The root mean square velocity of an ideal gas at constant pressure varies with density d as

Options

A. `d^(2)`
B. d
C. `sqrt(d)`
D. `1//sqrt(d)`

Answer

To find the relationship between the root mean square velocity (v_rms) of an ideal gas at constant pressure and its density (d), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Ideal Gas Law**:
The ideal gas law is given by the equation:
\[ PV = nRT \]
where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the universal gas constant, and \(T\) is the temperature.

2. **Root Mean Square Velocity Formula**:
The root mean square velocity (v_rms) of an ideal gas is defined as:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where \(M\) is the molar mass of the gas.

3. **Relate Molar Mass and Density**:
For one mole of gas, the mass (m) is equal to the molar mass (M). The density (d) of the gas can be expressed as:
\[ d = \frac{m}{V} \]
Since \(m = M\) for one mole, we can rewrite the density as:
\[ d = \frac{M}{V} \]

4. **Substituting for V**:
From the ideal gas law, we can express \(V\) as:
\[ V = \frac{nRT}{P} \]
For one mole of gas (\(n = 1\)), this simplifies to:
\[ V = \frac{RT}{P} \]

5. **Substituting V in the Density Equation**:
Now substituting \(V\) back into the density equation gives:
\[ d = \frac{M}{\frac{RT}{P}} = \frac{MP}{RT} \]

6. **Expressing v_rms in terms of Density**:
We can rearrange this to express \(M\) in terms of \(d\):
\[ M = \frac{dRT}{P} \]
Now substituting this into the v_rms equation:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3RT}{\frac{dRT}{P}}} = \sqrt{\frac{3P}{d}} \]

7. **Final Relationship**:
Thus, we find that:
\[ v_{rms} = \sqrt{\frac{3P}{d}} \]
This shows that the root mean square velocity is inversely proportional to the square root of the density (d) when pressure (P) is constant.

### Conclusion:
At constant pressure, the root mean square velocity (v_rms) of an ideal gas varies with density (d) as:
\[ v_{rms} \propto \frac{1}{\sqrt{d}} \]