Fundamentals of heat capacity

The measurement of heat capacity is a major technique of physical investigation for the understanding of materials, because heat capacity can be calculated ab initio from the model of a physical system. It is suggested that more significance can be attached to heat capacity measurements than to any other investigation, at least at low temperatures. Even at high temperatures, these measurements are useful in understanding many physio-chemical phenomenon. 

This is the first of a series of articles which focuses on this fundamental property of materials

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The heat capacity of a material is defined as the quanity of heat required to change its temperature by one degree. 

Mathematically, the heat capacity C of a system of arbitrary mass m is defined in terms of the following limit:

                                    

\begin{equation} C= \lim_{\Delta T\to 0}\frac{\Delta Q}{\Delta T}\end{equation}

where ∆Q is the quantity of heat that must be added to the system to raise its temperature by an amount of ∆T. 

In order to obtain a quantity that is independent of mass, the above equation is divided by the system mass to yield the specific heat capacity, or more simply the specific heat

\begin{equation} c= \frac{C}{m}=\frac{dq}{dT}\end{equation}

where dq is the quantity of heat required to raise the temperature of a unit mass of the system by an amount dT.  In general, the required quantity of heat will depend upon the temperature of the system as well as the changes that may occur in other physical properties of the system during the temperature rise. There are two principal specific heats, one defined at constant pressure and the other at constant volume:

   

\begin{equation}c_p=\left (\frac{dq}{dT}\right)_P \end{equation}

and

\begin{equation}c_v=\left(\frac{dq}{dT}\right)_V \end{equation}

respectively. In most theoretical calculations, the natural quantity to calculate is the ‘heat capacity per mole’ since this refers to a fixed number of particles. This quantity is also a ‘specific heat capacity’ and consequently, it is also referred to as the specific heat or molar specific heat. The molar specific heats will be denoted by upper case symbols and are defined as:

 \begin{equation} C_p= \left(\frac{dQ\prime}{dT\prime}\right)_P \end{equation}

and

 \begin{equation} C_v= \left(\frac{dQ\prime}{dT\prime}\right)_V \end{equation}

                          

where dQ’ is the quantity of heat required to raise the temperature of one mole of a substance by an amount dT’ under conditions of constant pressure and constant volume respectively. Though $\rm C_v$  is a more fundamental quantity than $\rm C_p$ but from the experimental point of view $\rm C_p$ is easier to determine. They are related by $\rm C_p - C_v =9\alpha^2\beta VT$    where $\rm \alpha$  is the temperature coefficient of thermal expansion, V is the volume and $\rm \beta$ is the bulk modulus of the solid. The difference between $\rm C_p$ and $\rm C_v$ is small for solids and may be neglected.