For an ideal gas, the graph representing the variation of the speed of sound (v) in a gas, with the r.m.s. speed (vrms) of its gas molecules with be

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Answer

To solve the problem of how the speed of sound (v) in an ideal gas varies with the root mean square speed (v_rms) of its gas molecules, we can follow these steps:

### Step 1: Understand the formulas
The root mean square speed (v_rms) of gas molecules is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature in Kelvin,
- \( M \) is the molar mass of the gas.

The speed of sound (v_s) in an ideal gas is given by:
\[ v_s = \sqrt{\frac{\gamma RT}{M}} \]
where:
- \( \gamma \) is the adiabatic index (ratio of specific heats, \( C_p/C_v \)).

### Step 2: Relate the two speeds
To find the relationship between \( v_{rms} \) and \( v_s \), we can divide the two equations:
\[ \frac{v_{rms}}{v_s} = \frac{\sqrt{\frac{3RT}{M}}}{\sqrt{\frac{\gamma RT}{M}}} \]
In this expression, \( R \), \( T \), and \( M \) will cancel out:
\[ \frac{v_{rms}}{v_s} = \sqrt{\frac{3}{\gamma}} \]

### Step 3: Express \( v_{rms} \) in terms of \( v_s \)
From the above relationship, we can express \( v_{rms} \) in terms of \( v_s \):
\[ v_{rms} = v_s \cdot \sqrt{\frac{3}{\gamma}} \]
This shows that \( v_{rms} \) is directly proportional to \( v_s \).

### Step 4: Graphical representation
Since \( v_{rms} \) is proportional to \( v_s \), we can represent this relationship graphically. If we let \( v_{rms} \) be on the x-axis and \( v_s \) be on the y-axis, the equation can be rearranged to:
\[ v_s = \frac{1}{\sqrt{\frac{3}{\gamma}}} v_{rms} \]
This is a linear equation of the form \( y = mx \), where \( m \) is a constant. Therefore, the graph will be a straight line passing through the origin.

### Conclusion
The graph representing the variation of the speed of sound (v) in a gas with the root mean square speed (v_rms) of its gas molecules will be a straight line.