Answer

Verified

53.4k+ views

**Hint:**The ratio of specific heat at constant pressure and the specific heat at constant volume is denoted by \[\gamma \]. Difference between specific heat at constant pressure and that at constant volume is the same as universal gas constant. Also, degrees of freedom of a molecule of the gas is proportional to its specific heat at constant volume.

**Formula Used:**

Definition of Heat capacity ratio:

\[\gamma = \dfrac{{{C_P}}}{{{C_V}}}\] (1)

Where,

\[\gamma \] is the heat capacity ratio,

\[{C_P}\] is the specific heat capacity at constant pressure,

\[{C_V}\] is the specific heat capacity at constant volume.

Relation between specific heat capacities and universal gas constant is given as:

\[{C_P} - {C_V} = R\] (2)

Where,

R is the universal gas constant.

The relationship between degrees of freedom and specific heat capacity at constant volume is known as:

\[{C_V} = \dfrac{{nR}}{2}\] (3)

Where,

n is the no. of degrees of freedom.

**Complete step by step answer:**

Step 1

First, rewrite the expression of eq.(2) to get an expression for \[{C_P}\]:

\[

{C_P} - {C_V} = R \\

\therefore {C_P} = {C_V} + R \\

\] (4)

Step 2

Now, use the eq.(3) in eq.(4) to get the form of \[{C_P}\] as:

\[{C_P} = \dfrac{{nR}}{2} + R = \dfrac{{nR + 2R}}{2} = \left( {\dfrac{{n + 2}}{2}} \right)R\] (5)

Step 3

Substitute the value of \[{C_P}\] from eq.(5) and value of \[{C_V}\]from eq.(3) in eq.(1) to get the value of n as:

\[

\gamma = \dfrac{{\left( {\tfrac{{n + 2}}{2}} \right)R}}{{\tfrac{{nR}}{2}}} \\

\Rightarrow \gamma = \dfrac{{n + 2}}{n} \\

\Rightarrow \gamma = 1 + \dfrac{2}{n} \\

\Rightarrow \dfrac{2}{n} = \gamma - 1 \\

\therefore n = \dfrac{2}{{\gamma - 1}} \\

\]

Hence, you will get the relationship between n and \[\gamma \].

Final answer:

The number of degrees of freedom of a molecule of the gas is (c) \[\dfrac{2}{{\gamma - 1}}\].

**Note:**This problem can be done in a tricky manner. If you just follow the values of \[\gamma \]and degrees of freedom then you will notice that as the number of atoms increases in a molecule degrees of freedom keeps increasing and \[\gamma \] keeps decreasing. So, clearly they are inversely related. Hence, in this question only possible inverse relation is given by option (c) which is the correct answer.

Recently Updated Pages

In a family each daughter has the same number of brothers class 10 maths JEE_Main

Which is not the correct advantage of parallel combination class 10 physics JEE_Main

If 81 is the discriminant of 2x2 + 5x k 0 then the class 10 maths JEE_Main

What is the value of cos 2Aleft 3 4cos 2A right2 + class 10 maths JEE_Main

If left dfracleft 2sinalpha rightleft 1 + cosalpha class 10 maths JEE_Main

The circumference of the base of a 24 m high conical class 10 maths JEE_Main

Other Pages

The resultant of vec A and vec B is perpendicular to class 11 physics JEE_Main

If a wire of resistance R is stretched to double of class 12 physics JEE_Main

The nitride ion in lithium nitride is composed of A class 11 chemistry JEE_Main

Electric field due to uniformly charged sphere class 12 physics JEE_Main

when an object Is placed at a distance of 60 cm from class 12 physics JEE_Main

According to classical free electron theory A There class 11 physics JEE_Main