An object must have a speed of at least 11.2 km / s to...

An object must have a speed of at least $11.2 \mathrm{~km} / \mathrm{s}$ to escape from the Earth's gravitational field. At what temperature will $v_{\mathrm{rms}}$ for $\mathrm{H}_{2}$ molecules equal the escape speed? Repeat for $\mathrm{N}_{2}$ molecules. $\left(M_{\mathrm{H} 2}=2.0 \mathrm{~kg} / \mathrm{kmol}\right.$ and $M_{\mathrm{N} 2}=28 \mathrm{~kg} / \mathrm{kmol}$.)

Key Concepts

Root-Mean-Square Speed

The root-mean-square (rms) speed represents the square root of the average of the squares of the speeds of gas molecules. It is given by the formula v_rms = ?(3RT/M), where R is the gas constant, T is the absolute temperature, and M is the molar mass. This measure links the microscopic kinetic energy of gas molecules to the macroscopic observable of temperature and is fundamental in understanding the motion of particles in a gas.

Kinetic Theory of Gases

Kinetic theory of gases explains the macroscopic properties of gases, such as pressure and temperature, based on the motion and collisions of individual molecules. It provides the framework for relating molecular speeds to temperature and is the basis for deriving the root-mean-square speed formula. Understanding this theory is key to predicting the behavior of gases under various thermodynamic conditions.

Escape Velocity

Escape velocity is the minimum speed required for an object to break free from a celestial body’s gravitational pull without further propulsion. It depends on the mass and radius of the body. In contexts where molecular speeds are compared to escape speeds, such as atmospheric retention studies, it highlights the relationship between thermal motion of particles and their ability to overcome gravitational forces.

Temperature and Molecular Kinetic Energy

Temperature in the context of kinetic theory is proportional to the average kinetic energy of the molecules in a gas. As temperature increases, the molecular speeds also increase, as described by kinetic energy relations. This concept is essential for linking thermodynamic temperature to molecular motion, thereby deriving speeds like v_rms, which in turn can be compared to critical speeds like escape velocity.

Molar Mass

Molar mass is the mass of one mole of a substance and plays a crucial role in the kinetic theory of gases. It appears in the denominator of the rms speed equation, indicating that heavier molecules (with a higher molar mass) will generally move slower at a given temperature compared to lighter molecules. Correctly accounting for molar mass is essential when calculating molecular speeds and understanding the behavior of different gases.

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