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Chapter 8: Problem 1107

A gas mixture consists of 2 mole of oxygen and 4 mole of argon at temperature \(\mathrm{T}\). Neglecting all vibrational modes, the total internal energy of the system is (A) \(11 \mathrm{RT}\) (B) \(9 \mathrm{RT}\) (C) \(15 \mathrm{RT}\) (D) \(4 \mathrm{RT}\)

Short Answer

Expert verified
The degree of freedom for Oxygen (neglecting vibrational modes) is 5, and for Argon, it is 3. Using the formula \(U = \frac{n f}{2} RT\), we find the internal energy for Oxygen to be \(5 RT\) and for Argon to be \(6 RT\). Adding them together, we get the total internal energy of the system as \(11 RT\). Therefore, the correct answer is (A) \(11 RT\).

Step by step solution

01

Determine the Degree of Freedom for Each Gas

Degree of freedom refers to the number of independent ways a particle in the system can move. For monotonic gases like Argon that consist of only one atom, the degree of freedom is 3 (1 translational motion in each of 3 dimensions x, y, z). Oxygen, being a diatomic gas, has 5 degrees of freedom (3 translational motions in x, y, z, and two rotational motions) as we are neglecting vibrational motions.
02

Compute the Total Internal Energy

The formula for the internal energy of a gas is given by \(U = \frac{n f}{2} RT\), where \(n\) is the number of moles of the gas, \(f\) is the degree of freedom, \(R\) is the gas constant and \(T\) is the temperature. For Oxygen (O2): \(U_{\text{Oxygen}} = n_{\text{Oxygen}} \times \frac{f_{\text{Oxygen}}}{2} \times RT = 2 \times \frac{5}{2} \times RT = 5 RT\) For Argon (Ar): \(U_{\text{Argon}} = n_{\text{Argon}} \times \frac{f_{\text{Argon}}}{2} \times RT = 4 \times \frac{3}{2} \times RT = 6 RT\)
03

Calculating the Total Internal Energy of System

The total internal energy of the system is the sum of the internal energies of Oxygen and Argon. Therefore, the total energy would be \( U_{\text{total}} = U_{\text{Oxygen}} + U_{\text{Argon}} = 5 RT + 6 RT = 11 RT\) So, the correct choice is (A) \(11 RT\).

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