The temperature at which the root mean squres speed of a gas will be half its value at `0^(@)C` is (assume the pressure remains constant)

Options

A. `-86.4^(@)C`
B. `-204.75^(@)C`
C. `-104.75^(@)C`
D. `-6825^(@)C`

Explanation

To find the temperature at which the root mean square speed (VRMS) of a gas is half its value at 0°C, we can follow these steps:

### Step 1: Understand the relationship between VRMS and temperature
The root mean square speed (VRMS) of a gas is directly proportional to the square root of the absolute temperature (T) in Kelvin. This relationship can be expressed as:
\[ V_{RMS} \propto \sqrt{T} \]

### Step 2: Establish the initial conditions
At 0°C, the absolute temperature is:
\[ T = 0°C + 273.15 = 273.15 \, K \]
Let the corresponding root mean square speed at this temperature be \( V \).

### Step 3: Set up the equation for the new speed
We are given that the new root mean square speed \( V' \) is half of the initial speed \( V \):
\[ V' = \frac{1}{2} V \]

### Step 4: Write the proportionality in terms of ratios
Using the relationship established in Step 1, we can write:
\[ \frac{V'}{V} = \sqrt{\frac{T'}{T}} \]
Substituting \( V' = \frac{1}{2} V \):
\[ \frac{1/2 V}{V} = \sqrt{\frac{T'}{T}} \]
This simplifies to:
\[ \frac{1}{2} = \sqrt{\frac{T'}{T}} \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides gives:
\[ \left(\frac{1}{2}\right)^2 = \frac{T'}{T} \]
\[ \frac{1}{4} = \frac{T'}{T} \]

### Step 6: Solve for the new temperature \( T' \)
Rearranging the equation gives:
\[ T' = \frac{T}{4} \]
Substituting \( T = 273.15 \, K \):
\[ T' = \frac{273.15}{4} = 68.2875 \, K \]

### Step 7: Convert the temperature back to Celsius
To convert the temperature from Kelvin to Celsius:
\[ T'_{Celsius} = T' - 273.15 \]
Substituting the value of \( T' \):
\[ T'_{Celsius} = 68.2875 - 273.15 \]
\[ T'_{Celsius} = -204.8625 \, °C \]

### Step 8: Round to the appropriate number of significant figures
Rounding gives:
\[ T'_{Celsius} \approx -204.75 \, °C \]

### Final Answer
The temperature at which the root mean square speed of the gas will be half its value at 0°C is approximately:
**-204.75°C**

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