The root mean square speed of one mole of a monoatomic gas having molecular mass `M` is `u_(rms)` The relation between the average kinetic energy `(E)` of the gas and `u_9rms)` is .

Options

A. `V_(r.m.s) = sqrt((3E)/(2M))`
B. `V_(r.m.s) = sqrt((2E)/(3M))`
C. `V_(r.m.s) = sqrt((2E)/(M))`
D. `V_(r.m.s) =sqrt((E )/(3M))`

Explanation

To establish the relationship between the root mean square speed (\( u_{rms} \)) of a monoatomic gas and its average kinetic energy (\( E \)), we can follow these steps:

### Step 1: Understand the formulas
The root mean square speed (\( u_{rms} \)) of a gas is given by the formula:
\[ u_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) is the gas constant,
- \( T \) is the temperature in Kelvin,
- \( M \) is the molecular mass of the gas.

The average kinetic energy (\( E \)) of one mole of a monoatomic gas is given by:
\[ E = \frac{3}{2} RT \]

### Step 2: Express \( RT \) in terms of \( E \)
From the average kinetic energy formula, we can express \( RT \) as:
\[ RT = \frac{2E}{3} \]

### Step 3: Substitute \( RT \) into the \( u_{rms} \) formula
Now, we will substitute \( RT \) in the \( u_{rms} \) formula:
\[ u_{rms} = \sqrt{\frac{3 \left(\frac{2E}{3}\right)}{M}} \]

### Step 4: Simplify the expression
Now, simplify the expression:
\[ u_{rms} = \sqrt{\frac{2E}{M}} \]

### Step 5: Final expression
Thus, we can express the root mean square speed in terms of average kinetic energy:
\[ u_{rms} = \sqrt{\frac{2E}{M}} \]

### Conclusion
The relationship between the average kinetic energy (\( E \)) of the gas and the root mean square speed (\( u_{rms} \)) is:
\[ u_{rms} = \sqrt{\frac{2E}{M}} \]