Gas molecules each of mass `10^(-26)` kg are taken in a container of volume `1 dm^(3)` . The root mean square speed of gas molecules is 1 km `sec^(-1)` .What is the temperature fo gas molecules.
(Given : `N_(A) =6xx10^(23),R=8J//mol.K`)

Options

A. 298 K
B. 25 K
C. 250 K
D. 2500 K

Explanation

To find the temperature of gas molecules given their mass, root mean square speed, and other constants, we can follow these steps:

### Step 1: Understand the given information
- Mass of each gas molecule, \( m = 10^{-26} \) kg
- Volume of the container, \( V = 1 \, \text{dm}^3 = 1 \, \text{L} = 10^{-3} \, \text{m}^3 \)
- Root mean square speed, \( V_{\text{rms}} = 1 \, \text{km/s} = 1000 \, \text{m/s} \)
- Avogadro's number, \( N_A = 6.022 \times 10^{23} \, \text{mol}^{-1} \)
- Gas constant, \( R = 8 \, \text{J/(mol K)} \)

### Step 2: Use the formula for root mean square speed
The root mean square speed of gas molecules is given by the formula:
\[ V_{\text{rms}} = \sqrt{\frac{3RT}{m}} \]
Where:
- \( R \) is the gas constant,
- \( T \) is the temperature in Kelvin,
- \( m \) is the mass of one molecule.

### Step 3: Rearrange the formula to solve for temperature \( T \)
Squaring both sides gives:
\[ V_{\text{rms}}^2 = \frac{3RT}{m} \]
Rearranging for \( T \):
\[ T = \frac{m V_{\text{rms}}^2}{3R} \]

### Step 4: Substitute the values into the equation
1. Convert the mass of one molecule to the appropriate form:
\[ m = 10^{-26} \, \text{kg} \]
2. Substitute \( V_{\text{rms}} = 1000 \, \text{m/s} \):
\[ V_{\text{rms}}^2 = (1000)^2 = 10^6 \, \text{m}^2/\text{s}^2 \]
3. Substitute \( R = 8 \, \text{J/(mol K)} \):
\[ T = \frac{(10^{-26}) (10^6)}{3 \times 8} \]

### Step 5: Calculate the temperature
\[ T = \frac{10^{-20}}{24} \]
\[ T = 4.1667 \times 10^{-22} \, \text{K} \]
Now, we need to multiply by Avogadro's number to convert it to per mole:
\[ T = \frac{(10^{-26} \times 6.022 \times 10^{23}) \times 10^6}{3 \times 8} \]
\[ T = \frac{6.022 \times 10^{-3}}{24} \]
\[ T \approx 250.95 \, \text{K} \]

### Step 6: Final answer
Thus, the temperature of the gas molecules is approximately:
\[ \boxed{250 \, \text{K}} \]