For one mole of gas the average kinetic energy is given as E.The `U_("rms")`of gas is `:`

Options

A. `[(2E)/(M)]^(1//2)`
B. `[(3E)/(M)]^(-1//2)`
C. `[(2E)/(2M)]^(1//2)`
D. `[(3E)/(2M)]^(1//2)`

Answer

To find the root mean square (RMS) velocity \( U_{\text{rms}} \) of a gas in terms of its average kinetic energy \( E \), we can follow these steps:

### Step 1: Understand the relationship between average kinetic energy and temperature
The average kinetic energy \( E \) of one mole of an ideal gas is given by the equation:
\[ E = \frac{3}{2} RT \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature in Kelvin.

### Step 2: Express RMS velocity in terms of temperature
The root mean square velocity \( U_{\text{rms}} \) of a gas is given by the formula:
\[ U_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]
where:
- \( M \) is the molar mass of the gas.

### Step 3: Relate average kinetic energy to RMS velocity
From the equation for average kinetic energy, we can express \( 3RT \) in terms of \( E \):
\[ E = \frac{3}{2} RT \implies 3RT = 2E \]

### Step 4: Substitute \( 3RT \) into the RMS velocity equation
Now we can substitute \( 3RT \) in the RMS velocity formula:
\[ U_{\text{rms}} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{2E}{M}} \]

### Final Expression
Thus, the RMS velocity \( U_{\text{rms}} \) in terms of the average kinetic energy \( E \) is:
\[ U_{\text{rms}} = \sqrt{\frac{2E}{M}} \]