1. What Is Gamma (γ)?
The ratio of specific heats, written as γ (gamma), is defined as:
\( \gamma = \dfrac{C_P}{C_V} \)
It tells how the specific heats at constant pressure and constant volume compare. Since \( C_P > C_V \), gamma is always greater than 1.
2. Why Cp Is Greater Than Cv
At constant volume, all the heat goes into increasing internal energy. At constant pressure, heat must also do expansion work. So Cp has to be larger.
This difference makes the ratio γ greater than 1.
2.1. Relation
C_P = C_V + R
3. Gamma in Terms of Degrees of Freedom
Using the equipartition results:
C_V = \dfrac{f}{2} R \quad\text{and}\quad C_P = C_V + R = \left( \dfrac{f}{2} + 1 \right)R
So:
\( \gamma = \dfrac{C_P}{C_V} = \dfrac{\dfrac{f}{2} + 1}{\dfrac{f}{2}} = 1 + \dfrac{2}{f} \)
3.1. Meaning
Gases with more degrees of freedom (larger f) have a smaller γ value.
4. Gamma Values for Different Types of Gases
- Monatomic gases: f = 3 →
\( \gamma = \dfrac{5}{3} \approx 1.67 \)
- Diatomic gases: f = 5 →
\( \gamma = \dfrac{7}{5} \approx 1.4 \)
- Polyatomic gases: f = 6 →
\( \gamma = \dfrac{4}{3} \approx 1.33 \)
4.1. Why These Differences Matter
Gases with more degrees of freedom have more ways to store energy, so their Cp and Cv are closer, giving a lower gamma.
5. Importance of Gamma in Gas Behaviour
The ratio γ affects how gases respond in many physical situations, including sound waves, engines, and adiabatic processes.
5.1. 1. Speed of Sound
The speed of sound in a gas depends on gamma:
v = \sqrt{ \gamma \dfrac{P}{\rho} }
Diatomic gases (γ ≈ 1.4) carry sound faster than polyatomic gases (γ ≈ 1.33).
5.2. 2. Adiabatic Expansion and Compression
Adiabatic processes use gamma in their equations:
P V^{\gamma} = \text{constant}
A higher gamma makes the gas heat up more when compressed.
5.3. 3. Engine Efficiency
In internal combustion engines, efficiency increases when gamma is higher. This is one reason lighter gases respond more strongly to pressure changes.
6. Everyday Examples Involving Gamma
- Helium balloons rise and cool off quickly because monatomic helium has a high γ.
- Air (mostly diatomic) responds moderately to compression, such as in bicycle pumps.
- Steam (polyatomic) heats and cools more slowly because of its lower γ.