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 solve the problem of finding 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 its absolute temperature (in Kelvin). The formula can be expressed as:
\[ V_{RMS} \propto \sqrt{T} \]
This means:
\[ V_{RMS} = k \sqrt{T} \]
where \( k \) is a constant.

### Step 2: Set up the initial condition
At 0°C, which is equivalent to 273 K, we denote the initial temperature as:
\[ T_1 = 273 \, \text{K} \]
At this temperature, let the VRMS be:
\[ V_{RMS1} = k \sqrt{T_1} = k \sqrt{273} \]

### Step 3: Define the condition for the new temperature
We need to find the temperature \( T_2 \) at which the VRMS is half of its value at 0°C:
\[ V_{RMS2} = \frac{1}{2} V_{RMS1} \]
Substituting the expression for VRMS gives:
\[ V_{RMS2} = \frac{1}{2} k \sqrt{273} \]

### Step 4: Relate the new temperature to the new VRMS
Using the relationship between VRMS and temperature, we can express \( V_{RMS2} \) in terms of \( T_2 \):
\[ V_{RMS2} = k \sqrt{T_2} \]
Setting the two expressions for \( V_{RMS2} \) equal gives:
\[ k \sqrt{T_2} = \frac{1}{2} k \sqrt{273} \]

### Step 5: Simplify the equation
We can cancel \( k \) from both sides (assuming \( k \neq 0 \)):
\[ \sqrt{T_2} = \frac{1}{2} \sqrt{273} \]

### Step 6: Square both sides to solve for \( T_2 \)
Squaring both sides results in:
\[ T_2 = \left(\frac{1}{2} \sqrt{273}\right)^2 \]
\[ T_2 = \frac{1}{4} \cdot 273 \]
\[ T_2 = 68.25 \, \text{K} \]

### Step 7: Convert the temperature back to Celsius
To convert from Kelvin to Celsius, we subtract 273:
\[ T_2 = 68.25 \, \text{K} - 273 = -204.75 \, \text{°C} \]

### Final Answer
The temperature at which the root mean square speed of a gas will be half its value at 0°C is approximately:
\[ -204.75 \, \text{°C} \]