# Dulong - Petit Law

## What is Dulong Petit Law?

In 1819, two french physicists, Pierre Louis Dulong and Alexis Petit, better known as Dulong and Petit proposed a thermodynamics law, which states the classical expression of Molar specific heat capacity.

According to the Dulong and Petit law, the gram-atomic heat capacity i.e. the product of the specific heat capacity and the atomic mass of an element remains constant. This law was then modified to apply to metallic elements only and now is used to calculate the approximation at intermediately high temperatures.

The modern theory however built upon the assumption by Einstein in 1907, tells us that the heat capacity of solids is due to the lattice vibrations in the solids. But despite its simplicity, the Dulong and Petit law offers a good prediction for the heat capacity of many elementary solids at higher temperatures.

### Dulong Petit Law Equation

Dulong Petit formula as given by Dulong and Petit, after numerous experiments showed that the heat capacity per weight for several elements was the same. The Dulong Petit derivation is as follows:

The law of Dulong and petit can be mathematically represented as:

cM = k

Where,

c = specific heat capacity

M = molar mass

k = constant

Dulong and Petit did not know about this constant k. They found that elements with higher atomic weights as given by Dalton and early atomists were smaller and when multiplied by the specific heat capacity, it came out to be a constant. In modern time, this constant came out to be equal to 3R, where R is the gas constant. The Dulong and petit law then became:

cM = 3R

For a mass m of the sample divided by its molar mass M, gives us the number of moles.

C(M/m) = 3R

C/n = 3R

Where,

C = heat capacity of the element

M = molar mass

m = mass of the sample

n = number of moles

At very low temperatures, the quantum mechanical energy stored within solids manifests itself with a larger and larger effect. The law then fails for substances in the cryogenic region.

### The Law of Dulong and Petit Formula

The Dulong and petit law can be used to find the valency and the atomic mass of the elements. The formula given by Dulong and petit is as follows.

cM = k

Where,

c = specific heat capacity

M = molar mass

k = constant

To find the atomic mass and valency of any element in its solid-state using Dulong petit law equation is given by-

Steps Involved

1. Calculate the approximate atomic mass using the formula.

Approx. atomic mass x specific heat capacity = 6.4

2. Find the valency of the element using the equation.

Approx. atomic mass = Equivalent mass x valency

3. Obtain the nearest whole number for the calculated valency. This is the valency of the element.

4. Calculate the corrected atomic mass of the element using:

Corrected atomic mass = Equivalent mass x valency

Numerical Problem- The equivalent mass of metal with a specific heat capacity of 0.03 is 69.66. Calculate the valency of the metal and its atomic mass using the law of Dulong and petit formula.

Solution-  To find the atomic mass and valency of the metal using Dulong and petit method we will-

Approx. atomic mass x specific heat capacity=6.4

Approx. atomic mass = 6.4/0.03

Approx. atomic mass = 213.33

Now that we have obtained the approximate atomic mass, we can calculate the valency of the element using the Dulong law-

Approx. atomic mass = Equivalent mass x valency

213.33/69.66 = valency

3.06 = valency

By correcting the valency to its nearest whole number we obtain the valency of the element - 3.

Corrected atomic mass = 69.66 x 3

Corrected atomic mass = 208.98

The valency of the metal is 3 and its atomic mass 208.98u.

### Conclusion

The Dulong and petit law give us an appropriate approximation of specific heat capacity and atomic masses of solids. Certain relationships which Dulong and petit could not explain were later discovered such as the kinetic theory of gases, which gave the value of the constant they defined as ‘k’. The Dulong petit law stands for metals in their solid-state at higher temperatures.

1. What are the Limitations of Dulong Petit Law?

Ans- The Dulong petit rules states that the product of specific heat capacity and the molar mass of any element in its solid state is a constant. But there are some necessary conditions required to fulfil the Dulong petit model. The Dulong petit limit is as follows:

• The law is not applicable for light atoms bonded strongly to each other at room temperature. It wrongly predicts a higher heat capacity than found because the energy vibrational modes are not populated at room temperature for these substances.

• The Dulong petit method can only provide an approximation of the atomic mass.

• It applies only to elements that are in solid-state.

2. What Do You Understand By the Term Specific Heat Capacity?

Ans- Specific heat capacity of any substance is the amount of heat required to raise the temperature of one unit mass of the substance by one unit. Mathematically it is represented as Heat capacity divided by the mass of the sample.

c = C/m

Where,

c = specific heat capacity

C = heat capacity

m = mass of the sample

SI unit of specific heat capacity is Joule per Kelvin per kilogram (J/(kg K).

The specific heat capacity for any element changes with the temperature change. During phase transitions, all the heat provided is used in changing the state of the element and the specific heat capacity at that moment comes out to be infinite. The specific heat capacity is also dependent on volume and pressure.