How Does the Dulong Petit Law Help Predict Heat Capacities?
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.
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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
Calculate the approximate atomic mass using the formula.
Approx. atomic mass x specific heat capacity = 6.4
Find the valency of the element using the equation.
Approx. atomic mass = Equivalent mass and valency
Obtain the nearest whole number for the calculated valency. This is the valency of the element.
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: Meaning, Formula, and Applications
1. What is the Dulong-Petit Law?
The Dulong-Petit law is a principle in thermodynamics which states that for most solid elements, the molar specific heat capacity at a constant volume is approximately constant, regardless of the element. This constant value is very close to 3R, where 'R' is the universal gas constant. In simple terms, it suggests that it takes roughly the same amount of heat to raise the temperature of one mole of any solid element by one degree.
2. What is the formula for the Dulong-Petit Law?
The formula for the Dulong-Petit law is expressed in terms of molar heat capacity (C) and the universal gas constant (R) as:
C ≈ 3R
Where R is approximately 8.314 J/(mol·K), making the molar heat capacity about 25 J/(mol·K). The law can also be written using the specific heat capacity (c) and the molar mass (M) of the element:
c × M ≈ 3R.
3. What are the main applications of the Dulong-Petit Law?
The Dulong-Petit law, while a historical model, has important applications, primarily in:
Estimating Atomic Mass: Historically, it was a crucial method for determining the approximate atomic mass of a newly discovered solid element by measuring its specific heat capacity.
Verifying Molar Mass: It can be used to check and validate the molar masses of metallic elements that were determined through other experimental means.
Predicting Heat Capacity: It provides a reasonable approximation for the molar heat capacity of many heavier metals at or above room temperature.
4. Under what conditions is the Dulong-Petit Law valid?
The Dulong-Petit law is an approximation and is generally valid only under the following conditions:
It applies specifically to solid elements, particularly metals with a simple crystalline structure.
It is most accurate for heavier elements (those with a high atomic mass).
It holds true at temperatures that are sufficiently high, well above a characteristic value for each substance known as the Debye temperature. It is not accurate at very low temperatures.
5. Why does the Dulong-Petit Law fail at low temperatures?
The Dulong-Petit law fails at low temperatures because it is based on classical physics, which assumes that atoms can vibrate with any amount of energy. However, quantum mechanics dictates that the vibrational energies of atoms in a crystal are quantized (exist in discrete levels). At very low temperatures, there is not enough thermal energy to excite these vibrational modes, so the solid cannot absorb heat as effectively. This causes the specific heat capacity to decrease and approach zero as the temperature nears absolute zero (0 Kelvin), a phenomenon the classical Dulong-Petit law cannot explain.
6. What is the physical principle that explains the Dulong-Petit Law?
The physical principle behind the Dulong-Petit law is the classical equipartition theorem. This theorem suggests that in thermal equilibrium, energy is shared equally among all available degrees of freedom. An atom in a solid lattice can oscillate in three dimensions (x, y, z). For each dimension, it has two degrees of freedom (one for kinetic energy and one for potential energy), making a total of six degrees of freedom. The equipartition theorem assigns (1/2)kT of energy to each. The total molar energy thus becomes 3RT, and the molar heat capacity (the rate of change of energy with temperature) is therefore 3R.
7. How does the Dulong-Petit Law relate to specific heat capacity and molar mass?
The Dulong-Petit law provides a direct relationship between an element's specific heat capacity (c) and its molar mass (M). The law states that the molar heat capacity (C) is approximately 3R. Since molar heat capacity is the product of specific heat capacity and molar mass (C = c × M), the law can be expressed as c × M ≈ 3R. This implies that for solid elements, the specific heat capacity is inversely proportional to the atomic mass; heavier elements will have lower specific heat capacities, and lighter elements will have higher ones.
