## Heat Capacity of Polymers

The heat capacity is a macroscopic thermodynamic property that is based on the molecular motion and vibration. It is one of the most important thermo-physical properties of polymers and is often used to calculate other calorimetric properties such as the enthalpy, entropy, and the Gibbs free energy.

The heat capacity is defined as the heat per amount material (mole, gram etc.) necessary to increase the temperature by one degree:

*C*(*T*) = (∂*Q* / ∂*T*)

It is usually measured in Joules per Kelvin and kilogram (or
mole).

Since d*U* = d*Q* - *p*d*V*, the heat capacity at constant volume is equal to the change in internal energy:

*C _{V}* = (∂

*U*/ ∂

*T*)

_{V}

At constant pressure, the change in enthalpy is d*H* = d*Q* + *V*d*p* and thus

*C _{p}* = (∂

*H*/ ∂

*T*)

_{p}=

*T*(∂

*S*/ ∂

*T*)

_{p}

The two heat capacities are related to each other:

*C _{p}* -

*C*=

_{p}*T V*

*α*

^{2}/

*β*

where *α* and *β* are the coefficient of thermal expansion and the isothermal compressibility.

In the case of polymers, we have to distinguish between the heat capacity of liquid, rubbery and glassy polymers. The heat capacity increases with increasing temperature, therefore, a liquid or rubbery polymer can hold more energy than a solid polymer. All materials show this increase in heat capacity with temperature.

The heat capacity of solid macromolecules at constant volume, *C _{v }*, can be described fully based on an approximate vibrational spectrum,
which can be approximated with the harmonic oscillator model. This
greatly simplifies the calculation of heat capacities. The description of the heat capacity of liquid macromolecules,
on the other hand, is much more complicated. Besides vibrational motions, large-amplitude rotations, internal
molecule rotations (conformational motions) and translational motions
have to be included in the calculation of the heat capacity. A detailed discussion of the heat capacities of liquid and solid macromolecules can be found in
Bernhard Wunderlich's monography "

*Thermal Analysis of Polymeric Materials*" (2005).

The specific heat capacity of polymers below and above the *T _{g}* is often estimated with group contribution (GC) schemes. In many cases, the correspondence between experimental and calculated
values is quite satisfactory (see table below). However, the GC models for heat capacities are only applicable to standard conditons (298 K, 1.0135 bar). Specific heat capacities as a function of
temperature have been published for only a limited number of polymers. In many cases, the heat capacity
(at constant pressure) as a function of temperature can be approximated by straight lines. The slope
of these straight lines is often related to the heat at 298K and shows an average value of

d*C _{p}^{s}* / d

*T*≈ 3 x 10

^{-3}K

^{-1}

*C*

_{p}^{s}d*C _{p}^{l}* / d

*T*≈ 1.3 x 10

^{-3}K

^{-1}

*C*

_{p}^{l}Only for very low temperatures (< 100 K), noticeable deviations from the straight line are observed.

Polymer / Heat Capacity of Solid J/mol-K: | Predicted C_{p}(298)** |
Measured C_{p}(298)* |

Polyethylene | (47.0) | 43.4 |

Polyisobutylene | 94.9 (91.5) | 94.0 |

Polystyrene | 128.3 (127.6) | 126.5 |

Poly(vinyl chloride) | 60.0 (64.6) | 59.0 |

Poly(vinyl acetate) | 101.2 (116.0) | 101.2 |

Poly(methyl acrylate) | 114.0 (116.0) | 115.0 |

Poly(methyl methacrylate) | 137.6 (137.5) | 137.0 |

Poly(butyl methacrylate) | 234.8 (221.5) | 235.9 |

Poly(1,4-butadiene) | (87.0) | 88.0 |

Polyisoprene | (108.4) | 108.0 |

Poly(oxy(2,6-diphenyl-1,4-phenylene) | 267.6 (252.6) | 272.8 |

Poly(hexamethylene adipamide) | 329.0 (333.0) | 329.2 |

When using these generic values, the expected standard deviation is often in the range of seven percent for solid polymers, whereas for rubbery or liquid polymers much larger deviations are observed. For this reason, we recommend to use experimental values for the slope. If the slope is unknown, the heat capacity at an arbitrary temperature may be approximated with following formulas:

*C _{p}^{s}*(

*T*) =

*C*(

_{p}^{s}*298 K*) · [1 + 3 x 10

^{-3}· (

*T*- 298)]

*C _{p}^{l}*(

*T*) =

*C*(

_{p}^{l}*298 K*) · [1 + 1.3 x 10

^{-3}· (

*T*- 298)]

The heat capacities of many common polymers have been measured by Wunderlich and others. The most comprehensive collection of heat capacity data can be found in the ATHAS data bank (Advanced THermal Analysis), which has been developed over the last 30 years by Wunderlich (Chemistry department of the University of Tennessee), and coworkers.