On earth, STP is based on the average atmospheric pressure at the surface and on a phase change of water that occurs at an easily produced temperature, being only slightly cooler than the average air temperature. The atmosphere of Venus is almost entirely carbon dioxide (CO₂), the pressure at the surface is a staggering 93 atm, and the average temperature is 470℃. Venusian scientists, if they existed, would certainly use the surface pressure as part of their definition of STP. To complete the definition, they would seek a phase change that occurs near the average temperature. Conveniently, the melting point of the element tellurium is 450℃. What are (a) the rms speed and (b) the mean free path of carbon dioxide molecules at Venusian STP based on this phase change in tellurium? The radius of a CO₂ molecule is 1.5 x 10^-10 m.

Verified step by step guidance

1. Step 1: To calculate the root mean square (rms) speed of carbon dioxide (CO₂) molecules, use the formula: $\sqrt{\frac{3kT}{m}}$, where $k$ is the Boltzmann constant ($1.38\times {10}^{-23} J·K-1$), $T$ is the temperature in kelvins, and $m$ is the mass of a CO₂ molecule. First, convert the temperature from Celsius to kelvins using $T=470+273$.
2. Step 2: Determine the mass of a single CO₂ molecule. The molar mass of CO₂ is approximately $44 g·\mathrm{mol}-1$. Convert this to kilograms per molecule by dividing by Avogadro's number ($6.022\times {10}^{23}$).
3. Step 3: Substitute the values of $k$, $T$, and $m$ into the rms speed formula to calculate the root mean square speed of CO₂ molecules.
4. Step 4: To calculate the mean free path of CO₂ molecules, use the formula: $\lambda =\frac{1}{\mathrm{\pi }\times 2\times r^2\times n}$, where $r$ is the radius of a CO₂ molecule and $n$ is the number density of molecules. The radius is given as $1.5\times {10}^{-10}$ m.
5. Step 5: Calculate the number density $n$ using the ideal gas law: $n=\frac{P}{k\times T}$, where $P$ is the pressure (93 atm, converted to pascals). Substitute the values of $r$ and $n$ into the mean free path formula to find the result.

Key Concepts

Root Mean Square Speed (rms speed)

The root mean square speed is a measure of the average speed of particles in a gas, calculated from the kinetic theory of gases. It is defined as the square root of the average of the squares of the speeds of the gas molecules. The rms speed is influenced by the temperature and molar mass of the gas, providing insight into the kinetic energy and behavior of gas molecules under different conditions.

Mean Free Path

Mean free path is the average distance a molecule travels between collisions with other molecules in a gas. It is influenced by the density of the gas and the size of the molecules. Understanding mean free path is crucial for predicting gas behavior, especially in high-pressure environments like Venus, where the density of CO₂ is significantly higher than on Earth.

Phase Change

A phase change refers to the transition of a substance from one state of matter to another, such as solid to liquid or liquid to gas. In the context of the question, the phase change of tellurium at 450℃ is relevant for defining standard temperature and pressure (STP) on Venus. This concept is essential for understanding how temperature and pressure conditions affect the physical properties and behavior of gases in different planetary atmospheres.