At what temperature is the rms speed of an atom in an argon gas cylinder equal to the rms speed of a helium gas atom at -20 °C?
The temperature at which the root-mean-square (RMS) speed of an argon atom matches that of a helium atom at -20 °C is approximately 2523.7 K. This calculation is based on the fundamental principles of the kinetic theory of gases, which relate the average kinetic energy of gas particles to their absolute temperature and mass.
Understanding RMS Speed
The root-mean-square (RMS) speed, denoted as $v_{rms}$, offers a representative measure of the average speed of particles within a gas. It's particularly significant because it's directly linked to the average translational kinetic energy of the gas particles. According to the kinetic theory of gases, the RMS speed of a gas atom or molecule is determined by the formula:
$v_{rms} = \sqrt{\frac{3RT}{M}}$
Where:
- $R$ represents the ideal gas constant (approximately 8.314 J/(mol·K)).
- $T$ denotes the absolute temperature of the gas in Kelvin (K).
- $M$ signifies the molar mass of the gas in kilograms per mole (kg/mol).
This formula highlights that for a given temperature, lighter gas particles will possess higher RMS speeds compared to heavier ones. Conversely, to achieve the same RMS speed, a heavier gas will require a higher temperature.
The Principle of Equal RMS Speeds
When the RMS speeds of two distinct gases are equivalent, a direct proportionality emerges between their absolute temperatures and their respective molar or atomic masses. If we set $v{rms, Ar} = v{rms, He}$, we can write:
$\sqrt{\frac{3RT{Ar}}{M{Ar}}} = \sqrt{\frac{3RT{He}}{M{He}}}$
By squaring both sides of the equation and canceling out the common terms (3R), the relationship simplifies to:
$\frac{T{Ar}}{M{Ar}} = \frac{T{He}}{M{He}}$
This simplified equation demonstrates that for their RMS speeds to be equal, the ratio of absolute temperature to molar mass must be identical for both gases. Consequently, a gas with a greater molar mass will need a proportionally higher temperature to achieve the same average particle speed as a gas with a smaller molar mass.
Calculating the Required Temperature for Argon
To determine the temperature at which argon atoms would have the same RMS speed as helium atoms at -20 °C, we utilize the known values for helium and the atomic masses of both elements.
Key Data Points:
| Parameter | Helium (He) | Argon (Ar) |
|---|---|---|
| Temperature ($T$) | -20 °C = 253 K | $T_{Ar}$ (to be found) |
| Atomic Mass ($M$) | 4.0 u | 39.9 u |
Note: While molar mass (in kg/mol) is typically used in the full RMS speed formula, the ratio of atomic masses in 'u' (atomic mass units) is directly proportional to the ratio of molar masses (in g/mol or kg/mol), making it suitable for comparative calculations like this.
Using the derived relationship $T{Ar} = T{He} \times \frac{M{Ar}}{M{He}}$:
-
Convert Helium temperature to Kelvin:
The given temperature for helium is -20 °C. To convert to Kelvin, add 273.15:
$T_{He} = -20^\circ C + 273.15 = 253.15 \text{ K}$. For the purpose of this calculation, 253 K is used. -
Substitute the values into the formula:
$T_{Ar} = 253 \text{ K} \times \frac{39.9 \text{ u}}{4.0 \text{ u}}$ -
Perform the calculation:
$T{Ar} = 253 \text{ K} \times 9.975$
$T{Ar} = 2523.675 \text{ K}$
Rounding this value, the required temperature for argon to have the same RMS speed as helium at -20 °C is approximately 2523.7 K.
Practical Insights
This calculation provides a clear illustration of how temperature directly impacts the kinetic energy and speed of gas particles. It underscores that significantly higher temperatures are necessary to impart the same average speed to heavier gas atoms, such as argon, compared to lighter ones like helium. This fundamental principle is vital in understanding various phenomena, including:
- Gas Diffusion: Lighter gases tend to diffuse more rapidly than heavier gases at the same temperature due to their higher average speeds.
- Atmospheric Composition: The distribution of gases in planetary atmospheres is influenced by their molecular masses and temperatures.
- Industrial Processes: In various chemical and physical processes involving gases, controlling temperature is critical for managing particle speeds and reaction rates.
For more detailed information on the behavior of gases, explore resources on the Kinetic Theory of Gases.
