Thermodynamic Properties: Monatomic and Diatomic Gases
Deriving Values of Cv, Cp, and γ for Monatomic and Diatomic Gases
Monatomic Gases
For monatomic gases, the internal energy (U) is given by:
U = (3/2) nRT
where n is the number of moles, R is the gas constant, and T is the temperature.
The specific heat capacity at constant volume (Cv) is defined as:
Cv = (∂U/∂T)v
Substituting the expression for U, we get:
Cv = (3/2) nR
The specific heat capacity at constant pressure (Cp) is defined as:
Cp = (∂H/∂T)p
where H is the enthalpy. For an ideal gas, H = U + pV = U + nRT.
Substituting the expression for H, we get:
Cp = (∂(U + nRT)/∂T)p
= Cv + nR
= (3/2) nR + nR
= (5/2) nR
The adiabatic index (γ) is defined as:
γ = Cp / Cv
= (5/2) nR / (3/2) nR
= 5/3
So, for monatomic gases:
- Cv = (3/2) nR
- Cp = (5/2) nR
- γ = 5/3
Diatomic Gases
For diatomic gases, the internal energy (U) is given by:
U = (5/2) nRT
where n is the number of moles, R is the gas constant, and T is the temperature.
The specific heat capacity at constant volume (Cv) is defined as:
Cv = (∂U/∂T)v
Substituting the expression for U, we get:
Cv = (5/2) nR
The specific heat capacity at constant pressure (Cp) is defined as:
Cp = (∂H/∂T)p
where H is the enthalpy. For an ideal gas, H = U + pV = U + nRT.
Substituting the expression for H, we get:
Cp = (∂(U + nRT)/∂T)p
= Cv + nR
= (5/2) nR + nR
= (7/2) nR
The adiabatic index (γ) is defined as:
γ = Cp / Cv
= (7/2) nR / (5/2) nR
= 7/5
So, for diatomic gases:
- Cv = (5/2) nR
- Cp = (7/2) nR
- γ = 7/5
Derivation of Planck’s Law and its Implications
Planck’s Law of Radiation
Planck’s law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature T. The spectral radiance B(ν, T) is given by:
\( B(\nu, T) = \frac{2h\nu^3}{c^2} \cdot \frac{1}{e^{h\nu / kT} – 1} \)
where:
- \( B(\nu, T) \) is the spectral radiance,
- \( h \) is Planck’s constant,
- \( \nu \) is the frequency of the radiation,
- \( c \) is the speed of light,
- \( k \) is the Boltzmann constant,
- \( T \) is the absolute temperature.
Wien’s Distribution Law
Wien’s distribution law is an approximation of Planck’s law for high frequencies (or equivalently, short wavelengths). For high frequencies, \( h\nu \gg kT \), the exponential term \( e^{h\nu / kT} \) becomes very large, and we can approximate:
\( e^{h\nu / kT} – 1 \approx e^{h\nu / kT} \)
Substituting this into Planck’s law:
\( B(\nu, T) \approx \frac{2h\nu^3}{c^2} \cdot \frac{1}{e^{h\nu / kT}} = \frac{2h\nu^3}{c^2} \cdot e^{-h\nu / kT} \)
This is Wien’s distribution law:
\( B(\nu, T) \approx \frac{2h\nu^3}{c^2} \cdot e^{-h\nu / kT} \)
Rayleigh-Jeans Law
Rayleigh-Jeans law is an approximation of Planck’s law for low frequencies (or equivalently, long wavelengths). For low frequencies, \( h\nu \ll kT \), the exponential term \( e^{h\nu / kT} \) can be expanded in a Taylor series:
\( e^{h\nu / kT} \approx 1 + \frac{h\nu}{kT} \)
Substituting this into Planck’s law:
\( B(\nu, T) \approx \frac{2h\nu^3}{c^2} \cdot \frac{1}{\frac{h\nu}{kT}} = \frac{2\nu^2 kT}{c^2} \)
This is the Rayleigh-Jeans law:
\( B(\nu, T) \approx \frac{2\nu^2 kT}{c^2} \)
We’ve derived Planck’s law of radiation and deduced both Wien’s distribution law and Rayleigh-Jeans law from it.
Entropy and the Entropy-Temperature Diagram
Definition of Entropy:
Entropy (S) is a thermodynamic property that measures the disorder, randomness, or uncertainty of a system. It can also be thought of as a measure of the amount of thermal energy unavailable to do work in a system. Entropy is typically denoted by the symbol “S” and is measured in units of joules per kelvin (J/K).
Entropy-Temperature Diagram:
An entropy-temperature diagram, also known as a T–S diagram, is a graphical representation of the relationship between entropy and temperature for a particular system. The diagram is a useful tool for visualizing and analyzing thermodynamic processes.
Here’s a detailed description of the entropy-temperature diagram:
Axes:
- The horizontal axis represents entropy (S) in units of J/K.
- The vertical axis represents temperature (T) in units of kelvin (K).
Thermodynamic Processes:
The entropy-temperature diagram can be used to illustrate various thermodynamic processes, such as:
- Isothermal Process: A process that occurs at constant temperature is represented by a horizontal line on the T–S diagram.
- Adiabatic Process: A process that occurs without heat transfer is represented by a vertical line on the T–S diagram.
- Isentropic Process: A process that occurs at constant entropy is represented by a vertical line on the T–S diagram.
- Reversible Process: A reversible process is represented by a smooth, continuous curve on the T–S diagram.
In conclusion, the entropy-temperature diagram is a powerful tool for visualizing and analyzing thermodynamic processes. It provides a clear understanding of the relationship between entropy and temperature, which is essential for understanding various thermodynamic concepts.