# Why gamma is not equal to 2?

In my text book, I came across a ratio called heat capacity ratio or gamma which is equal to

$$\gamma = 1+\frac{2}{f}$$

where $$f$$ is degree of freedom. Therefore

$$C_p = \gamma C_v$$

where $$C_p$$ and $$C_v$$ are molar heat capacity at constant pressure and constant volume.

In an online lecture I came to know that energy given to a piston gas system is equally shared by piston and gas (when piston is movable).Amount of energy gained by piston is work done and by gas is change in internal energy.let us assume that change in internal energy dU cause change in temperature by 1 kelvin. let us say that we have 1 mole of gas in a cylinder piston system So Cv is amount of energy needed to change temperature of our system by 1 kelvin when piston is immovabe ( constant volume) therefore Cv=dU and Cp be energy needed to do the same temperature change when piston is movable (constant pressure)so Cp=2dU in later scenario Cp is equally shared by gas and piston therefore Cp must equal to 2Cv since Cv =dU .By definition I came to know that R is amount of energy needed to rise temperature of one mole of substance by one kelvin so Cv must equal to R(gas constant) since Cv also defined in the same way .But in my book It is said that gamma is greater than one and less than two (usually).why degree of freedom affect Cp Cv relation and Cv R relation? why Cp is not equal to 2 Cv ? why Cv is not equal to R ?why with increase in temperature gamma value decrease?

• I recommend that you edit your question for readability. I've made a partial edit to get you started. Sep 3, 2019 at 11:22
• Thanks for the edit I don't have computer to edit like this .I am asking this questions in an old smart phone. Sep 3, 2019 at 11:46
• Can you provide a source for that claim? An idealized piston should not absorb any energy from the system during a quasistatic process like this; the additional energy is transferred to the environment as work, and the amount of work done is generically not simply half of the heat added to the system. Sep 3, 2019 at 11:49