An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas is . The internal energy of one mole of the gas is and the speed of sound in the gas is . At a fixed temperature and pressure, which of the following

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 $v_{n}$ . At a fixed temperature and pressure, which of the following is the correct option? [2023]

Smart Achievers · Accepted Answer

(3)

$The internal energy of one mole of the gas,$

$U_{n} = \frac{1 \times n \times R T}{2} = \frac{n R T}{2}$

$∴$ $U_{7} > U_{5} > U_{3} and U_{7} > U_{6} > U_{3}$

$Speed of the sound in the gas (of number of degree of freedom of a molecule = n)$

$v_{n} = \sqrt{\frac{γ R T}{M}} = \sqrt{(\frac{2}{n} + 1) \frac{R T}{M}}$

$Clearly, more n, less v$

$∴$ $v_{5} > v_{7}$

Solution

1 $v_{3} < v_{6}$ and $U_{3} > U_{6}$

2 $v_{5} > v_{3}$ and $U_{3} > U_{5}$

3 $v_{5} > v_{7}$ and $U_{5} < U_{7}$

4 $v_{6} < v_{7}$ and $U_{6} < U_{7}$

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Solution

Q. 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 Un and the speed of sound in the gas is vn. At a fixed temperature and pressure, which of the following is the correct option [2023]

1 v3<v6 and U3>U6

2 v5>v3 and U3>U5

3 v5>v7 and U5<U7

4 v6<v7 and U6<U7

Ans. (3) The internal energy of one mole of the gas, Un=1×n×RT2=nRT2 ∴ U7>U5>U3 and U7>U6>U3 Speed of the sound in the gas (of number of degree of freedom of a molecule =n) vn=γRTM=(2n+1)RTM Clearly, more n, less v ∴ v5>v7

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1 $v_{3} < v_{6}$ and $U_{3} > U_{6}$

2 $v_{5} > v_{3}$ and $U_{3} > U_{5}$

3 $v_{5} > v_{7}$ and $U_{5} < U_{7}$

4 $v_{6} < v_{7}$ and $U_{6} < U_{7}$