If 42.0 \(\mathrm{kJ}\) of heat is added to a \(32.0-\mathrm{g}\) sample of liquid methane under 1 \(\mathrm{atm}\) of pressure at a temperature of \(-170^{\circ} \mathrm{C}\) , what are the final state and temperature of the methane once the system equilibrates? Assume no heat is lost to the surroundings. The normal boiling point of methane is \(-161.5^{\circ} \mathrm{C}\) The specific heats of liquid and gaseous methane are 3.48 and \(2.22 \mathrm{J} / \mathrm{g}-\mathrm{K}\) , respectively. [ Section 11.4\(]\)

First, we need to find the amount of heat needed to raise the temperature of the liquid methane to its boiling point. We will use the specific heat of liquid methane and the temperature difference. We have: Specific heat of liquid methane, \(c_l = 3.48 \frac{J}{g \cdot K}\) Mass of methane, \(m = 32.0g\) Initial temperature, \(T_i = -170 \ °C\) Boiling point of methane, \(T_{bp} = -161.5 \ °C\) The heat needed, \(q_{heat}\), can be calculated by using the formula: \(q_{heat} = m \cdot c_l \cdot (T_{bp} - T_i)\)

Now, we will compare the heat needed to raise the temperature to the boiling point with the amount of heat added. The given heat added is: \(q_{added} = 42.0 kJ = 42000 J\) We will check if \(q_{added} >= q_{heat}\) or not.

If the added heat is not enough to reach the boiling point, we will calculate the final temperature using the formula: \(T_f = T_i + \frac{q_{added}}{m \cdot c_l}\) If the added heat is enough to reach the boiling point and change the phase, we need to calculate the amount of heat required for complete vaporization: Heat of vaporization of methane, \(L_v = 8.17 \times 10^2 \ \frac{J}{g}\) Vaporization heat needed, \(q_{vaporization} = m \times L_v\)

Based on the calculations in Step 3, we will determine if the final state is liquid or gaseous. If the heat added is not enough to vaporize the methane, the final state will be liquid and the final temperature can be calculated using the formula in Step 3. If the heat added is enough to vaporize the methane completely, then the final state is gaseous, and we will need to find the final temperature using the specific heat of gaseous methane and the remaining added heat after vaporization: Specific heat of gaseous methane, \(c_g = 2.22 \frac{J}{g \cdot K}\) Remaining heat after vaporization, \(q_{remaining} = q_{added} - q_{vaporization}\) Final temperature, \(T_f = T_{bp} + \frac{q_{remaining}}{m \cdot c_g}\)