Gas Mixture:
The gas mixture consists of 2 moles of oxygen (O2) and 4 moles of argon (Ar).

Total Internal Energy:
The total internal energy of a gas system is the sum of the kinetic and potential energies of its individual gas molecules.

Internal Energy for a Monoatomic Gas:
For a monoatomic gas, such as argon, the internal energy is solely due to the kinetic energy of the gas molecules. The kinetic energy of a gas molecule is given by the equation:

KE = (3/2) * k * T,

where KE is the kinetic energy, k is the Boltzmann constant, and T is the temperature.

Internal Energy for a Diatomic Gas:
For a diatomic gas, such as oxygen, the internal energy consists of both kinetic and potential energies. In addition to the translational kinetic energy, diatomic molecules also have rotational kinetic energy. The total internal energy of a diatomic gas molecule can be expressed as:

U = (5/2) * k * T,

where U is the internal energy, k is the Boltzmann constant, and T is the temperature.

Calculating the Total Internal Energy:
To calculate the total internal energy of the gas mixture, we need to consider the contributions from both oxygen and argon.

Internal Energy of Oxygen:
For 2 moles of oxygen, the internal energy can be calculated as:

U_O2 = (5/2) * k * T * 2,

where U_O2 is the internal energy of oxygen.

Internal Energy of Argon:
For 4 moles of argon, the internal energy can be calculated as:

U_Ar = (3/2) * k * T * 4,

where U_Ar is the internal energy of argon.

Total Internal Energy of the Gas Mixture:
The total internal energy of the gas mixture is the sum of the internal energies of oxygen and argon:

U_total = U_O2 + U_Ar.

Substituting the respective equations, we have:

U_total = (5/2) * k * T * 2 + (3/2) * k * T * 4.

Simplifying the equation, we get:

U_total = 5 * k * T + 6 * k * T.

Combining like terms, we have:

U_total = 11 * k * T.

Since we are given that the internal energy is to be expressed in units of RT, we can divide the equation by RT:

U_total/RT = 11.

Therefore, the total internal energy of the gas mixture in units of RT is 11.