At what temperature the most probable velocity of `O_(2)` gas is equal to the RMS velocity of `O_(3)` at 't'°C?

Answer

To solve the problem of finding the temperature at which the most probable velocity of \( O_2 \) gas equals the RMS velocity of \( O_3 \) at \( t \)°C, we can follow these steps:

### Step 1: Understand the formulas for velocities
The most probable velocity (\( v_{mp} \)) and the root mean square velocity (\( v_{rms} \)) for gases can be expressed as:
- Most Probable Velocity (\( v_{mp} \)) for \( O_2 \):
\[ v_{mp} = \sqrt{\frac{2RT}{M}} \]
- RMS Velocity (\( v_{rms} \)) for \( O_3 \):
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]

Where:
- \( R \) = universal gas constant
- \( T \) = absolute temperature in Kelvin
- \( M \) = molar mass of the gas in kg/mol

### Step 2: Determine molar masses
- Molar mass of \( O_2 \) = \( 2 \times 16 = 32 \, \text{g/mol} = 0.032 \, \text{kg/mol} \)
- Molar mass of \( O_3 \) = \( 3 \times 16 = 48 \, \text{g/mol} = 0.048 \, \text{kg/mol} \)

### Step 3: Express the velocities in terms of temperature
For \( O_2 \) at temperature \( T \) (in Kelvin):
\[ v_{mp} = \sqrt{\frac{2RT}{0.032}} \]

For \( O_3 \) at temperature \( t + 273 \) (in Kelvin):
\[ v_{rms} = \sqrt{\frac{3R(t + 273)}{0.048}} \]

### Step 4: Set the two velocities equal
According to the problem, we set the most probable velocity of \( O_2 \) equal to the RMS velocity of \( O_3 \):
\[ \sqrt{\frac{2RT}{0.032}} = \sqrt{\frac{3R(t + 273)}{0.048}} \]

### Step 5: Square both sides to eliminate the square roots
Squaring both sides gives:
\[ \frac{2RT}{0.032} = \frac{3R(t + 273)}{0.048} \]

### Step 6: Cancel out \( R \) and rearrange
Cancelling \( R \) from both sides:
\[ \frac{2T}{0.032} = \frac{3(t + 273)}{0.048} \]

### Step 7: Cross-multiply to solve for \( T \)
Cross-multiplying gives:
\[ 2T \cdot 0.048 = 3(t + 273) \cdot 0.032 \]
\[ 0.096T = 0.096(t + 273) \]

### Step 8: Simplify and solve for \( T \)
Dividing both sides by \( 0.096 \):
\[ T = t + 273 \]

### Step 9: Convert back to Celsius
Since \( T \) is in Kelvin, to find the temperature in Celsius:
\[ T - 273 = t \]

Thus, the temperature \( T \) in Celsius is:
\[ T = t \]

### Final Answer
The temperature at which the most probable velocity of \( O_2 \) gas is equal to the RMS velocity of \( O_3 \) at \( t \)°C is \( t \)°C.

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