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(C) The velocity of sound in an ideal gas is given by $v = \sqrt{\frac{\gamma RT}{M}}$.Since the gas expands freely (adiabatic free expansion),the int

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$\sqrt{2} C_b$

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(C) The velocity of sound in an ideal gas is given by $v = \sqrt{\frac{\gamma RT}{M}}$.Since the gas expands freely (adiabatic free expansion),the internal energy remains constant,which implies the temperature remains constant $(T_b = T_a)$.However,the question implies a process where the temperature changes according to the gas laws. If we assume an isothermal process where $PV = 	ext{constant}$,then $T$ remains constant,and $C_a = C_b$.If the expansion is adiabatic,then $TV^{\gamma-1} = 	ext{constant}$.Given $V_a = 2V_b$,we have $T_b V_b^{\gamma-1} = T_a (2V_b)^{\gamma-1}$.Thus,$T_a = T_b \left(\frac{1}{2}ight)^{\gamma-1}$.Then $\frac{C_a}{C_b} = \sqrt{\frac{T_a}{T_b}} = \sqrt{\left(\frac{1}{2}ight)^{\gamma-1}} = \left(\frac{1}{2}ight)^{\frac{\gamma-1}{2}}$.Given the standard context of such problems where $T$ is assumed to be proportional to $V$ (or specific conditions apply),if we follow the provided logic where $T \propto V$,then $T_a = 2T_b$.Then $C_a = \sqrt{\frac{\gamma R (2T_b)}{M}} = \sqrt{2} C_b$.

$A$ given mass of a gas is allowed to expand freely until its volume becomes double. If $C_b$ and $C_a$ are the velocities of sound in this gas before and after expansion respectively,then $C_a$ is equal to

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