Heat capacity

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 ${\displaystyle C}$, is the limit $${\displaystyle C=\underset{\mathrm{\Delta }T\to 0}{lim}\frac{Q}{\mathrm{\Delta }T},}$$ where ${\displaystyle Q}$ is the amount of heat that must be added to the object (of mass M) in order to raise its temperature by ${\displaystyle \mathrm{\Delta }T}$.

The value of this parameter usually varies considerably depending on the starting temperature ${\displaystyle T}$ of the object and the pressure ${\displaystyle p}$ 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 ${\displaystyle C(p,T)}$ 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 ${\displaystyle C_p}$ and defined as:

$${\displaystyle C_p={\frac{dQ}{dT}|}_{p=\text{const}}}$$

From the first law of thermodynamics follows ${\displaystyle dQ=dU+pdV}$ and the inner energy as a function of ${\displaystyle p}$ and ${\displaystyle T}$ is:

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

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

$${\displaystyle C_p={\frac{dQ}{dT}|}_{p=\text{const}}={\left(\frac{\mathrm{\partial }U}{\mathrm{\partial }T}\right)}_p+p{\left(\frac{\mathrm{\partial }V}{\mathrm{\partial }T}\right)}_p={\left(\frac{\mathrm{\partial }H}{\mathrm{\partial }T}\right)}_p}$$

where the final equality follows from the appropriate Maxwell relations, and is commonly used as the definition of the isobaric heat capacity.

At constant volume, dV = 0, dQ = dU (isochoric process)

A system undergoing a process at constant volume implies that no expansion work is done, so the heat supplied contributes only to the change in internal energy. The heat capacity obtained this way is denoted ${\displaystyle C_V.}$ The value of ${\displaystyle C_V}$ is always less than the value of ${\displaystyle C_p}$. (${\displaystyle C_V<C_p}$.)

Expressing the inner energy as a function of the variables ${\displaystyle T}$ and ${\displaystyle V}$ gives:

$${\displaystyle dQ={\left(\frac{\mathrm{\partial }U}{\mathrm{\partial }T}\right)}_{V}dT+{\left(\frac{\mathrm{\partial }U}{\mathrm{\partial }V}\right)}_{T}dV+pdV}$$

For a constant volume (${\displaystyle dV=0}$) the heat capacity reads:

$${\displaystyle C_V={\frac{dQ}{dT}|}_{V=\text{const}}={\left(\frac{\mathrm{\partial }U}{\mathrm{\partial }T}\right)}_V}$$

The relation between ${\displaystyle C_V}$ and ${\displaystyle C_p}$ is then:

$${\displaystyle C_p=C_V+\left({\left(\frac{\mathrm{\partial }U}{\mathrm{\partial }V}\right)}_T+p\right){\left(\frac{\mathrm{\partial }V}{\mathrm{\partial }T}\right)}_p}$$

Calculating Cp and CV for an ideal gas

Mayer's relation:

$${\displaystyle C_p-C_V=nR.}$$ $${\displaystyle C_p/C_V=\gamma ,}$$

where:

• ${\displaystyle n}$ is the number of moles of the gas,
• ${\displaystyle R}$ is the universal gas constant,
• ${\displaystyle \gamma }$ is the heat capacity ratio (which can be calculated by knowing the number of degrees of freedom of the gas molecule).

Using the above two relations, the specific heats can be deduced as follows:

$${\displaystyle C_V=\frac{nR}{\gamma -1},}$$ $${\displaystyle C_p=\gamma \frac{nR}{\gamma -1}.}$$ Following from the equipartition of energy, it is deduced that an ideal gas has the isochoric heat capacity

$${\displaystyle C_V=nR\frac{N_f}{2}=nR\frac{3+N_i}{2}}$$

where ${\displaystyle N_f}$ is the number of degrees of freedom of each individual particle in the gas, and ${\displaystyle N_i=N_f-3}$ is the number of internal degrees of freedom, where the number 3 comes from the three translational degrees of freedom (for a gas in 3D space). This means that a monoatomic ideal gas (with zero internal degrees of freedom) will have isochoric heat capacity ${\displaystyle C_v=\frac{3nR}{2}}$.

At constant temperature (Isothermal process)

No change in internal energy (as the temperature of the system is constant throughout the process) leads to only work done by the total supplied heat, and thus an infinite amount of heat is required to increase the temperature of the system by a unit temperature, leading to infinite or undefined heat capacity of the system.

At the time of phase change (Phase transition)

Heat capacity of a system undergoing phase transition is infinite, because the heat is utilized in changing the state of the material rather than raising the overall temperature.

Calculating changes in entropy using heat capacity

The change in entropy of a system is generally not easy to measure directly, and hence it is common to measure the isobaric and isochoric heat capacities as functions of temperature, which are much easier to measure, allowing the change in entropy to be calculated as follows:

Given an isochoric system, ${\displaystyle C_v={\left(\frac{\mathrm{\partial }U}{\mathrm{\partial }T}\right)}_{N,V}}$, which can be rewritten as ${\displaystyle {dU|}_{N,V=const}=C_{v}dT}$.

where ${\displaystyle N}$ is the particle number.

The fundamental thermodynamic relation ${\displaystyle dU=TdS-pdV+\mu dN}$ can be restricted to obtain ${\displaystyle {dU|}_{N,V=const}=TdS}$

where:

• ${\displaystyle S}$ is the entropy of the system
• ${\displaystyle \mu }$ is the chemical potential of the system

Hence ${\displaystyle C_{v}dT=TdS}$ and ${\displaystyle dS=\frac{C_v}{T}dT}$. Integrating both sides, keeping in mind that ${\displaystyle C_v}$ is a function of ${\displaystyle T}$, the following relation is obtained:

${\displaystyle S_2-S_1=\mathrm{\Delta }S={\int }_{T_1}^{T_2}\frac{C_v(T)}{T}dT}$

where:

• ${\displaystyle S_1,T_1}$ are the initial entropy and temperature respectively
• ${\displaystyle S_2,T_2}$ are the final entropy and temperature respectively
• ${\displaystyle \mathrm{\Delta }S}$ is the change in entropy of the system

Similarly, for an isobaric system, using ${\displaystyle C_p={\left(\frac{\mathrm{\partial }H}{\mathrm{\partial }T}\right)}_{N,p}}$ and ${\displaystyle dH=TdS+Vdp+\mu dN}$, it can also be derived that

${\displaystyle \mathrm{\Delta }S={\int }_{T_1}^{T_2}\frac{C_p(T)}{T}dT}$

Heterogeneous objects

The heat capacity may be well-defined even for heterogeneous objects, with separate parts made of different materials; such as an electric motor, a crucible with some metal, or a whole building. In many cases, the (isobaric) heat capacity of such objects can be computed by simply adding together the (isobaric) heat capacities of the individual parts.

However, this computation is valid only when all parts of the object are at the same external pressure before and after the measurement. That may not be possible in some cases. For example, when heating an amount of gas in an elastic container, its volume and pressure will both increase, even if the atmospheric pressure outside the container is kept constant. Therefore, the effective heat capacity of the gas, in that situation, will have a value intermediate between its isobaric and isochoric capacities ${\displaystyle C_p}$ and ${\displaystyle C_V}$.

For complex thermodynamic systems with several interacting parts and state variables, or for measurement conditions that are neither constant pressure nor constant volume, or for situations where the temperature is significantly non-uniform, the simple definitions of heat capacity above are not useful or even meaningful. The heat energy that is supplied may end up as kinetic energy (energy of motion) and potential energy (energy stored in force fields), both at macroscopic and atomic scales. Then the change in temperature will depend on the particular path that the system followed through its phase space between the initial and final states. Namely, one must somehow specify how the positions, velocities, pressures, volumes, etc. changed between the initial and final states; and use the general tools of thermodynamics to predict the system's reaction to a small energy input. The "constant volume" and "constant pressure" heating modes are just two among infinitely many paths that a simple homogeneous system can follow.