Question

A sealed rigid vessel contains air at STP. It is heated to bring the air to a temperature of 80 °C.

What will be the ratio of the mean free path of the air molecules at 80 °C to their mean free path at STP?

Answer 1

The ratio of the mean free path of the air molecules at 80 °C to their mean free path at STP is approximately 1.293, calculated by taking the ratio of the temperatures in Kelvin; this reflects the fact that mean free path increases with temperature.

Step-by-step explanation:

The student is inquiring about the mean free path of air molecules when a sealed rigid vessel containing air at Standard Temperature and Pressure (STP) is heated to 80 °C.

Mean free path is the average distance a molecule travels between collisions.

As temperature increases, the mean free path increases as well because molecules move faster and collide less frequently due to the expansion of the gas.

To find the ratio of the mean free path at 80 °C to the mean free path at STP, we can use the formula that states the mean free path is proportional to the temperature and inversely proportional to the molar density and the molecules' cross-sectional area.

At STP, the temperature is 273 K, and at 80 °C, it is 353 K.

Since the vessel is sealed and rigid, the number of moles and the volume of the gas remain constant, meaning the pressure increases proportionally with temperature based on Charles' Law.

The ratio of mean free path at 80 °C to STP is therefore directly proportional to the ratio of temperatures (in Kelvin) as the pressure and number of molecules per unit volume increase proportionally with temperature.

Thus, the ratio would be:

Mean free path at 80 °C/Mean free path at STP = Temperature at 80 °C/Temperature at STP = 353 K / 273 K ≈ 1.293.

The ratio of the mean free path of the air molecules at 80 °C to their mean free path at STP is approximately 1.293.

Answer 2

The mean free path of air molecules at 80 °C is approximately 1.29 times larger than their mean free path at STP.

To determine the ratio of the mean free path of air molecules at 80 °C to their mean free path at STP, we need to consider the factors affecting mean free path:

* **Temperature (T):** Higher temperature increases the kinetic energy of molecules, leading to more frequent collisions and a **shorter** mean free path.

* **Pressure (P):** Higher pressure increases the density of molecules, leading to more frequent collisions and a **shorter** mean free path.

* **Molecular size (d):** Larger molecules collide more frequently, resulting in a **shorter** mean free path.

In this case, the pressure and the type of gas remain constant (air), so we only need to consider the change in temperature.

The mean free path (λ) can be estimated using the following equation:

λ = k * T / (√2 * π * P * d²)

where:

* k is Boltzmann's constant (approx. 1.38 × 10^-23 J/K)

* T is the absolute temperature (K)

* P is the pressure (Pa)

* d is the effective diameter of the molecule (m)

Since the pressure and molecule size remain constant, the ratio of the mean free paths at different temperatures will be:

λ_80°C / λ_STP = (k * T_80°C) / (k * T_STP)

where:

* T_80°C = 353 K (80 °C + 273.15 K)

* T_STP = 273.15 K (Standard Temperature and Pressure)

Substituting the values, we get:

λ_80°C / λ_STP = (1.38 × 10^-23 J/K * 353 K) / (1.38 × 10^-23 J/K * 273.15 K) ≈ 1.29

Therefore, the mean free path of air molecules at 80 °C is approximately 1.29 times larger than their mean free path at STP. This means the molecules travel about 29% farther on average before colliding at the higher temperature.