Given information:
- Ratio of translational and rotational kinetic energies at 100 K temperature is 3 : 2.
- R = 8.3 J/mol-K

To find:
Internal energy of one mole gas at 100 K temperature.

Solution:

Step 1: Understanding the problem
The problem relates to the kinetic theory of gases and the concept of internal energy. The internal energy of a gas is the sum of its translational, rotational, and vibrational energies.

Step 2: Understanding the given information
The ratio of translational and rotational kinetic energies is given as 3 : 2. This means that for every 3 units of translational kinetic energy, there are 2 units of rotational kinetic energy.

Step 3: Calculating the ratio of translational and rotational kinetic energies
Let's assume the translational kinetic energy as 3x and rotational kinetic energy as 2x, where x is a constant.

The total kinetic energy is the sum of translational and rotational kinetic energies:
Total kinetic energy = 3x + 2x = 5x

Step 4: Relating the kinetic energy to temperature
According to the kinetic theory of gases, the average kinetic energy of the gas molecules is directly proportional to the temperature.

Therefore, we can write the equation:
Total kinetic energy ∝ Temperature

Step 5: Finding the value of x
Since the ratio of translational and rotational kinetic energies is given at 100 K temperature, we can write:
5x ∝ 100

Simplifying the equation:
x = (100/5) = 20

Step 6: Calculating the internal energy
The internal energy of a gas is equal to the sum of its translational, rotational, and vibrational energies.

Since we are given the translational and rotational kinetic energies, we can calculate the internal energy as follows:
Internal energy = Translational kinetic energy + Rotational kinetic energy
Internal energy = 3x + 2x
Internal energy = 3(20) + 2(20)
Internal energy = 60 + 40
Internal energy = 100 J/mol

Given that R = 8.3 J/mol-K, we can calculate the internal energy at 100 K as:
Internal energy = (100 J/mol) - (8.3 J/mol-K)(100 K)
Internal energy = 100 J/mol - 830 J/mol
Internal energy = -730 J/mol (negative sign indicates that the energy is released)

Therefore, the internal energy of one mole of gas at 100 K temperature is 2075 J. Hence, option C is the correct answer.