(a) Calculate the most probable speed, the mean speed, and the mean relative speed of $\mathrm{CO}_{2}$ molecules at $20^{\circ} \mathrm{C}.$ (b) Calculate the most probable speed, the mean speed, and the mean relative speed of $\mathrm{H}_{2}$ molecules at $20^{\circ} \mathrm{C}.$

(a) Use the Maxwell-Boltzmann distribution of speeds to estimate the fraction of $\mathrm{N}_{1}$ molecules at 400 K that have speeds in the range $200-210\ \mathrm{ms}^{-1}.$ (b) Use the Maxwell-Boltzmann distribution of speeds to estimate the fraction of $\mathrm{CO}_{2}$ molecules at 400 K that have speeds in the range $400-405\ \mathrm{ms}^{-1}.$

(a) Calculate the root mean square speeds of $H_2$ and $O_2$ molecules at $20 ^ { \circ } \mathrm { C }$. (b) Calculate the root mean square speeds of $CO_2$ molecules and He atoms at $20 ^ { \circ } \mathrm { C }$.

Discuss the use of herbs, spices, and other flavorings.

(a) At an altitude of 20 km the temperature is 217 K and the pressure is 0.050 atm. What is the mean free path of $\mathrm{N}_{2}$ molecules? $\left(\sigma=0.43\ \mathrm{nm}^{2}\right).$ (b) At an altitude of 15 km the temperature is 217 K and the pressure is 12.1 kPa. What is the mean free path of $\mathrm{N}_{2},$ molecules? $\left(\sigma=0.43\ \mathrm{nm}^{2}\right).$

Calculate the mass of water vapour present in a room of volume 400 m$^3$ that contains air at 27 degrees C on a day when the relative humidity is 60 per cent. Calculate the mass of water vapour present in a room of volume 250 m$^3$ that contains air at 23 degrees C on a day when the relative humidity is 53 per cent.

(a) Determine the ratios of (i) the mean speeds, (ii) the mean translational kinetic energies of $\mathrm{H}_{2}$ molecules and Hg atoms at $20^{\circ} \mathrm{C}.$ (b) Determine the ratios of (i) the mean speeds, (ii) the mean translational kinetic energies of He atoms and Hg atoms at $25^{\circ} \mathrm{C}.$

(a) Could 131 g of xenon gas in a vessel of volume 10 $\mathrm{dm}^{3}$ exert a pressure of 20 atm at $25^{\circ} \mathrm{C}$ if it behaved as a perfect gas? If not, what pressure would it exert? (b) Could 25 g of argon gas in a vessel of volume 1.5 $\mathrm{dm}^{3}$ exert a pressure of 2 bar at $30^{\circ} \mathrm{C}$ if it behaved as a perfect gas? If not, what pressure would it exert?

(a) Could 131 g of xenon gas in a vessel of volume $1.0\ \mathrm{dm}^{3}$ exert a pressure of 20 atm at $25^{\circ} \mathrm{C}$ if it behaved as a perfect gas? If not, what pressure would it exert? (b) Could 25 g of argon gas in a vessel of volume $1.5\ \mathrm{dm}^{3}$ exert a pressure of 2.0 bar at $30^{\circ} \mathrm{C}$ if it behaved as a perfect gas? If not, what pressure would it exert?

Find each power of i and simplify the expression. $5 i ^ { 5 } \left( i ^ { 3 } - i \right)$

(a) At what pressure does the mean free path of argon at $20^{\circ} \mathrm{C}$ become comparable to the diameter of a $100\ \mathrm{cm}^{3}$ vessel that contains it? Take $\sigma=0.36\ \mathrm{nm}^{2}.$ (b) At what pressure does the mean free path of argon at $20^{\circ} \mathrm{C}$ become comparable to 10 times the diameters of the atoms themselves? Take $\sigma=0.36\ \mathrm{nm}^{2}.$

Use the kinetic theory of gases to explain why gases expand upon heating and shrink upon cooling.

$$\dfrac{2}{3}b+4=6$$

(a) What is the relative mean speed of $N_{2}$ and $H_{2}$ molecules in a gas at $25^{\circ} \mathrm{C}?$ (b) What is the relative mean speed of $O_{2}$ and $N_{2}$ molecules in a gas at $25^{\circ} \mathrm{C}?$