# Dulong - Petit Law

## An Introduction to Dulong - Petit Law

Physics is an interesting and thought-provoking subject. Students should not have the habit of rote learning, rather they should study the subject with an open and fresh mind. Students should constantly keep profiling their minds and stay curious while learning the subject of Physics.

The team of experts at Vedantu has framed the teaching method in a fashion where they keep provoking the young minds to keep making them learn new concepts, to make them stay curious and open their minds towards the world of science.

This particular topic explains one such concept of Physics for the students. This will help the students have a strong and easy grasp of the topic and also increase their scores in the exams.

### The Article Explains The Following Concepts -

• What is Dulong Petit Law?

•  Dulong Petit Law Equation

• The Law of Dulong and Petit Formula

• Numerical problem with solution

• Conclusion

• FAQs

### 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:

$c \times M = k$

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 times, this constant came out to be equal to 3R, where R is the gas constant. The Dulong and petit law then became:

$c \times M = 3R$

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

$C \times (\frac{M}{m}) = 3R$

$\frac{C}{n} = 3R$

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.

$c \times M = k$

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

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

Approx. atomic mass = Equivalent mass and valency

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

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

Corrected atomic mass = Equivalent mass x valency

### Limitation of Dulong Petit Law -

Every set of theories comes with its own challenges and limitations. Similarly, the Dulong Petit Law has the following limitations -

• The law is applicable only to those elements which are in solid-state

• The law is not applicable to lighter elements having a high melting point. Thus applicable only to the heavier elements.

• The law gives only an approximate atomic mass. Thus, no exact number is given.

### 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 = $\frac{6.4}{0.03}$

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 and valency

$\frac{213.33}{69.66} = valency$

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 is 208.98u.

### Conclusion

The Dulong and petit law gives 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 gasses, 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.

## FAQs on Dulong - Petit Law

1. What are the Limitations of Dulong Petit Law?

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:

1. 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.

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

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

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

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.

$c = \frac{C}{m}$

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.

3. What is the meaning of the term Specific Heat capacity?

The very simple dictionary definition states that the term specific heat capacity is the heat required to raise the temperature of the unit mass of a given substance by a given amount (which is usually one degree).

Here, heat capacity indicates the amount of energy required to raise some unit quantity of a substance by 1 degree C.

To know more about the topic visit the website of Vedantu and refer to the free PDFs made available for the benefit of the students.

4. What is the reason that different materials have different specific heat capacities?

The reason behind the difference in the specific heat capacity is that objects with different molecular structures have different ways of storing energy. Since the specific heat is a macroscopic counting property that counts how much energy certain materials can store, they are different for different materials.

For example - the gas argon is monoatomic because it is composed of several molecules of a single atom and a single atom can store energy only in three ways.

Whereas, oxygen is composed of many molecules of two atoms of oxygen and this molecule has more ways of storing energy. Thus, the specific energy of the gas oxygen should be higher than argon's because it has more ways of storing energy.

Also, Bigger molecules have many other ways of storing energy so their specific heat is even higher.

So, the difference in a material due to different atoms, different molecules, and different packaging result in differences in heat storage.

5. How relevant is the topic Dulong Petit Law for competitive exams?

The concept is first introduced in the syllabus of Physics for class 11th students. The significance of the topic can be understood by a student once they analyze and observe the papers of competitive exams.

Questions have been asked from the topic in JEE Mains also there were  a few questions on the NEET paper.

Students cannot skip the topic and have to pay greater emphasis on getting a conceptual clarity of the particular topic.

The study materials provided at Vedantu will guide a student in the right direction as the team members scientifically analyze the significance and weightage given to every chapter and topic.

For more such important topics visit the website of Vedantu and gear up your level of preparation.

6. Can we say that the Dulong Petit Law is accurate?

Dulong Petit Law has been able to explain certain phenomena to a greater extent. Its relevance is still held high in the field of science. Despite these achievements, we cannot say that the law gives an accurate result of the findings. As the law and its methods have its limitations (refer to the article) due to which they can predict an approximate result instead of accurate results.

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