Question

Speed of sound at \(T_1 = 0^\circ C\) is \(V_0\) and at \(T_2 = \alpha^\circ C\) speed becomes \(2V_0\). Find \(\alpha\):

• A. \(819^\circ C\)
• B. \(918^\circ C\)
• C. \(546^\circ C\)
• D. \(1092^\circ C\)

Hint:

A very common mistake in thermodynamics and gas-related problems is forgetting to convert temperatures from Celsius to Kelvin.
All formulas involving temperature (\(T\)) in gas laws (like ideal gas law, speed of sound) require the absolute temperature in Kelvin.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question relates the speed of sound in a gas to its temperature. We are given the speed at \(0^\circ C\) and are told it doubles at a new temperature \(\alpha^\circ C\). We need to find \(\alpha\).
Step 2: Key Formula or Approach:
The speed of sound (\(V\)) in an ideal gas is proportional to the square root of its absolute temperature (\(T\)).
\[ V = \sqrt{\frac{\gamma R T}{M}} \] This implies that \(V \propto \sqrt{T}\). Therefore, we can write the relation:
\[ \frac{V_1}{V_2} = \sqrt{\frac{T_1}{T_2}} \] where \(T_1\) and \(T_2\) are in Kelvin.
Step 3: Detailed Explanation:
First, convert the given temperatures to the absolute scale (Kelvin). Remember, \(T(K) = T(^\circ C) + 273\).
- Initial temperature: \(T_1 = 0^\circ C = 0 + 273 = 273\) K.
- Final temperature: \(T_2 = \alpha^\circ C = \alpha + 273\) K.
The speeds are given as:
- Initial speed: \(V_1 = V_0\).
- Final speed: \(V_2 = 2V_0\).
Now, substitute these values into the ratio formula:
\[ \frac{V_0}{2V_0} = \sqrt{\frac{273}{\alpha + 273}} \] \[ \frac{1}{2} = \sqrt{\frac{273}{\alpha + 273}} \] To solve for \(\alpha\), we square both sides of the equation:
\[ \left(\frac{1}{2}\right)^2 = \frac{273}{\alpha + 273} \] \[ \frac{1}{4} = \frac{273}{\alpha + 273} \] Now, cross-multiply to solve for \(\alpha\):
\[ \alpha + 273 = 4 \times 273 \] \[ \alpha = (4 \times 273) - 273 \] \[ \alpha = 3 \times 273 \] \[ \alpha = 819 \] Step 4: Final Answer:
The value of \(\alpha\) is 819. Since the final temperature was given as \(\alpha^\circ C\), the answer is \(819^\circ C\).