What is the ratio of the total energy of all the molecules of one mole of `O_(2)` to the total energy of all the molecules of two moles of helium at the same temperature?

Options

A. `1:2`
B. `2:1`
C. `2:3`
D. `3:2`

Explanation

To find the ratio of the total energy of all the molecules of one mole of \( O_2 \) to the total energy of all the molecules of two moles of helium at the same temperature, we can follow these steps:

### Step 1: Understand the formula for total energy
The total energy \( E \) of a gas can be expressed using the formula:
\[ E = N \cdot \frac{3}{2} k_B T \]
where:
- \( N \) is the number of moles of the gas,
- \( k_B \) is the Boltzmann constant,
- \( T \) is the temperature in Kelvin.

### Step 2: Calculate the total energy for one mole of \( O_2 \)
For one mole of \( O_2 \):
\[ E_{O_2} = 1 \cdot \frac{3}{2} k_B T = \frac{3}{2} k_B T \]

### Step 3: Calculate the total energy for two moles of helium
For two moles of helium:
\[ E_{He} = 2 \cdot \frac{3}{2} k_B T = 3 k_B T \]

### Step 4: Find the ratio of the total energies
Now, we can find the ratio of the total energy of \( O_2 \) to the total energy of helium:
\[ \text{Ratio} = \frac{E_{O_2}}{E_{He}} = \frac{\frac{3}{2} k_B T}{3 k_B T} \]

### Step 5: Simplify the ratio
The \( k_B T \) terms cancel out:
\[ \text{Ratio} = \frac{\frac{3}{2}}{3} = \frac{3}{2} \cdot \frac{1}{3} = \frac{1}{2} \]

### Final Answer
Thus, the ratio of the total energy of all the molecules of one mole of \( O_2 \) to the total energy of all the molecules of two moles of helium at the same temperature is:
\[ \text{Ratio} = 1 : 2 \]

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