2.4 Distribution of Molecular Speeds

One cylinder contains helium gas and another identical cylinder contains krypton gas at the same temperature. Mark each of these statements true, false, or impossible to determine from the given information. (a) The rms speeds of atoms in the two gases are the same. (b) The average kinetic energies of atoms in the two gases are the same. (c) The internal energies of 1 mole of gas in each cylinder…

An ideal gas is at a temperature of 300 K. To double the average speed of its molecules, what does the temperature need to be changed to?

Using the approximation $\int_{v_{1}}^{v_{1} + \Delta v} f \left(v\right) d v \approx f \left(v_{1}\right) \Delta v$ for small $\Delta v$, estimate the fraction of nitrogen molecules at a temperature of $3.00 \times 10^{2} \text{K}$ that have speeds between 290 m/s and 291 m/s.

Find (a) the most probable speed, (b) the average speed, and (c) the rms speed for nitrogen molecules at 295 K.

At what temperature is the average speed of carbon dioxide molecules $\left(M = 44.0 \text{g}/\text{mol}\right)$ 510 m/s?

a) At what temperature do oxygen molecules have the same average speed as helium atoms $\left(M = 4.00 \text{g}/\text{mol}\right)$ have at 300 K? b) What is the answer to the same question about most probable speeds? c) What is the answer to the same question about rms speeds?

(a) Find the density in SI units of air at a pressure of 1.00 atm and a temperature of $20 °\text{C}$, assuming that air is $78 \% \text{N}_{2} , 21 \% \text{O}_{2} , \text{and} 1 \% \text{Ar}$, (b) Find the density of the atmosphere on Venus, assuming that it’s $96 \% \text{CO}_{2} \text{and} 4 \% \text{N}_{2}$, with a temperature of 737 K and a pressure of 92.0 atm.

When an air bubble rises from the bottom to the top of a freshwater lake, its volume increases by $80 \%$. If the temperatures at the bottom and the top of the lake are 4.0 and 10 $° \text{C}$, respectively, how deep is the lake?

One process for decaffeinating coffee uses carbon dioxide $\left(M = 44.0 \text{g}/\text{mol}\right)$ at a molar density of about $14,600 \text{mol}/\text{m}^{3}$ and a temperature of about $60 ° \text{C}$. (a) Is CO₂ a solid, liquid, gas, or supercritical fluid under those conditions? (b) The van der Waals constants for carbon dioxide are $a = 0.3658 \text{Pa} \cdot \text{m}^{6} / \text{mol}^{2}$ and $b = 4.286 \times 10^{−5} \text{m}^{3} / \text{mol} .$ Using the van der Waals equation, estimate the pressure of $\text{CO}_{2}$ at that temperature and density.

On a warm day when the air temperature is $30 °\text{C}$, a metal can is slowly cooled by adding bits of ice to liquid water in it. Condensation first appears when the can reaches $15 °\text{C}$. What is the relative humidity of the air?

The mean free path for helium at a certain temperature and pressure is $2.10 \times 10^{−7} \text{m} .$ The radius of a helium atom can be taken as $1.10 \times 10^{−11} \text{m}$. What is the measure of the density of helium under those conditions (a) in molecules per cubic meter and (b) in moles per cubic meter?

In the chapter on fluid mechanics, Bernoulli’s equation for the flow of incompressible fluids was explained in terms of changes affecting a small volume dV of fluid. Such volumes are a fundamental idea in the study of the flow of compressible fluids such as gases as well. For the equations of hydrodynamics to apply, the mean free path must be much less than the linear size of such a volume, $a \approx d V^{1 / 3} .$ For…

(a) Estimate the specific heat capacity of sodium from the Law of Dulong and Petit. The molar mass of sodium is 23.0 g/mol. (b) What is the percent error of your estimate from the known value, $1230 \text{J}/\text{kg} \cdot ° \text{C}$?

Find the ratio $f \left(v_{\text{p}}\right) / f \left(v_{\text{rms}}\right)$ for hydrogen gas $\left(M = 2.02 \text{g}/\text{mol}\right)$ at a temperature of 77.0 K.

Unreasonable results. (a) Find the average speed of hydrogen sulfide, $\text{H}_{2} \text{S}$, molecules at a temperature of 250 K. Its molar mass is 31.4 g/mol (b) The result isn’t very unreasonable, but why is it less reliable than those for, say, neon or nitrogen?

Eight bumper cars, each with a mass of 322 kg, are running in a room 21.0 m long and 13.0 m wide. They have no drivers, so they just bounce around on their own. The rms speed of the cars is 2.50 m/s. Repeating the arguments of Pressure, Temperature, and RMS Speed, find the average force per unit length (analogous to pressure) that the cars exert on the walls.

Verify the normalization equation $\int_{0}^{\infty} f \left(v\right) d v = 1 .$ In doing the integral, first make the substitution $u = \sqrt{\frac{m}{2 k_{\text{B}} T}} v = \frac{v}{v_{p}} .$ This “scaling” transformation gives you all features of the answer except for the integral, which is a dimensionless numerical factor. You’ll need the formula

$\int_{0}^{\infty} x^{2} e^{− x^{2}} d x = \frac{\sqrt{\pi}}{4}$

to find the numerical factor and verify the normalization.

Verify that $v_{\text{rms}} = \sqrt{\overset{–}{v^{2}}} = \sqrt{\frac{3 k_{\text{B}} T}{m}}$.