2.4 Distribution of molecular speeds  (Page 4/9)

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?

In a sample of hydrogen sulfide $(M=34.1\text{g/mol})$ at a temperature of $3.00\times {10}^2\text{K},$ estimate the ratio of the number of molecules that have speeds very close to $v_{\text{rms}}$ to the number that have speeds very close to $2v_{\text{rms}}.$

Using the approximation ${\displaystyle {\int }_{v_1}^{v_1+\text{Δ}v}f(v)dv\approx f(v_1)\text{Δ}v}$ for small $\text{Δ}v$ , 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.

Using the method of the preceding problem, estimate the fraction of nitric oxide (NO) molecules at a temperature of 250 K that have energies between $3.45\times {10}^{\mathrm{-21}}\text{J}$ and $3.50\times {10}^{\mathrm{-21}}\text{J}$ .

By counting squares in the following figure, estimate the fraction of argon atoms at $T=300\text{K}$ that have speeds between 600 m/s and 800 m/s. The curve is correctly normalized. The value of a square is its length as measured on the x -axis times its height as measured on the y -axis, with the units given on those axes.

Using a numerical integration method such as Simpson’s rule, find the fraction of molecules in a sample of oxygen gas at a temperature of 250 K that have speeds between 100 m/s and 150 m/s. The molar mass of oxygen $\left({\text{O}}_2\right)$ is 32.0 g/mol. A precision to two significant digits is enough.

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

Repeat the preceding problem for nitrogen molecules at 2950 K.

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

The most probable speed for molecules of a gas at 296 K is 263 m/s. What is the molar mass of the gas? (You might like to figure out what the gas is likely to be.)

a) At what temperature do oxygen molecules have the same average speed as helium atoms $(M=4.00\text{g/mol})$ 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?

In the deep space between galaxies, the density of molecules (which are mostly single atoms) can be as low as ${10}^6{\text{atoms/m}}^3,$ and the temperature is a frigid 2.7 K. What is the pressure? (b) What volume (in ${\text{m}}^3$ ) is occupied by 1 mol of gas? (c) If this volume is a cube, what is the length of its sides in kilometers?

(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 $78\text{\%}{\text{N}}_2,21\text{\%}{\text{O}}_2,\text{and}1\text{\%}\text{Ar}$ , (b) Find the density of the atmosphere on Venus, assuming that it’s $96\text{\%}{\text{CO}}_2\text{and}4\text{\%}{\text{N}}_2$ , with a temperature of 737 K and a pressure of 92.0 atm.

The air inside a hot-air balloon has a temperature of 370 K and a pressure of 101.3 kPa, the same as that of the air outside. Using the composition of air as $78\text{\%}{\text{N}}_2,21\text{\%}{\text{O}}_2,\text{and}1\text{\%}\text{Ar}$ , find the density of the air inside the balloon.

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 $\text{°}\text{C}$ , respectively, how deep is the lake?