# Heat Capacities

Since we have looked at both C_{v} and C_{p} already, now is a good time to compare these heat capacities.

Heat Capacity at constant Volume, C_{v} |

The equipartition theory states that translational and rotational degrees of freedom each contribute RT/2 JK^{-1} to the constant volume heat capacity, whilst vibrational degrees each contribute RT JK^{-1}.

Mode (Type) of Freedom |
Number possessed by a molecule |

Translational | 3 |

Rotational (a) linear(b) non-linear |
2 3 |

Vibrational (a) linear(b) non-linear |
3n – 5 3n – 6 |

Rotational and Vibrational degrees of freedom can only be accessed at higher temperatures (as these energy levels are separated by a greater amount than translational levels), resulting in a graph somewhat like this one. ie: when the system becomes hot enough, rotational degrees may be accessed and at even higher temperatures, vibrational degrees. This produces the steps on the graph – a particular mode will only contribute to the heat capacity at a temperature high enough for it to be fully excited. |

The more ‘perfect’ the gas is, the more it will adhere to these guidelines. So argon shows good similarities. The more molecular interactions there are, the more the theory falls apart.

We can see then, how C_{v} changes with temperature. What about C_{p}?

Heat Capacity at constant Pressure, C_{p} |

You’ll be relieved to hear that C_{p} changes with temperature in a more simple fashion.

We can model it with the equation **C _{p,m }= a + bT + c/T^{2}**. a, b and c are constants and independent of temperature, and at the lowest temperatures

**C**. Once again, a is a constant independent of temperature.

_{p,m}= aT^{3}Finally, there is an important relationship between C_{v} and C_{p} **for perfect gases**:

Let us consider C_{p} – C_{v}.

C_{p} – C_{v} = (dH/dT)_{p} – (dU/dT)_{p}

and remembering that H = U + nRT

**\** dH/dT = dU/dT + nR

**\** C_{p} – C_{v} = (dU/dT)_{p} + nR – (dU/dT)_{p} = nR

**C _{p} – C_{v} = nR**