Problem 3: Available degrees of freedom and heat capacity As you saw in group problems, the heat capacity per molecule of a i ii: diatomic gas smoothly transitions between three different regions: E M o with only translational degrees of freedom active, 2 2 . o with both translational and rotational degrees active, ii\" 0 with translational, rotational, and vibrational all active. g For this problem assume the transition to be very sudden, as in the graph. in} 10 50 100 800 A. First, recall that heat capacity is dened as the amount of heat Temperature (K) required to change the temperature of an object by a certain amount: C = 51, => Q = CAT. Find the heat necessary to increase 1 mol of this diatomic gas from 10 K to 50 K. B. Now suppose we mix two 1 mol samples of this ideal gas: MIN] 0 A cool sample at 50 K. o A hot sample at some temperature TH between 100 K and 800 K. The mixture eventually reaches thermal equilibrimn at 115 K. H|U1 * 1. On the graph shown, shade the area corresponding to the heat lost by the hot sample. In a different color or pattern, shade the area corresponding to the heat gained by the cool sample. 50 100 115 T 300 H Temperature (K) Heat Capacity per molecule le er 2. Find the area of the cool sample, and that of the hot sample in terms of TH. Use your results to solve for the initial temperature of the hot sample. MN C. The previous analysis is an oversimplication because the change in heat capacity is actually gradual. A more accurate graph is shown here. Mm Briey explain how you could use a similar method, with this gradual graph. Shade the two areas you would use to nd the temperature both the warmer and cooler samples (you do not need to calculate anything). Heal Capadty per molecule lulu h" 50 100 11 5 T 800 H Temperature (K)