Choose the substance with the larger positional probability in each case. a. 1 mole of \(\mathrm{H}_{2}\) (at \(\mathrm{STP} )\) or 1 mole of $\mathrm{H}_{2}\left(\text { at } 100^{\circ} \mathrm{C}, 0.5 \mathrm{atm}\right)$ b. 1 mole of \(\mathrm{N}_{2}(\text { at } \mathrm{STP})\) or 1 mole of \(\mathrm{N}_{2}(\text { at } 100 \mathrm{K}, 2.0 \mathrm{atm})\) c. 1 mole of \(\mathrm{H}_{2} \mathrm{O}(s)\) (at \(0^{\circ} \mathrm{C} )\) or 1 \(\mathrm{mole}\) of $\mathrm{H}_{2} \mathrm{O}(l)\left(\mathrm{at} 20^{\circ} \mathrm{C}\right)$

In this case, we need to compare 1 mole of \(\mathrm{H}_{2}\) at STP to 1 mole of \(\mathrm{H}_{2}\) at 100°C and 0.5 atm. Since higher temperatures and lower pressures lead to higher positional probabilities, let's compare both the conditions: 1. Temperature: At STP, the temperature is 0°C, while in the second case, it is 100°C. The second case has a higher temperature, which leads to higher positional probability. 2. Pressure: At STP, the pressure is 1 atm, while in the second case, it is 0.5 atm. The second case has lower pressure, which leads to higher positional probability. As both the temperature and pressure lead to higher positional probability in the second case, we can conclude that 1 mole of \(\mathrm{H}_{2}\) at 100°C and 0.5 atm has a larger positional probability than at STP.

Now, we need to compare 1 mole of \(\mathrm{N}_{2}\) at STP to 1 mole of \(\mathrm{N}_{2}\) at 100 K and 2.0 atm: 1. Temperature: At STP, the temperature is 273.15 K, while in the second case, it is 100 K. The first case has a higher temperature, which leads to higher positional probability. 2. Pressure: At STP, the pressure is 1 atm, while in the second case, it is 2.0 atm. The first case has lower pressure, which leads to higher positional probability. Since both the temperature and pressure lead to higher positional probability in the first case, we can conclude that 1 mole of \(\mathrm{N}_{2}\) at STP has a larger positional probability than at 100 K and 2.0 atm.

Finally, we need to compare 1 mole of \(\mathrm{H}_{2} \mathrm{O}(s)\) at 0°C to 1 mole of \(\mathrm{H}_{2} \mathrm{O}(l)\) at 20°C. In this case, we are comparing solid water (ice) to liquid water. It is important to note that the positional probability of particles in a liquid is higher than that of particles in a solid due to the increased freedom of movement in the liquid state. Since liquid water has a higher positional probability than solid water, we can conclude that 1 mole of \(\mathrm{H}_{2} \mathrm{O}(l)\) at 20°C has a larger positional probability than 1 mole of \(\mathrm{H}_{2} \mathrm{O}(s)\) at 0°C.