# The Dulong-Petit law and its meaning

Do you remember the working principle of desert coolers? Why do the places close to the sea have a moderate climate? Now, both of these phenomena have something to do with water. If you remember, we used to read in high school that water has a high specific heat capacity due to which it can absorb a large amount of heat before raising its temperature. Similarly, it takes a lot of time for water to cool down and to lose all that accumulated heat. Hence, the moderate climate of coastal areas. Now today, I am going to explain a famous law-the Dulong-Petit law, which relates the specific heat of different elements.

In 1819, French physicists Pierre Louis Dulong and Alexis Thérèse Petit proposed a thermodynamic law. This is the Dulong-Petit's law. This law gives the classical expression for the molar specific heat capacity of certain chemical elements. It also explains the similarity between molar-specific heat capacities of different elements. Dulong-Petit's law is as follows:

## The Dulong-Petit law

Dulong–Petit law states that the gram-atomic heat capacity (specific heat times atomic weight) of an element is a constant. In other words, the gram-atomic specific heat is the same for all solid elements, about six calories per gram atom.

Specific heat capacity: The amount of heat required to raise the temperature of a unit mass of that substance by unit degree.

Gram-atomic heat capacity: The amount of heat required to raise the temperature of one mole of an element by unit degree.

Let us try to understand this law with an example:

The specific heat of copper is 0.093 cal/ gm K and that of lead is only 0.031 cal/gm K. But according to Dulong and Petit, the molar specific heats of copper and lead are essentially the same. In fact, this law states that the molar-specific heats of all elements are constant and equal to a value 3R. Here, R is the universal gas constant.

## Limitations:

• The Dulong-Petit's law is only valid at high temperatures and not at low temperatures. This is because this law is based on Maxwell-Boltzmann statistics which is a classical law. However, at low temperatures, quantum mechanical principles must be applied. Hence, Einstein-Bose statistics can better explain specific heat capacities at low temperatures.
• The Dulong-Petit law fails at room temperatures for light atoms bonded strongly to each other, such as in metallic beryllium and in carbon as the diamond. The reason for this is that Dulong-Petit's law is based on the principle of equipartition of energy. This means that an atom has different modes of vibration-translational and rotational. all these modes are equally activated at high temperatures only. Hence, this leads to a mismatch in results at low temperatures.

Dulong-Petit law was helpful in obtaining the atomic weights of various elements. If the specific heat of an element is measured, we can calculate its atomic weight using this empirical law. Later, this law was modified to apply only to metallic elements at low temperatures.

## Author's message

In this article, I have explained a famous law from Condensed matter physics. I hope you find it amusing that not all laws of physics are universal. Just like, the Dulong-Petit's law is not applicable always and to all elements and at all temperatures. Hope you like this article and enjoy reading it.

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