Calculate the ratio of average K.E. of one molecule of hydrogen `(H_2)` gas and oxygen `(O_2)` gas at `0^@ C`.

Answer

To calculate the ratio of the average kinetic energy of one molecule of hydrogen gas (H₂) and oxygen gas (O₂) at 0°C, we can follow these steps:

### Step 1: Understand the formula for average kinetic energy
The average kinetic energy (KE) of a single molecule of an ideal gas is given by the formula:
\[ E = \frac{3}{2} k T \]
where:
- \(E\) is the average kinetic energy,
- \(k\) is Boltzmann's constant,
- \(T\) is the absolute temperature in Kelvin.

### Step 2: Convert the temperature to Kelvin
The given temperature is 0°C. To convert this to Kelvin:
\[ T = 0 + 273 = 273 \, K \]

### Step 3: Apply the formula for both gases
For both hydrogen (H₂) and oxygen (O₂), the average kinetic energy can be calculated using the same formula since they are both at the same temperature:
- For hydrogen:
\[ E_{H_2} = \frac{3}{2} k (273) \]
- For oxygen:
\[ E_{O_2} = \frac{3}{2} k (273) \]

### Step 4: Calculate the ratio of average kinetic energies
Now we can find the ratio of the average kinetic energy of hydrogen to that of oxygen:
\[ \text{Ratio} = \frac{E_{H_2}}{E_{O_2}} = \frac{\frac{3}{2} k (273)}{\frac{3}{2} k (273)} \]
Since both \(E_{H_2}\) and \(E_{O_2}\) are equal, the ratio simplifies to:
\[ \text{Ratio} = 1 \]

### Final Answer
The ratio of the average kinetic energy of one molecule of hydrogen gas to that of oxygen gas at 0°C is:
\[ \text{Ratio} = 1 : 1 \]

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