Problem 40 - Physics

Question

A sound wave travels twice as far in neon (Ne) as it does in krypton (Kr) in the same time interval. Both neon and krypton can be treated as monatomic ideal gases. The atomic mass of neon is \(20.2 \mathrm{u}\), and that of krypton is \(83.8 \mathrm{u}\). The temperature of the krypton is \(293 \mathrm{~K}\). What is the temperature of the neon?

Step-by-Step Solution

Answer

The temperature of the neon is approximately 282 K.

Step 1: Understand Sound Wave Speed in Gases

The speed of sound in an ideal monatomic gas is given by \( v = \sqrt{\frac{\gamma k T}{m}} \), where \( \gamma = \frac{5}{3} \) for monatomic gases, \( k \) is the Boltzmann constant, \( T \) is the temperature, and \( m \) is the mass of one molecule of the gas.

Step 2: Expression for Sound Speed Ratio

Since sound travels twice as far in neon as in krypton in the same time interval, the speed of sound in neon (\( v_{Ne} \)) is twice that in krypton (\( v_{Kr} \)). Therefore: \( \frac{v_{Ne}}{v_{Kr}} = 2 \).

Step 3: Substitute Speed Formula

Using the speed formula in both cases: \( \frac{\sqrt{\frac{\gamma k T_{Ne}}{m_{Ne}}}}{\sqrt{\frac{\gamma k T_{Kr}}{m_{Kr}}}} = 2 \). Simplify this to: \( \sqrt{\frac{T_{Ne} m_{Kr}}{T_{Kr} m_{Ne}}} = 2 \).

Step 4: Solve for Temperature of Neon

Square both sides to eliminate the square root: \( \frac{T_{Ne} m_{Kr}}{T_{Kr} m_{Ne}} = 4 \). Then rearrange for \( T_{Ne} \): \( T_{Ne} = 4 \times T_{Kr} \times \frac{m_{Ne}}{m_{Kr}} \).

Step 5: Plug in Given Values

Substitute the given values: \( m_{Ne} = 20.2 \mathrm{u} \), \( m_{Kr} = 83.8 \mathrm{u} \), and \( T_{Kr} = 293 \mathrm{K} \). Calculate \( T_{Ne} = 4 \times 293 \mathrm{K} \times \frac{20.2}{83.8} \).

Step 6: Calculate Result

Perform the calculations: \( T_{Ne} = 4 \times 293 \times \frac{20.2}{83.8} \approx 282 \mathrm{K} \).

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation describing the behavior of gases under different conditions. It is expressed as \( PV = nRT \), where:

• \( P \) is the pressure of the gas
• \( V \) is the volume
• \( n \) is the number of moles
• \( R \) is the ideal gas constant
• \( T \) is the temperature in Kelvin

This law assumes that gas molecules do not interact and occupy no volume. Although these assumptions are idealized, the law provides a good approximation of real gas behavior under most conditions.
If you increase the temperature, you increase the speed and kinetic energy of the gas molecules. This principle is crucial when considering the speed of sound in gases.

Monatomic Gases

Monatomic gases are composed of single atoms. Examples include noble gases like helium, neon, and krypton. These gases have a simple structure with no bonds between atoms.
The simplicity of monatomic gases allows for straightforward calculations. For instance, the speed of sound in monatomic gases can be calculated using the expression \( v = \sqrt{\frac{\gamma k T}{m}} \). Here:

• \( \gamma = \frac{5}{3} \) is the adiabatic index for monatomic gases
• \( k \) is the Boltzmann constant
• \( T \) is the temperature
• \( m \) is the atomic mass

Their lack of molecular complexity means the calculations often result in more predictable behavior compared to more complex multi-atomic gases.

Temperature and Sound Speed

The speed of sound in a gas depends on its temperature. As the temperature increases, so does the kinetic energy of the gas particles, which in turn increases the speed of sound. This relationship can be seen in the formula \( v = \sqrt{\frac{\gamma k T}{m}} \).
Because the speed of sound in liquid and solid media is generally less sensitive to temperature changes than in gases, temperature is a key factor in gas dynamics. Understanding this allows us to predict how sound will travel through different gases and under different temperature conditions.
In the context of the exercise, knowing the temperature of krypton and the relationship of sound speeds allows us to calculate the temperature at which neon must be to achieve the observed speed ratio.

Atomic Mass

Atomic mass affects the speed of sound in gases significantly. It represents the mass of a single atom and is typically measured in atomic mass units (u).
In the context of sound speed, a lighter gas (lower atomic mass) generally allows sound to travel faster than a heavier gas. This is due to the lower inertia of lighter gas particles. The formula \( v = \sqrt{\frac{\gamma k T}{m}} \) highlights this inverse relationship.
Thus, the atomic mass of a gas is a crucial determinant of how quickly sound waves can propagate through it. In the exercise, the differing atomic masses of neon and krypton directly affect the speed of sound and, consequently, the temperature calculations.