Question 127: Thermodynamics & KTG

Question

An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas is $n$. The internal energy of one mole of the gas is $U_n$ and the speed of sound in the gas is $\mathrm{v}_n$. At a fixed temperature and pressure, which of the following is the correct option?

Accepted Answer

1. Formula for Sound Velocity:
- The velocity of sound, denoted as $$ v $$, is determined by the equation $$ v = \sqrt{\frac{\gamma R T}{M}} $$, where:
- $$ \gamma $$ is a factor given by $$ \gamma = 1 + \frac{2}{n} $$.
- $$ R $$ is the gas constant.
- $$ T $$ is the temperature.
- $$ M $$ is the molar mass of the gas.
2. Relation of $$ \gamma $$ with $$ n $$:
- The value of $$ \gamma $$ is inversely related to $$ n $$. This means as $$ n $$ increases, $$ \gamma $$ decreases.
3. Internal Energy Relation:
- The internal energy, denoted as $$ U $$, for a mole of gas is given by $$ U = \frac{n R T}{2} $$.
4. Effect of Increasing $$ n $$ on Sound Velocity and Internal Energy:
When the value of $$ n $$ increases:
- The factor $$ \gamma $$ decreases, according to its formula.
- As a result, the velocity of sound $$ v $$ also decreases, as it is dependent on $$ \sqrt{\gamma} $$.
- Conversely, the internal energy $$ U_n $$ increases with the increase in $$ n $$, as evident from the internal energy formula.
Therefore, in summary, a higher value of $$ n $$ leads to a lower velocity of sound and higher internal energy in the context of these formulas.

Answer

$v_3 < v_6$ and $U_3 > U_6$

Answer

$v_5 > v_3$ and $U_3 > U_5$

Answer

$v_5 > v_7$ and $U_5 < U_7$

Answer

$v_6 < v_7$ and $U_6 < U_7$

Thermodynamics & KTG

An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas is n. The internal energy of one mole of the gas is U_n and the speed of sound in the gas is \mathrm{v}_n. At a fixed temperature and pressure, which of the following is the correct option?

Choose the correct answer: