An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the

Q: Why did we take the number of moles as 1?

The question explicitly states 'The internal energy of one mole of the gas is $U_n$'. Therefore, we substitute $n_{\text{moles}} = 1$ into the internal energy formula.

Q: Why is the temperature $T$ considered constant in our proportionalities?

The problem specifies 'At a fixed temperature and pressure', which means $T$ and $P$ do not change when we compare the functional dependence of these properties on the degrees of freedom $n$.

Q: Does the molar mass $M$ change for different gases?

While different real gases have different molar masses, the problem asks us to evaluate the mathematical dependence of $v_n$ purely as a function of $n$. We treat $M$ as a constant parameter to isolate the effect of changing the degrees of freedom.

Q: Why does the speed of sound decrease when degrees of freedom increase?

The speed of sound depends on the adiabatic index $\gamma$. Gases with more degrees of freedom have more ways to store internal energy (like rotational or vibrational modes) rather than just translational kinetic energy. This makes them less 'stiff' adiabatically, lowering $\gamma$ and thus reducing the speed of sound.

Q: Can the degrees of freedom $n$ actually be 7?

Yes! While monoatomic gases have 3 and rigid diatomic gases have 5, non-linear polyatomic gases at high temperatures can have active vibrational modes, leading to higher degrees of freedom like 6 or 7.

Question

An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas in n. The internal energy of one mole of the gas is $U_n$ and the speed of sound in the gas is $v_n$. At a fixed temperature and pressure, which of the following is the correct option ? Options: (A) $v_3 < v_6$ and $U_3 > U_6$ (B) $v_5 > v_3$ and $U_3 > U_5$ (C) $v_5 > v_7$ and $U_5 < U_7$ (D) $v_6 < v_7$ and $U_6 < U_7$

Sigma Percentile · Accepted Answer

### Analyzing the Setup Imagine a container filled with exactly one mole of an ideal gas, resting in perfect thermodynamic equilibrium. The problem introduces us to a fascinating scenario where we need to analyze how the macroscopic properties of this gas change purely based on its microscopic degrees of freedom, denoted by $n$. We are given two primary observables to track: the internal energy of the gas, $U_n$, and the speed of sound propagating through it, $v_n$. The temperature $T$ and pressure $P$ are held strictly constant. Our mission is to determine the correct inequalities comparing these properties for different values of $n$. ### The Master Equation for Internal Energy Let's start by unlocking the internal energy. From the kinetic theory of gases, we know that the internal energy of an ideal gas is directly tied to how many ways its molecules can move, rotate, or vibrate—its degrees of freedom. The formula for the internal energy of $n_{ ext{moles}}$ of an ideal gas is given by: $$U = \frac{f}{2} n_{ ext{moles}} R T$$ In our specific case, the problem states we have exactly $1$ mole of gas, and it uses $n$ to represent the degrees of freedom instead of the traditional $f$. Substituting these into our equation, we get: $$U_n = \frac{n}{2} R T$$ Since the universal gas constant $R$ and the temperature $T$ are fixed, we can clearly see a direct proportionality: $$U_n \propto n$$ This is a beautifully simple relationship. It tells us that as the degrees of freedom increase, the internal energy increases linearly. Therefore, we can confidently write the inequality: $$U_3 < U_5 < U_6 < U_7$$ ### The Speed of Sound Dynamics Now, let's shift our focus to the speed of sound, $v_n$. The speed at which a mechanical wave travels through a gas depends on its adiabatic elasticity and its inertia. The standard formula is: $$v_n = \sqrt{\frac{\gamma R T}{M}}$$ Here, $\gamma$ is the adiabatic index (the ratio of specific heats, $C_p/C_v$), and $M$ is the molar mass. But how does $\gamma$ connect to our degrees of freedom, $n$? The kinetic theory provides the bridge: $$\gamma = 1 + \frac{2}{n}$$ Substituting this back into our speed of sound equation, we reveal the hidden dependence on $n$: $$v_n = \sqrt{\left(1 + \frac{2}{n} ight) \frac{R T}{M}}$$ Assuming the molar mass $M$ is constant for the sake of functional comparison, we observe the proportionality: $$v_n \propto \sqrt{1 + \frac{2}{n}}$$ **Here is where we must be careful.** Look at the term $\frac{2}{n}$. As the degrees of freedom $n$ *increases*, the fraction $\frac{2}{n}$ *decreases*. Consequently, the entire term inside the square root becomes smaller. This means that the speed of sound actually **decreases** as the degrees of freedom increase! This leads us to our second master inequality: $$v_3 > v_5 > v_6 > v_7$$ ### Final Calculation and Conclusion Armed with our two derived relationships, we can now evaluate the given options like a detective. - **Option (A):** $v_3 < v_6$ (False, we found $v_3 > v_6$) - **Option (B):** $v_5 > v_3$ (False, we found $v_5 < v_3$) - **Option (C):** $v_5 > v_7$ and $U_5 < U_7$ (True! Both conditions perfectly match our inequalities) - **Option (D):** $v_6 < v_7$ (False, we found $v_6 > v_7$) The elegance of thermodynamics shines through here. By simply understanding how microscopic degrees of freedom influence macroscopic formulas, we effortlessly arrive at **Option (C)** as the correct answer.

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Visualized Solution

\text{Analyzing the Given Parameters}

\text{Internal Energy Formula}

\text{Dependence of } U_n \text{ on } n

\text{Speed of Sound Formula}

\text{Dependence of } v_n \text{ on } n

\text{Evaluating the Options}

The Sigma Insight: Kinetic Theory of Gases

Analyzing the Setup

The Master Equation for Internal Energy

The Speed of Sound Dynamics

Final Calculation and Conclusion

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