Discussion
If T and ρ represent the temperature and density of a gas ,then the velocity of sound in the gas is directly proportional to
Answer: A
The speed of sound is proportional to \(\sqrt{T}\).
Laplace's formula gives \(v = \sqrt{\frac{\gamma P}{\rho}}\). For an ideal gas \(\frac{P}{\rho} = \frac{RT}{M}\), so
\[ v = \sqrt{\frac{\gamma RT}{M}} \propto \sqrt{T} \]
At a given temperature, changing the pressure changes the density in the same proportion, so the speed does not depend on density on its own. Speed increases with temperature, but as its square root, not as T or \(T^{2}\). \(\sqrt{\rho}\) has the density on the wrong side; in the formula it appears in the denominator.
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