21. Kinetic Theory of Ideal Gases

By what factor does the rms speed of a molecule change if the temperature is increased from 10℃ to 1000℃?

Dust particles are ≈ 10 μm in diameter. They are pulverized rock, with ρ ≈ 2500 kg/m³. If you treat dust as an ideal gas, what is the rms speed of a dust particle at 20℃?

(II) A group of 25 particles have the following speeds: two have speed 10 m/s, seven have 15 m/s, four have 20 m/s, three have 25 m/s, six have 30 m/s, one has 35 m/s, and two have 40 m/s. Determine the rms speed.

(I) By what factor will the rms speed of gas molecules increase if the temperature is increased from 0°C to 160°C?

(I) Calculate the rms speed of helium atoms near the surface of the Sun at a temperature of about 6000 K.

Estimate the rms speed of an amino acid, whose molecular mass is 89 u, in a living cell at 37°C.

The escape speed from the Earth is 1.12 x 10⁴ m/s (Section 8–7). So a gas molecule traveling away from Earth near the outer boundary of the Earth’s atmosphere would, at this speed, be able to escape from the Earth’s gravitational field and be lost to the atmosphere. At what temperature is the rms speed of oxygen molecules?

For an ideal gas at temperature T show that$\frac{d{v}_{rms}}{dT}=\frac{1}{2}\left(\frac{v_{rms}}{T}\right)$, and using the approximation $\Delta v_{rms}\approx \left(\frac{d{v}_{rms}}{dT}\right)\Delta T$, show that $\frac{\Delta v_{rms}}{v_{rms}}\approx \frac{1}{2}\left(\frac{\Delta T}{T}\right)$.

Calculate the total water vapor pressure in the air on the following day: a hot summer day, with the temperature 30°C and the relative humidity at 75%.

A space vehicle returning from the Moon enters the Earth’s atmosphere at a speed of about 42,000 km/h. Molecules (assume nitrogen) striking the nose of the vehicle with this speed correspond to what temperature? (Because of this high temperature, the nose of a space vehicle must be made of special materials; indeed, part of it does vaporize, and this is seen as a bright blaze upon reentry.)

The escape speed from the Earth is 1.12 x 10⁴ m/s (Section 8–7). So a gas molecule traveling away from Earth near the outer boundary of the Earth’s atmosphere would, at this speed, be able to escape from the Earth’s gravitational field and be lost to the atmosphere. Can you explain why our atmosphere contains oxygen but not helium?

(I) Twelve molecules have the following speeds, given in arbitrary units: 6.0, 2.0, 4.0, 6.0, 0.0, 4.0, 1.0, 8.0, 5.0, 3.0, 7.0, and 8.0. Calculate the mean speed.