Heat capacity

Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat that must be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K). It quantifies the ability of a material or system to store thermal energy. Heat capacity is an extensive property. The corresponding intensive property is the specific heat capacity, found by dividing the heat capacity of an object by its mass. Dividing the heat capacity by the amount of substance in moles yields its molar heat capacity. The volumetric heat capacity measures the heat capacity per volume. In architecture and civil engineering, the heat capacity of a building is often referred to as its thermal mass.

Definition

Basic definition The heat capacity of an object, denoted by C {\displaystyle C} , is the limit

C = lim Δ T → 0 Q Δ T , {\displaystyle C=\lim _{\Delta T\to 0}{\frac {Q}{\Delta T}},}

where Q {\displaystyle Q} is the amount of heat that must be added to the object (of mass M) in order to raise its temperature by Δ T {\displaystyle \Delta T} . The value of this parameter usually varies considerably depending on the starting temperature T {\displaystyle T} of the object and the pressure p {\displaystyle p} applied to it. In particular, it typically varies dramatically with phase transitions such as melting or vaporization (see enthalpy of fusion and enthalpy of vaporization). Therefore, it is considered a function C ( p , T ) {\displaystyle C(p,T)} of those two variables.

Variation with temperature The variation can be ignored in contexts when working with objects in narrow ranges of temperature and pressure. For example, the heat capacity of a block of iron weighing one pound is about 204 J/K when measured from a starting temperature T = 25 °C and P = 1 atm of pressure. That approximate value is adequate for temperatures between 15 °C and 35 °C, and surrounding pressures from 0 to 10 atmospheres, because the exact value varies very little in those ranges. One can trust that the same heat input of 204 J will raise the temperature of the block from 15 °C to 16 °C, or from 34 °C to 35 °C, with negligible error.

Heat capacities of a homogeneous system undergoing different thermodynamic processes

At constant pressure, dQ = dU + pdV (isobaric process) At constant pressure, heat supplied to the system contributes to both the work done and the change in internal energy, according to the first law of thermodynamics. The heat capacity is called C p {\displaystyle C_{p}} and defined as:

C p = d Q d T | p = const {\displaystyle C_{p}=\left.{\frac {dQ}{dT}}\right|_{p={\text{const}}}}

From the first law of thermodynamics follows d Q = d U + p d V {\displaystyle dQ=dU+p\,dV} and the inner energy as a function of p {\displaystyle p} and T {\displaystyle T} is:

d Q = ( ∂ U ∂ T ) p d T + ( ∂ U ∂ p ) T d p + p [ ( ∂ V ∂ T ) p d T + ( ∂ V ∂ p ) T d p ] {\displaystyle dQ=\left({\frac {\partial U}{\partial T}}\right)_{p}dT+\left({\frac {\partial U}{\partial p}}\right)_{T}dp+p\left[\left({\frac {\partial V}{\partial T}}\right)_{p}dT+\left({\frac {\partial V}{\partial p}}\right)_{T}dp\right]}

For constant pressure ( d p = 0 ) {\displaystyle (dp=0)} the equation simplifies to:

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