91Ó°ÊÓ

The root-mean-square speed, commonly abbreviated to \( v_{\text{rms}} \), is an important concept when discussing the behavior of gases. In the world of physics, the rms speed refers to the square root of the average of the squares of the speeds of all the molecules in a gas sample.
This measure gives us a meaningful average speed since it's directly related to kinetic energy. Unlike average speed, which is simply the mean of velocities, the rms speed considers the velocities squared, which is crucial for deriving accurate results.
For an ideal gas, the formula for rms speed is:

• \[ v_{\mathrm{rms}} = \sqrt{\frac{3kT}{m}} \]

where \( k \) is the Boltzmann constant, \( T \) is the absolute temperature in Kelvin, and \( m \) is the molecular mass of the gas. This equation links microscopic particle behavior to macroscopic properties like temperature.

Temperature plays a decisive role in the behavior of gases. As per the ideal gas law, when the temperature of a gas changes, so does the speed of its molecules. A rise in temperature causes the gas molecules to move more energetically, increasing their speed.
In the context of this problem, the temperature change from winter to summer affects the root-mean-square speed of air molecules. Converting temperatures from \(^\circ\)C to Kelvin is crucial because the Kelvin scale directly relates temperature to energy. This conversion accounts for the absolute measure of thermal energy:

• Winter: \(-5^{\circ}C = 268 K\)
• Summer: \(25^{\circ}C = 298 K\)

In this situation, \( \Delta T \), the change in temperature, is 30 K (i.e., 298 K - 268 K). This difference is essential for understanding how temperature variation can impact molecular speeds in gases.

Understanding how changes in temperature affect \( v_{\text{rms}} \) involves calculus differentiation. By differentiating the rms speed formula with respect to temperature, we can predict how small changes in temperature could influence molecular speed.
The derivation begins with the formula \( v_{\text{rms}} = \sqrt{\frac{3kT}{m}} \). Applying the chain rule in differentiation allows us to find:

• \[ \frac{d v_{\mathrm{rms}}}{d T} = \frac{1}{2} \frac{v_{\mathrm{rms}}}{T} \]

This expression tells us how the rms speed changes concerning the temperature of the gas. It shows that rms speed is proportional to the square root of temperature, a crucial point for understanding any dynamic process involving gases.

To estimate practical changes in rms speed, we use an approximation method. The small changes in rms speed \( \Delta v_{\mathrm{rms}} \) in response to temperature changes can be related using a linear approximation. The expression is:

• \[ \Delta v_{\mathrm{rms}} \approx \frac{1}{2} \frac{v_{\mathrm{rms}}}{T} \Delta T \]

This holds under the assumption that \( \Delta T \) is small relative to \( T \). From this, we find the relative change in rms speed:

• \[ \frac{\Delta v_{\mathrm{rms}}}{v_{\mathrm{rms}}} \approx \frac{1}{2} \frac{\Delta T}{T} \]

This approximation simplifies calculations by relating changes in speed directly to proportional changes in temperature, making it easy to estimate percentages, such as a 5.3% increase in rms speed due to the temperature rise from winter to summer.