[exp(hω/kT) - 1]²[exp(hω/kT)](-hω/kT²)

With a little rearrangement this can be put into the form

Cp = 3R (hω/kT) ²exp(hω/kT)/[exp(hω/kT) - 1]²

Now if the numerator and denominator are separated via [exp(hω/kT)]² the consequence is

Cp = 3R(hω/kT)²exp(-hω/kT)/[1 - exp(-hω/kT)]²

The attempt to attain the limit of this expression as T→0 generates the ambiguous result of ∞/∞. The application of l'Hospital's Rule 2 time's finally produces the consequence that the limit of C_p is zero as T→0.  

The Heat Capacity Function for Low Temperatures

As mentioned above the empirical heat capacity curve for silver seems to be proportional to T² or T³ for tiny values of T. The graph of the quantum mechanical heat capacity function obtained above specifies that for values of T near zero the heat capacity function is zero and flat.

Fig:  Heat Capacity for Solids at Different Temperatures

To examine the behaviour of the heat capacity function for small T it simplify matters let hω/k, that has the dimensions of temperature, be signified as θ. This parameter is sometimes termed the Einstein temperature since it was Einstein who 1^st formulated this line of analysis. (Einstein was far more adept at realizing the implications of the quantization of energy establish via Planck than Planck himself.) The heat capacity function is then:

C_p = 3R (θ/T)²exp(-θ/T)/[1 - exp(-θ/T)]²

Matters can be made even simpler through replacing θ/T with z. Then

C_p = 3Rz²exp (-z)/[1 - exp(-z)]²

and

ln(Cp) = ln(3R) + 2ln(z) - z - 2ln[1 - exp(-z)]

This means that ln (Cp) →-∞ as z→+∞; i.e., Cp→0 as T→0.

Limit using l'Hospital's Rule

C_p = 3R(hω/kT)²exp(-hω/kT)/[1 - exp(-hω/kT)]²

The term [1 - exp(-hω/kT)]² approaches the limit of 1 as T→0 so let us ignore that term. Thus

C_p = 3R(hω/kT)²/exp(hω/kT)

and to simplify matters let hω/k, which has the dimensions of temperature, be denoted as θ. Then

C_p = 3R(θ/T)²/exp(θ/T)

When T goes to zero the above term goes to ∞/∞. By l'Hospital's Rule we should consider the limit of the derivatives of the numerator and denominator. Therefore

3R(2(θ²)(-1/T³)/(exp(θ/T)(-θ/T²) = 6Rθ(1/T)/exp(θ/T)

Which, as T→0, goes to ∞/∞                                                                               

Therefore l'Hospital's rule must be applied again, which gives

6Rθ (-1/T²)/(exp(θ/T)(θ/T²) = 6R/exp(θ/T) which has a limit of zero as T→0.

Therefore C_p → 0 as T→0.

Heat Capacity at Absolute Zero

From the definition of entropy

TdS = δQ

We can compute the absolute entropy via integrating from zero kelvin temperature to the final temperature T_f

S(T_f) = ∫_t=0 ^Tf SQ/T = ∫₀ ^Tf δQ/dT*dT/T =  ∫₀^Tf C(T)* dT/T

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

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