In equation ` p = 1/3 alpha v_(r m s)^(2) `, the term (prop) represents dencity of gas.
`v_(r m s) = sqrt (3 R T)/(M)`.

Options

A. If both Assertion and Reason are true and the reason is correct explanation of the Assertion.
B. If both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
C. If Assertion is true, but the Reason is false.
D. If Assertion is false but the Reason is true.

Explanation

To solve the problem step by step, we will analyze the given equation and the relationships between the variables involved.

### Step 1: Understand the given equation
The equation given is:
\[ P = \frac{1}{3} \alpha v_{rms}^2 \]
where \( P \) is the pressure, \( \alpha \) represents the density of the gas, and \( v_{rms} \) is the root mean square velocity of gas molecules.

### Step 2: Express \( v_{rms} \)
We are given that:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, and \( M \) is the molar mass of the gas.

### Step 3: Substitute \( v_{rms} \) into the equation
Now, we substitute \( v_{rms} \) into the pressure equation:
\[ P = \frac{1}{3} \alpha \left(\sqrt{\frac{3RT}{M}}\right)^2 \]
This simplifies to:
\[ P = \frac{1}{3} \alpha \left(\frac{3RT}{M}\right) \]

### Step 4: Simplify the equation
We can simplify further:
\[ P = \alpha \frac{RT}{M} \]

### Step 5: Relate density \( \alpha \) to mass and volume
Using the ideal gas law, we know:
\[ PV = nRT \]
where \( n \) is the number of moles. We can express \( n \) as:
\[ n = \frac{m}{M} \]
where \( m \) is the mass of the gas. Thus:
\[ PV = \frac{m}{M} RT \]
Rearranging gives us:
\[ \frac{RT}{M} = \frac{PV}{m} \]

### Step 6: Relate density to the equation
The density \( \alpha \) can be expressed as:
\[ \alpha = \frac{m}{V} \]
Substituting this into our equation gives:
\[ P = \alpha \frac{PV}{m} \]
This shows that:
\[ P = \frac{1}{3} \alpha v_{rms}^2 \]
is consistent with the definitions of pressure and density.

### Conclusion
Thus, we have shown that the assertion \( P = \frac{1}{3} \alpha v_{rms}^2 \) is true, and the reason \( v_{rms} = \sqrt{\frac{3RT}{M}} \) is also true. However, the reason does not correctly explain the assertion, so the conclusion is that the assertion is true, and the reason is true but does not explain the assertion.