Question

At what temperature will He atoms have the same \(u_{\mathrm{rms}}\) value as \(\mathrm{N}_{2}\) molecules at \(25^{\circ} \mathrm{C} ?\)

Short answer

Helium atoms must be at approximately 42.57 K.

Step-by-step solution

Understand the RMS Speed Formula

The root mean square (RMS) speed of gas particles is given by the formula: \( u_{\text{rms}} = \sqrt{ \frac{3kT}{m} } \), where \( k \) is the Boltzmann constant, \( T \) is the absolute temperature in Kelvin, and \( m \) is the molar mass of the gas in kilograms per mole.

Express RMS Speed for Both Gases

For helium (He) atoms and nitrogen (N₂) molecules to have the same RMS speed, their speeds must equal: \( u_{\text{rms, He}} = u_{\text{rms, N_2}} \). This translates to the equation: \( \sqrt{ \frac{3kT_{\text{He}}}{m_{\text{He}}} } = \sqrt{ \frac{3kT_{\text{N}_2}}{m_{\text{N}_2}} } \).

Rearrange and Solve for Temperature

Since the RMS speeds are equal, we can square both sides of the equation: \( \frac{3kT_{\text{He}}}{m_{\text{He}}} = \frac{3kT_{\text{N}_2}}{m_{\text{N}_2}} \). By canceling the 3k from both sides (because they appear in both numerators), we get: \( \frac{T_{\text{He}}}{m_{\text{He}}} = \frac{T_{\text{N}_2}}{m_{\text{N}_2}} \). Rearranging gives: \( T_{\text{He}} = T_{\text{N}_2} \times \frac{m_{\text{He}}}{m_{\text{N}_2}} \).

Convert Given Temperature to Kelvin and Substitute Values

The temperature \( T_{\text{N}_2} \) is given as 25°C, which is 298 K. The molar mass of helium (He) is approximately 4.00 g/mol, and that of nitrogen (N₂) is approximately 28.02 g/mol. Convert these molar masses to kilograms by dividing by 1000: \( m_{\text{He}} = 0.004 \) kg/mol, \( m_{\text{N}_2} = 0.02802 \) kg/mol. Substitute these values into the equation: \( T_{\text{He}} = 298 K \times \frac{0.004}{0.02802} \).

Calculate the Temperature for Helium

Solve \( T_{\text{He}} = 298 \times \frac{0.004}{0.02802} \). This gives \( T_{\text{He}} \approx 42.57 \) K. Therefore, helium atoms would need to be at approximately 42.57 Kelvin to have the same RMS speed as nitrogen molecules at 25°C.

Key concepts

Boltzmann Constant

The Boltzmann constant, represented as \( k \), is a fundamental physical constant that plays a crucial role in the kinetic theory of gases. It serves as a bridge between macroscopic and microscopic physics.
This constant relates the average kinetic energy of particles in a gas with the temperature of the gas. The value of the Boltzmann constant is approximately \( 1.38 \times 10^{-23} \text{ J/K} \).

The primary significance of the Boltzmann constant in the context of gases is its use in the formula for the root mean square (RMS) speed:

• It represents a component of the energy equation for particles, allowing us to calculate temperature-related phenomena at a molecular level.
• In the RMS speed formula \( u_{\text{rms}} = \sqrt{ \frac{3kT}{m} } \), it links temperature (\( T \)) with motion (speed) of particles faster than most others on average.

By connecting the temperature of a gas with kinetic energy, the Boltzmann constant provides insights into how fast molecules are moving at a given temperature.

Molar Mass

Molar mass is the mass of one mole of a substance. It's typically expressed in grams per mole (g/mol).
Molar mass determines how much a given number of molecules (one mole) weigh, allowing us to relate microscopic scale properties to macroscopic quantities.

In the context of gas particles and their speeds, molar mass is crucial because:

• It appears in the RMS speed formula: \( u_{\text{rms}} = \sqrt{ \frac{3kT}{m} } \), where it's necessary to convert molar mass to kilograms per mole.
• The RMS speed inversely varies with the square root of the molar mass, meaning lighter molecules move faster on average than heavier ones at the same temperature.

Understanding molar mass allows us to compute gas-related calculations like the speed of gas particles at various temperatures.

Absolute Temperature

Absolute temperature is measured in Kelvin (K), which is the SI base unit of temperature. The Kelvin scale uses absolute zero as its null point, which is approx -273.15°C. At this point, theoretical molecular motion ceases.
This absolute measure is essential in thermodynamics and gas laws because it makes all temperatures positive, avoiding confusion encountered with scales like Celsius.

Why is absolute temperature so important in gas calculations?

• It directly relates to the average kinetic energy of particles: a higher absolute temperature means higher average kinetic energy.
• In the RMS speed formula \( u_{\text{rms}} = \sqrt{ \frac{3kT}{m} } \), \( T \) always needs to be expressed in Kelvin to ensure consistency with kinetic energy relations.

Using Kelvin simplifies equations in thermodynamics and ensures a consistent, straightforward representation of thermal dynamics.

Kinetic Theory of Gases

The kinetic theory of gases is a model that explains the behavior of gases in terms of the motion of their particles. This theory helps us understand and predict the properties of gases, such as pressure and temperature, using macroscopic concepts.
The core assumptions of kinetic theory include:

• Gas particles are in continuous, random motion.
• Collisions between gas particles are perfectly elastic, meaning there's no energy loss in such collisions.
• The volume of the individual particles is negligible compared to the volume the gas occupies.
• The average kinetic energy of gas particles is proportional to the absolute temperature of the gas.

This theory is significant for bridging macroscopic observations with microscopic actions. It underpins the derivation of important equations like the ideal gas law and RMS speed expression \( u_{\text{rms}} = \sqrt{ \frac{3kT}{m} } \). Through this theory, gas behavior becomes predictable and quantifiable in terms of temperature, pressure, and volume.