The rms speed at NTP of a gas can be calculated from the expression:

Options

A. `sqrt(3P//d)`
B. `sqrt(3PV//M)`
C. `sqrt(3RT//M)`
D. `sqrt(3d//P)`

Answer

To calculate the root mean square (rms) speed of a gas at Normal Temperature and Pressure (NTP), we can use the following steps:

### Step-by-Step Solution:

1. **Understanding RMS Speed**: The rms speed (v_rms) of a gas is given by the formula:
\[ v_{rms} = \sqrt{\frac{3P}{D}} \]
where P is the pressure of the gas and D is the density of the gas.

2. **Relating Density to Mass and Volume**: We know that density (D) can be expressed as:
\[ D = \frac{m}{V} \]
where m is the mass of the gas and V is its volume.

3. **Substituting Density into the RMS Speed Formula**: By substituting the expression for density into the rms speed formula, we get:
\[ v_{rms} = \sqrt{\frac{3P}{\frac{m}{V}}} = \sqrt{\frac{3PV}{m}} \]

4. **Using the Ideal Gas Law**: The ideal gas law states that:
\[ PV = nRT \]
where n is the number of moles, R is the universal gas constant, and T is the temperature in Kelvin.

5. **Substituting PV in the RMS Speed Formula**: From the ideal gas law, we can express PV as:
\[ PV = nRT \]
Therefore, we can substitute this into our rms speed equation:
\[ v_{rms} = \sqrt{\frac{3(nRT)}{m}} \]

6. **Relating Moles to Mass**: The number of moles (n) can be related to mass (m) using the molar mass (M):
\[ n = \frac{m}{M} \]
Substituting this back into the equation gives:
\[ v_{rms} = \sqrt{\frac{3\left(\frac{m}{M}\right)RT}{m}} = \sqrt{\frac{3RT}{M}} \]

7. **Final Expression for RMS Speed**: Thus, the final expression for the rms speed of a gas at NTP is:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]

### Summary of the Expression:
The rms speed of a gas at NTP can be calculated using the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]