Boltzmann Constant
The Boltzmann constant is vital for comprehending phenomena in thermodynamics, statistical mechanics, and quantum mechanics. It provides insights into concepts such as temperature, entropy, and the energy distribution among particles within a system. By applying the Boltzmann constant, physicists can illustrate how macroscopic properties arise from the interactions of individual particles, establishing it as a fundamental element in the study of physical systems.
1.0Definition of Boltzman Constant
- The Boltzmann constant serves as the proportionality factor that relates the thermodynamic temperature of a gas to the average K.E. of its particles.
- It is represented by kb.
2.0Ideal Gas Equation In Terms of Boltzman Constant
PV=nRT …..(1)
If v is the volume of 1 gram mass of the gas and M0 is the molecular mass ,then number of moles is given as,
n= Molecular Mass Mass of the gas (gm)=M01
PV=nRT
PV=M01RT (r=M0R)
PV=rT, perfect gas equation for 1gm of the gas
kb= Boltzman Constant, it is the gas constant per molecule.
kb=NAR⇒R=kbNA
n=NAN= Avogadro’s Number no.of molecules
PV=nRT
PV=NAN⋅kbNA⋅T=kbNT
PV=kbNT
3.0Formula of Boltzman Constant
Kb=NAR=mole−1JMole−1K−1=JK−1
Dimensional Formula: [ML2T−2K−1]
4.0Value of Boltzman Constant
- One mole of gas at Standard Temperature Pressure
- Calculating Value of Universal Gas Constant,
R=TPV
P=0.76 m of Hg column = 0.76 ×13.6×103×9.8Nm−2
Standard Temperature(T) = 273.15 K
Volume of one mole of gas = 22. 4 litre =22.4×10−3 m3
R=273.150.76×13.6×103×9.8×22.4×10−3=8.31Jmole−1 K−1
- Value of R in C.G.S R=4.28.31cal mole−1∘C−1
- kb=NAR=6.023×10238.31Jmole−1K−1=1.38×10−23JK−1
5.0Role in Thermodynamics
E=23kbT
This equation implies that mean kinetic energy per molecule is commensurate to the absolute temperature of the gas. Faster the motion of the molecules of a gas, higher will be their kinetic energy and hence higher will be the temperature of the gas
6.0Application of Boltzman Constant
In Law of Equipartition of Energy
- This law realms that in any dynamical system in thermal equilibrium, the energy is equally disseminated amongst its various degree of freedom and the energy linked with each degree of freedom per molecule is 21kbT, where kb = Boltzmann constant and T is the absolute temperature.
- The Law of Equipartition holds good for all degrees of freedom whether translational, rotational, vibrational.
- In the microscopic formulation of the ideal gas law, the Boltzmann constant connects macroscopic properties like pressure, volume, and temperature to the behavior of individual gas molecules.
7.0Sample Questions On Boltzman Constant
Q-1. In a definite region of space there are only 5 molecules per cm3 on an average. The temperature there is 3 K. What is the pressure of this gas?
Sol.
PV=kbNT
P=VkbNT=(VN)kbT
(VN=5 cm−3=5×106 m−3)
P=5 ×106×1.38×10−23×3=20.7×10−17Nm−2
Q-2. There are N molecules of a gas in a vessel. If the number of molecules is intensified to 2N, what will be the total energy of the gas?
Sol.
Average K.E per molecule, 21mv2=23kbT
Total energy of N molecules=, 21mNv2=23kbNT
When the number of molecules is increased from N to 2N,total energy of the gas is doubled, though the average K.E. per molecule remains the same.
Q-3. At what temperature does all molecular motion cease. Explain?
Sol.
E=23kbT
All molecular motion ceases at absolute zero or at zero Kelvin,
Absolute Temperature ∝ Average K.E of the molecules
T=0, Average K.E=0
Q-4. A prototype of an ideal gas occupies a volume V at a pressure P and absolute temperature T. The mass of each molecule is m, kb is Boltzman Constant, write the expression for the density of the gas.
Sol.
P=31ρV2=32mρ⋅21mv2=32⋅mρ⋅E=23kbT
Density ρ=kbTPm
Q5. What are the limitations of Boltzman Constant?
Solution:
- It is applicable to ideal gases.
2.The Boltzmann constant is primarily applicable at standard and moderate temperatures. However, at extremely high temperatures (like those found in plasmas) or very low temperatures (close to absolute zero), different physical phenomena can arise that necessitate alternative descriptions.