The root mean square speed 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)`

Explanation

To solve the problem of how the root mean square speed of an ideal gas at constant pressure varies with density, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definition of Root Mean Square Speed**:
The root mean square speed (\( u_{rms} \)) of an ideal gas is given by the formula:
\[ u_{rms} = \sqrt{\frac{3RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, and \( M \) is the molar mass of the gas.

2. **Relate Density to the Ideal Gas Law**:
From the ideal gas law, we know:
\[ PV = nRT \]
where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, and \( R \) is the gas constant. The density \( d \) of the gas can be expressed as:
\[ d = \frac{m}{V} = \frac{nM}{V} \]
Thus, we can rearrange the ideal gas law to find density:
\[ d = \frac{PM}{RT} \]

3. **Express Molar Mass in Terms of Density and Pressure**:
Rearranging the equation for density gives:
\[ M = \frac{dRT}{P} \]

4. **Substitute Molar Mass into the RMS Speed Equation**:
Substituting \( M \) into the root mean square speed equation:
\[ u_{rms} = \sqrt{\frac{3RT}{\frac{dRT}{P}}} \]
This simplifies to:
\[ u_{rms} = \sqrt{\frac{3P}{d}} \]

5. **Analyze the Relationship at Constant Pressure**:
Since the pressure \( P \) is constant, we can see that:
\[ u_{rms} \propto \sqrt{\frac{1}{d}} \]
This indicates that the root mean square speed is inversely proportional to the square root of the density.

6. **Conclusion**:
Therefore, we conclude that the root mean square speed of an ideal gas at constant pressure varies inversely with the square root of its density:
\[ u_{rms} \propto \frac{1}{\sqrt{d}} \]

### Final Answer:
The root mean square speed of an ideal gas at constant pressure varies with density \( d \) as:
\[ u_{rms} \propto \frac{1}{\sqrt{d}} \]