The correct relation between $\gamma=\frac{ c _{ p }}{ c _{ v }}$ and temperature $T$ is :

Question

In a BCC (Body-Centered Cubic) structure, the radius of the atoms is:

Answer 1

To determine the correct relationship between $\gamma = \frac{c_p}{c_v}$ and temperature $T$, we need to understand what $\gamma$ represents and how it is influenced by temperature.

$\gamma$ is known as the adiabatic index or heat capacity ratio, where $c_p$ is the specific heat at constant pressure, and $c_v$ is the specific heat at constant volume.

For an ideal gas, the relationship between $\gamma$ and temperature depends largely on the molecular structure of the gas:

For monoatomic gases, $\gamma$ is generally constant over a wide temperature range because both $c_p$ and $c_v$ are constant.
For diatomic and polyatomic gases, $\gamma$ can change with temperature due to additional degrees of freedom becoming accessible, such as rotational and vibrational modes.

As temperature increases, more rotational and vibrational modes are excited, which can affect $c_v$ more than $c_p$. However, $\gamma$ tends to decrease with increasing temperature for non-monoatomic gases because $c_v$ increases faster than $c_p$.

In the options provided:

 implies direct proportionality.
$\gamma \propto T^{\circ}$ suggests that $\gamma$ is independent of temperature, which suits monoatomic gases like noble gases where $\gamma$ is fairly constant regardless of temperature.
$\gamma \propto \frac{1}{T}$ and $\gamma \propto \frac{1}{\sqrt{T}}$ imply an inverse relationship, which isn't generally applicable to a wide range of gases.

Therefore, the correct relation is $\gamma \propto T^{\circ}$, indicating $\gamma$ is mostly independent of temperature for ideal monoatomic gases.

Hence, the answer is: $\gamma \propto T^{\circ}$.

The correct relation between $\gamma=\frac{ c _{ p }}{ c _{ v }}$ and temperature $T$ is :

Show Hint

The Correct Option is A

Solution and Explanation

Top Questions on Ideal gas equation

Questions Asked in JEE Main exam