Problem 29 - Physics

Question

The speed of a sound in a container of hydrogen at \(201 \mathrm{~K}\) is \(1220 \mathrm{~m} / \mathrm{s}\). What would be the speed of sound if the temperature were raised to \(405 \mathrm{~K}\) ? Assume that hydrogen behaves like an ideal gas.

Step-by-Step Solution

Answer

The speed of sound at 405 K is approximately 1731 m/s.

Step 1: Understand the Relationship

The speed of sound in a gas is proportional to the square root of its absolute temperature, provided the gas behaves ideally. The formula is \( v \propto \sqrt{T} \), where \( v \) is the speed of sound and \( T \) is the temperature in Kelvin.

Step 2: Setup Equation

We can write the proportional relationship as an equation: \( \frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}} \), where \( v_1 \) and \( v_2 \) are the speeds of sound at temperatures \( T_1 \) and \( T_2 \) respectively.

Step 3: Substitute Known Values

Substitute the given values:\[ \frac{v_2}{1220} = \sqrt{\frac{405}{201}} \]

Step 4: Calculate the Square Root

Calculate the square root on the right side: \( \sqrt{\frac{405}{201}} = \sqrt{2.0149} \approx 1.419 \).

Step 5: Solve for the Unknown Speed

Rearrange the equation to solve for \( v_2 \): \( v_2 = 1220 \times 1.419 \approx 1730.98 \).

Step 6: Conclusion

The speed of sound at \( 405 \text{ K} \) is approximately \( 1731 \text{ m/s} \).

Key Concepts

Ideal Gas

In order for us to properly explore the speed of sound within a gas, it's important to first understand what it means for a gas to behave "ideally." An ideal gas is a theoretical concept used to simplify and predict the behavior of gases. In this model:

• The gas particles do not interact with each other, meaning there are no forces of attraction or repulsion between them.
• The particles occupy no volume themselves, even though they fill up the container they're in.

These assumptions, while not exactly true for real gases, allow us to predict behaviors under a variety of conditions. For calculations about sound speed in gases, treating hydrogen as an ideal gas ensures we can use simplified mathematical models like the one for temperature dependence. This approach provides us with a more straightforward way to explore complex phenomena.

Temperature Dependence

The speed of sound in a gas has a direct relationship with the gas's temperature. This is known as temperature dependence. When we talk about temperature in the context of sound speed, we use the Kelvin scale. This is essential because the Kelvin scale measures absolute temperature, which starts at absolute zero, the point at which all atomic movement stops.
The relationship is governed by the equation:

• \( v \propto \sqrt{T} \)

This formula signifies that as temperature rises, the speed of sound in the gas also increases, assuming the gas remains ideal. It's important to remember that this dependency is expressed as a square root function, which gives us a manageable way to estimate changes in speed with temperature shifts.

Proportional Relationship

Within the context of an ideal gas, the speed of sound exhibits a proportional relationship with the square root of the temperature. Proportional relationships in mathematics describe how one quantity will change in response to another. Here,
\[ \frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}} \] explains how the ratio of two different speeds of sound is equal to the square root of the ratio of their corresponding temperatures.
This mathematical tool uses our understanding that as the temperature increases, the gas particles move faster. In turn, they transmit sound waves more efficiently, leading to an increase in the speed of sound. So, when the temperature of our hydrogen gas rises significantly, like from 201 K to 405 K, the speed of sound increases predictably based on this proportional equation.

Square Root Function

In mathematical terms, a square root function is critical for understanding the relationship between speed of sound and temperature. The square root function transforms numbers by extracting the number that, when multiplied by itself, returns the original number.
This is significant because in our equation,

• \( v \propto \sqrt{T} \)

the square root denotes the change in speed as not directly proportional to the temperature itself, but rather to its square root. It means that for a doubling in temperature, the speed of sound doesn't double; instead, it increases by the square root of 2.
This moderates how drastic changes in temperature affect the speed, giving us a balanced approach to calculating real-world scenarios, like that of hydrogen gas at different temperatures.