Find the escape speed on Titan. What is the limiting molecular weight of gases that could be retained by Titan's gravity? (Hint: Use the ideas presented in Box 7-2 and assume an average atmospheric temperature of \(95 \mathrm{~K}\).)

Gravitational acceleration is a measure of the force of gravity at a specific location. On Earth, this value is approximately 9.8 m/s², but this can vary on other celestial bodies like Titan, one of Saturn's moons. Understanding gravitational acceleration is crucial when figuring out the escape speed from a planet or moon because it determines how much speed an object needs to break free from its gravitational pull.

For Titan, the gravitational acceleration is considerably smaller at 1.352 m/s². This lower value is due to Titan's smaller mass relative to Earth. To find the escape velocity, one must use the formula: \(v_{e} = \sqrt{2gr}\), where \(g\) represents the gravitational acceleration and \(r\) the radius of Titan. This formula illustrates that the required escape speed is dependent on both the value of \(g\) and the radius of the moon. The lower gravitational acceleration on Titan means that objects require less speed to escape its gravity compared to Earth, facilitating exploration and potential future missions.

The atmospheric temperature plays an important role when calculating the limiting molecular weight of gases that a celestial body can retain. Titan has a much colder average atmospheric temperature of about 95 K (-178.15°C or -288.67°F), in contrast to Earth's much warmer climate. The temperature of an atmosphere affects the speed at which gas molecules move; colder temperatures result in slower moving molecules.

When the kinetic energy of the gas molecules is lower due to reduced temperatures, the gravitational pull of the moon or planet can more easily retain these gases. For Titan, this means that its cooler atmosphere allows it to hold onto lighter gases than Earth could at the same temperatures, a fascinating aspect for those studying planetary atmospheres and possible extraterrestrial life. By using the temperature in the formula to calculate the limiting molecular weight, \(M = \frac {2kT}{v_{e}^{2}}\), students can understand how Titan's temperature influences which gases remain in its atmosphere.

Limiting molecular weight is the threshold below which a gas will escape from a planet's or moon's atmosphere due to the thermal speed of its molecules. It combines concepts of both temperature and gravity: the higher the temperature or the lower the gravity, the lighter the molecules that can escape from the atmosphere.

The calculation of Titan's limiting molecular weight involves the escape speed and the atmospheric temperature. The formula \(M = \frac {2kT}{v_{e}^{2}}\) includes the Boltzmann constant (\(k\)), representing the relationship between temperature and kinetic energy, and the square of the escape velocity (\(v_{e}^{2}\)). The lower escape velocity on Titan allows it to retain molecules of a lower molecular weight. With a calculated escape speed, the limiting molecular weight suggests what types of gases are likely to be found in Titan's atmosphere. Despite its distance from the Sun and colder environment, these factors contribute to Titan having a dense atmosphere with a diverse chemical composition.