Heat Capacity

Heat Capacity

Heat capacity (usually denoted by a capital C, often with subscripts), or thermal capacity, is the measurable physical quantity that characterizes the amount of heat required to change a substance's temperature by a given amount. In SI units, heat capacity is expressed in units of joules per kelvin (J/K).

An object's heat capacity (symbol C) is defined as the ratio of the amount of heat energy transferred to an object to the resulting increase in temperature of the object .

$\displaystyle{C=\frac{Q}{ \Delta T}.}$ $

Heat capacity is an extensive property, so it scales with the size of the system. A sample containing twice the amount of substance as another sample requires the transfer of twice as much heat (Q) to achieve the same change in temperature (ΔT). For example, if it takes 1,000 J to heat a block of iron, it would take 2,000 J to heat a second block of iron with twice the mass as the first.

The Measurement of Heat Capacity

The heat capacity of most systems is not a constant. Rather, it depends on the state variables of the thermodynamic system under study. In particular, it is dependent on temperature itself, as well as on the pressure and the volume of the system, and the ways in which pressures and volumes have been allowed to change while the system has passed from one temperature to another. The reason for this is that pressure-volume work done to the system raises its temperature by a mechanism other than heating, while pressure-volume work done by the system absorbs heat without raising the system's temperature. (The temperature dependence is why the definition a calorie is formally the energy needed to heat 1 g of water from 14.5 to 15.5 °C instead of generally by 1 °C. )

Different measurements of heat capacity can therefore be performed, most commonly at constant pressure and constant volume. The values thus measured are usually subscripted (by p and V, respectively) to indicate the definition. Gases and liquids are typically also measured at constant volume. Measurements under constant pressure produce larger values than those at constant volume because the constant pressure values also include heat energy that is used to do work to expand the substance against the constant pressure as its temperature increases. This difference is particularly notable in gases where values under constant pressure are typically 30% to 66.7% greater than those at constant volume.

Thermodynamic Relations and Definition of Heat Capacity

The internal energy of a closed system changes either by adding heat to the system or by the system performing work. Recalling the first law of thermodynamics,

$dU=\delta Q-\delta W$.

For work as a result of an increase of the system volume we may write,

$dU=\delta Q-PdV$.

If the heat is added at constant volume, then the second term of this relation vanishes and one readily obtains

$\displaystyle{\left( \frac{\partial U}{\partial T}\right) _{V}=\left( \frac{\partial Q}{\partial T}\right) _{V}=C_{V}}$.

This defines the heat capacity at constant volume, C_V. Another useful quantity is the heat capacity at constant pressure, C_P. With the enthalpy of the system given by

$H=U+PV$,

our equation for dU changes to

$dH=\delta Q+VdP$,

and therefore, at constant pressure, we have

$(\frac{\partial H}{\partial T})_{P}=(\frac{\partial Q}{\partial T})_{P}=C_{P}$.