The root mean square speed of 8 g of He is 300 `ms_(-1)`. Total kinetic energy of He gas is :

Options

A. 120 J
B. 240 J
C. 360 J
D. None of these

Explanation

To find the total kinetic energy of 8 g of helium gas given that the root mean square (RMS) speed is 300 m/s, we can follow these steps:

### Step 1: Understand the RMS Speed Formula
The RMS speed (v_rms) of a gas is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) = universal gas constant (8.314 J/(mol·K))
- \( T \) = temperature in Kelvin
- \( M \) = molar mass of the gas in kg/mol

### Step 2: Rearrange the Formula to Find RT
We can square both sides of the RMS speed formula to eliminate the square root:
\[ v_{rms}^2 = \frac{3RT}{M} \]
Rearranging gives us:
\[ RT = \frac{M v_{rms}^2}{3} \]

### Step 3: Calculate Molar Mass of Helium
The molar mass of helium (He) is approximately 4 g/mol. Converting this to kg/mol:
\[ M = 4 \text{ g/mol} = 0.004 \text{ kg/mol} \]

### Step 4: Substitute Values into the Equation
Now we can substitute the values into the equation:
\[ RT = \frac{0.004 \cdot (300)^2}{3} \]
Calculating \( (300)^2 \):
\[ (300)^2 = 90000 \]
Now substituting this value:
\[ RT = \frac{0.004 \cdot 90000}{3} = \frac{360}{3} = 120 \text{ J} \]

### Step 5: Calculate Total Kinetic Energy
The total kinetic energy (KE) of the gas can be calculated using the formula:
\[ KE = \frac{3}{2} nRT \]
Where \( n \) is the number of moles. To find \( n \):
\[ n = \frac{\text{mass}}{\text{molar mass}} = \frac{8 \text{ g}}{4 \text{ g/mol}} = 2 \text{ mol} \]

Now substituting \( n \) and \( RT \) into the kinetic energy formula:
\[ KE = \frac{3}{2} \cdot 2 \cdot 120 = 3 \cdot 120 = 360 \text{ J} \]

### Final Answer
The total kinetic energy of the helium gas is **360 Joules**.

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