Question

The graph drawn between the velocity of sound in a gas and pressure of the gas at a given temperature is

• A. a straight line with negative slope
• B. a straight line with positive slope
• C. a straight line parallel to pressure axis
• D. a parabola
• E. an exponential curve

Hint:

Logic Tip: A classic trick question! Always remember that isothermal changes in pressure cause identical proportional changes in density. Thus, $P/\rho$ remains constant, meaning sound velocity only depends on temperature, not pressure.

Answer 1

The Correct Option is C

Concept:
The velocity of sound $v$ in an ideal gas is given by the Newton-Laplace formula: $$v = \sqrt{\frac{\gamma P}{\rho}}$$ where $\gamma$ is the adiabatic index, $P$ is the pressure, and $\rho$ is the density of the gas. From the ideal gas law, $PV = nRT$, we can write $\rho = \frac{PM}{RT}$.
Step 1: Substitute density into the velocity formula.
Replace $\rho$ in the velocity equation: $$v = \sqrt{\frac{\gamma P}{\left(\frac{PM}{RT}\right)}}$$
Step 2: Simplify the expression.
The pressure $P$ in the numerator and denominator cancels out: $$v = \sqrt{\frac{\gamma RT}{M}}$$
Step 3: Analyze the dependency.
At a constant temperature ($T = \text{constant}$), for a given gas ($\gamma, M = \text{constant}$), the velocity of sound $v$ is entirely independent of the pressure $P$. Changing the pressure changes the density proportionally, leaving the ratio $P/\rho$ constant.
Step 4: Determine the graph shape.
Since velocity is independent of pressure, plotting velocity (y-axis) against pressure (x-axis) will result in a horizontal straight line. This line is parallel to the pressure axis.