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A diatomic molecule has how many degrees of freedom (For rigid rotator)
A diatomic molecule has 3 translational and 2 rotational degrees of freedom. Hence total degrees of freedom , f = 3 + 2 = 5
The degrees of freedom for triatomic gas is (At room temperature)
For a triatomic gas f = 6 ( 3 translation + 3 rotational )
If the degrees of freedom of a gas are f, then the ratio of two specific heats $ { C_p \over C_v}$ is given by
$ { C_p \over C_v } = \gamma = 1 + { 2 \over f } $
A diatomic gas molecule has translational, rotational and vibrational degrees of freedom. The $ { C_p \over C_v}$ is
Degrees of freedom = 3 ( translatory ) + 2 ( rotatary ) + 1 ( vibratory_ = 6 $ { C_p \over C_v} = \gamma = 1 + { 2 \over f } = 1 + { 2 \over 6 } = 1 + { 1 \over 3} = { 4 \over 3 } = 1.33 $
The value of $C_v$ for one mole of neon gas is
Neon gas is mono atomic and for mono atomic gases $ C_v = {3 \over 2 } R $
The relation between two specific heats of a gas is
When $ C_p $ and $ C _ V$ are given with calorie and R with Joule then $ C_p - C_v = R/J $
The molar specific heat at constant pressure for a monoatomic gas is
$ C_p -C_v = R \Rightarrow C_p =R + C_v \Rightarrow R + { f \over 2} R = R + {3 \over 2 } R = { 5 \over 2} R $ ( f = 3 )
For a gas $ { R \over C_v } = 0.67 $ .This gas is made up of molecules which are
$C_v = { R \over 0.67 } = 1.5 R = {3 \over 2 } R $ This is the value for mono atomic gases
The specific heat of an ideal gas is
According to the equilibrium theorem, the molar heat capacities should be independent of temperature How ever, variations in $C_v$ and $C_p$ are observed as the temperature changes. At very high temperatures, vibrations are also inportant and that affects the values of $C_v$ and $C_p$ for diatomic and poly atomic gases. Here in the question according to given information (D) may be correct answer.
The specific heats at constant pressure is greater than that of the same gas at constant volume because
At constant pressure, the gas does work in expanding against the external pressure. This work requires additional energy, which is why the specific heat at constant pressure (C_p) is greater than the specific heat at constant volume (C_v).
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