Molecules of an ideal gas are known to have three translational degrees of freedom and two rotationa

Q: Why don't we count the third rotational degree of freedom for a diatomic gas?

A diatomic molecule is linear. The moment of inertia along the axis connecting the two atoms is extremely small because the mass is concentrated almost exactly on the axis. Therefore, the rotational kinetic energy $\frac{1}{2}I\omega^2$ along this axis is negligible at normal temperatures.

Q: What about vibrational degrees of freedom? Shouldn't $f$ be $7$?

Vibrational modes in diatomic molecules are usually 'frozen' out at room temperature due to quantum mechanical effects. They only become active at very high temperatures. Unless the problem specifically mentions high temperatures or vibrational modes, always assume $f=5$ for a diatomic gas.

Q: How is the formula $\gamma = 1 + 2/f$ derived?

From the equipartition theorem, the molar specific heat at constant volume is $C_V = \frac{f}{2}R$. Using Mayer's relation ($C_p - C_V = R$), we find $C_p = \frac{f}{2}R + R = \left(\frac{f+2}{2}\right)R$. Therefore, $\gamma = \frac{C_p}{C_V} = \frac{(f+2)/2}{f/2} = \frac{f+2}{f} = 1 + \frac{2}{f}$.

Q: Does this logic apply to all ideal gases?

Yes, the formulas $U = \frac{f}{2}\mu RT$ and $\gamma = 1 + \frac{2}{f}$ apply to all ideal gases. However, the value of $f$ changes based on atomicity: $f=3$ for monoatomic, $f=5$ for diatomic, and $f=6$ for non-linear polyatomic gases.

Q: What if the question asked for $2$ moles instead of $1$ mole?

The internal energy $U$ is an extensive property, so it would double: $U = \frac{5}{2}(2)RT = 5RT$. However, $\gamma$ is an intensive property (a ratio of specific heats), so it would remain exactly the same: $\frac{7}{5}$.

Question

Molecules of an ideal gas are known to have three translational degrees of freedom and two rotational degrees of freedom. The gas is maintained at a temperature of $T$. The total internal energy $U$ of a mole of this gas, and the value of $\gamma = \left( \frac{C_p}{C_V} \right)$ are given respectively, by

Options:

(A) $U = \frac{5}{2}RT$ and $\gamma = \frac{6}{5}$

(B) $U = 5RT$ and $\gamma = \frac{7}{5}$

(C) $U = \frac{5}{2}RT$ and $\gamma = \frac{7}{5}$

(D) $U = 5RT$ and $\gamma = \frac{6}{5}$

Sigma Percentile · Accepted Answer

### The Anatomy of a Diatomic Molecule

Imagine a diatomic gas molecule, like oxygen ($O_2$) or nitrogen ($N_2$). It looks like a tiny dumbbell flying through space. To understand its energy, we need to know all the independent ways it can move—these are called its **degrees of freedom** ($f$).

First, it can translate (move in a straight line) along the $x$, $y$, and $z$ axes. This gives us **$3$ translational degrees of freedom** ($f_{	ext{trans}} = 3$).

But it doesn't just fly straight; it tumbles! It can rotate end-over-end in two independent ways (imagine spinning a baton). We don't count rotation along its own internuclear axis because the atoms are essentially point masses, and the moment of inertia is negligible. So, that gives us **$2$ rotational degrees of freedom** ($f_{	ext{rot}} = 2$).

Adding them up, the total degrees of freedom for our diatomic molecule is:

$$f = f_{	ext{trans}} + f_{	ext{rot}} = 3 + 2 = 5$$

### The Law of Equipartition of Energy

Now, let's talk about how this molecule stores thermal energy. The **Law of Equipartition of Energy** is a beautiful principle that states that every active degree of freedom contributes exactly $\frac{1}{2}RT$ to the internal energy per mole of the gas.

Since our gas has $5$ degrees of freedom, the total internal energy $U$ for $\mu$ moles is given by:

$$U = \frac{f}{2} \mu RT$$

In our specific problem, we are dealing with exactly $1$ mole of gas ($\mu = 1$). Substituting our values, we get:

$$U = \frac{5}{2} (1) RT = \frac{5}{2} RT$$

Half the problem is solved! We now know the internal energy.

### The Magic Ratio: Gamma ($\gamma$)

Next, we need to find $\gamma$, which is the ratio of molar specific heat at constant pressure ($C_p$) to the molar specific heat at constant volume ($C_V$).

There is a direct, incredibly useful relationship between $\gamma$ and the degrees of freedom $f$:

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

Let's plug in our value of $f = 5$:

$$\gamma = 1 + \frac{2}{5}$$

Taking the lowest common multiple, we get:

$$\gamma = \frac{5 + 2}{5} = \frac{7}{5}$$

And there we have it! The internal energy is **$\frac{5}{2} RT$**, and the ratio of specific heats $\gamma$ is **$\frac{7}{5}$**. This perfectly matches option (c). Always remember that the atomicity of the gas (monoatomic, diatomic, polyatomic) dictates the degrees of freedom, which in turn dictates all its thermodynamic properties!

Select Answer:

Visualized Solution

$f_{trans} = 3$

$f_{rot} = 2$

$f = f_{trans} + f_{rot}$

$U = \frac{f}{2} μ R T$

$U = \frac{5}{2} R T$

$γ = 1 + \frac{2}{f}$

$γ = \frac{7}{5}$

$Result$

The Sigma Insight: Degree of Freedom and Law of Equipartition of Energy

The Anatomy of a Diatomic Molecule

The Law of Equipartition of Energy

The Magic Ratio

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Select Answer:

Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ftrans​=3

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } frot​=2

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f=ftrans​+frot​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } U=2f​μRT

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } U=25​RT

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } γ=1+f2​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } γ=57​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Result