Root-Mean-Square Velocity of Gases Practice Problems

A nitrogen (N₂, molar mass = 28.01 g/mol) molecule in the atmosphere has a root mean square speed of 509 m/s. Determine its momentum as it moves at that speed. 

Constant pressure of a gas sample can be attained by fitting a container with a movable piston. A gas in such a container is maintained at a constant pressure of 800 Pa. If the gas is oxygen (molar mass 32.00 g/mol) and its temperature varies between 40°C and -70°C, calculate the range of root-mean-square speeds for oxygen molecules.

Monoatomic gases helium (He), argon (Ar), and xenon (Xe) are isolated from the atmosphere and placed in the same tank. How does the root-mean-square speed vary between these three species?

Nitrogen (N₂) is a diatomic gas with a molar mass of 28.01 g/mol. What is its root-mean-square speed (v_rms) at room temperature (298K)?

Brownian motion refers to the random motion of particles as they collide with fluid particles. A dust particle of mass 2.15 × 10^-17 kg is suspended in air whose temperature is 308 K. Determine the root mean square speed of its Brownian motion.

Oxygen and Chlorine are among the gaseous elements in the periodic table. Determine the temperature when the root-mean-square speed of chlorine molecules is the same as the root-mean-square speed of oxygen molecules at room temperature (25.0°C). Hint: The molar mass of O₂ is 32.00 g/mol; Cl₂ is 70.90 g/mol.

The molar mass of Nitrogen is 28.01 g/mol. If the root-mean-square speed of the nitrogen molecules is 509 m/s, how many molecules with this speed moving between the left and right faces of a cube of length 0.2 m are required to produce a pressure of 1.5 atm?

A 0.15m long cube filled with nitrogen (N₂, molar mass 28.01 g/mol) at a rms velocity of 524m/s. Imagine a molecule repeatedly moving between the left and right faces, then determine the average force exerted on one of the cube's faces. Take the motion to be perpendicular to the left and right faces.

Consider that in the future a very high spatial resolution microscope is developed to study the motion of gas molecules.  Suppose that a scientist is using the microscope to study the motion of oxygen molecules at standard atmospheric pressure inside a small container. The microscope detects two oxygen molecules separated by a distance of 22 molecular radii, which then collide head-on after 2.3 seconds. Assume that the oxygen molecules in the container have the same root mean square speed and that there are no other gases present in the container. The molecular radius of oxygen is 1.52 angstroms, and its molecular weight is 32 g/mol. Determine the temperature of the oxygen gas in Kelvin.

A sample of 15 nitrogen molecules has speeds ranging from 36 m/s, 37 m/s, 38 m/s, ....., and 50 m/s. Determine the i) average speed and ii) the root-mean-square speed of the nitrogen molecules in the sample.

Consider a cylindrical container of radius r filled with an ideal gas that has a movable piston. The gas undergoes an isobaric process while the piston is moving at a constant speed v and the gas expands. Determine the expression for the rate of change of the root mean square speed of the gas in terms of v, the instantaneous value of the root mean square speed (v_rms), and the instantaneous value of the cylinder's height h.

A sample of hydrogen gas contains hydrogen (¹₁H) and deuterium (²₁H) atoms. A scientist developed a method to separate the two isotopes based on the technique of isotope separation by gas diffusion. The sample of hydrogen is introduced into a series of diffusion-permeable porous cells at a fixed temperature T. ¹₁H diffuses through the porous membrane more rapidly than ²₁H, resulting in a separation of the two isotopes. The scientist collects the enriched gas stream, which contains ²₁H. Calculate the ratio of the root mean square velocities of ¹₁H and ²₁H (v_rms(¹₁H)/v_rms(²₁H)). ¹₁H has a mass of 1.008 u, and ²₁H has a mass of 2.014 u.

A sealed container has a pressure of 1.5 atm and contains gas molecules with an rms speed of 343 m/s at a temperature of 300 K. Name the gas in the container if it has a number density of 5.8 × 10²⁵ m^-3.

The root-mean-square speed of molecules in a gas is 700 m/s. If the volume of the gas is suddenly doubled, what will be the new rms speed of the molecules, assuming the pressure remains constant?

A gas molecule is at a temperature of 15°C and is moving at a random rms speed. Find the ratio of new rms speed to old rms speed if the temperature of the molecule is raised to 2000°C.

Assuming that pollen grains are spherical and have a density of 1100 kg/m³, calculate the rms speed of a pollen grain at a temperature of 25°C if it behaves like an ideal gas. The diameter of a typical pollen grain is 20 μm.

A physics lab experiment involves sixteen pendulums with different lengths swinging under gravity. Calculate their root mean square velocity if three pendulums are swinging with a speed of 2 m/s, six with 3 m/s, two with 4 m/s, three with 5 m/s, and two more pendulums with 6 m/s.

A balloon filled with helium gas is initially at a temperature of 25°C. If the temperature of the balloon is increased to 85°C, by what factor will the rms speed of the helium atoms increase as a result?

Ten particles are moving with velocities as follows (in arbitrary units) : 20.0, 25.0, 30.0, 35.0, 40.0, 45.0, 50.0 and three particles are stationary. What is their mean velocity?

What is the root mean square speed of oxygen molecules at a temperature of 3 × 10² K?

Estimate the root mean square speed for nitrogen gas molecules (N₂), with an approximate molecular weight of about 28 u, in ambient conditions at sea level where temperatures are typically around 20°C.

The escape speed from Jupiter is approximately 6.0 × 10⁴ m/s. What would be the temperature at which the rms speed of hydrogen molecules equals this escape velocity?

Consider nitrogen gas (N₂) in a container that is being heated increasing its temperature by ΔT. The molecules of the nitrogen gas have a root mean square (rms) speed given by $v_{rms}=\sqrt{\frac{3kT}{m}}$ where k is the Boltzmann constant, T is the temperature in Kelvin, and m is the mass of a nitrogen molecule. i) Derive expressions for $\frac{dv_{rms}}{dT}$ and, as an approximation, ii) $\frac{\Delta v_{\mathrm{rms}}}{v_{\mathrm{rms}}}$ in terms of ΔT and T.

On a spring day with a temperature of 15°C and relative humidity of about 60%, what would be the total water vapor pressure? Hint: saturated water vapor pressure at 15°C = 1710 Pa.

A meteorite enters the Earth’s atmosphere at a speed of about 6.0 × 10⁴ km/h. Assuming the molecular mass of the air colliding with the meteorite is 29 u, calculate the temperature of the colliding air at this speed.

For a gas molecule to escape from Titan's gravitational field, it must reach the escape speed of 2.64 x 10³ m/s. The surface temperature of Titan, one of Saturn's moons, is approximately 94 K. The molecular mass of methane (CH₄) is 16.0 g/mol, and the atomic mass of hydrogen (H₂) is 2.0 g/mol. By comparing the average speed of these gases at the surface temperature of Titan, explain why Titan's atmosphere contains methane but not hydrogen. Hint: Boltzmann constant k = $1.38 \times 10^{-23} \mathrm{~J} / \mathrm{K}$.