Kerala KEAM 5 June 2024 Solved Paper

The total energy for an ideal gas is given by:$$E = \frac{f}{2} n R T$$where:- $f$ is the number of degrees of freedom of the gas molecules,- $n$ is the number of moles of the gas,- $R$ is the universal gas constant,- $T$ is the temperature.For a diatomic molecule like oxygen (O$_2$):- The number of degrees of freedom $f = 5$ (3 translational and 2 rotational, neglecting vibrational modes).For a monoatomic molecule like argon (Ar):- The number of degrees of freedom $f = 3$ (3 translational, no rotation in ideal cases for argon).Now, calculate the total energy for the mixture:- The energy for 1 mole of oxygen:$$E_{O_2} = \frac{5}{2} \cdot 1 \cdot R T = \frac{5}{2} R T$$- The energy for 3 moles of argon:$$E_{Ar} = \frac{3}{2} \cdot 3 \cdot R T = \frac{9}{2} R T$$The total energy for the mixture is the sum of the energies of oxygen and argon:$$E_{total} = E_{O_2} + E_{Ar} = \frac{5}{2} R T + \frac{9}{2} R T = \frac{14}{2} R T = 7 R T$$Thus, the total energy of the gas mixture is:$$\boxed{7 \, RT}$$ Quick Tip: For ideal gases, the total energy is calculated by the formula $E = \frac{f}{2} n R T$, where $f$ depends on the type of gas (monoatomic or diatomic). The total energy for a mixture is simply the sum of the individual energies for each component.