Question

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\(]\)

Short answer

The final state of the methane after the system equilibrates is liquid with a temperature of approximately \(-163.6^{\circ}C\).

Step-by-step solution

Calculate the heat needed to raise the temperature to boiling point

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: \(q_{heat} = m \cdot c_l \cdot (T_{bp} - T_i)\)

Check if the added heat is enough to change the phase

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.

Calculate the final temperature or vaporization heat

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, \(q_{vaporization} = m \times L_v\)

Determine the final state and temperature of the methane

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, \(T_f = T_{bp} + \frac{q_{remaining}}{m \cdot c_g}\)

Key concepts

Specific Heat Capacity

Understanding how substances absorb heat is crucial in chemistry. Specific heat capacity is a property that allows us to calculate how much heat energy is needed to change the temperature of a substance. It is defined as the amount of heat per unit mass required to raise the temperature by one degree Celsius (or one Kelvin). For instance, the specific heat capacity of liquid methane is given as \(3.48 \frac{J}{g \cdot K}\). This means that each gram of methane requires \(3.48\) joules of heat to increase its temperature by one degree Kelvin.

When calculating the heat needed to reach a certain temperature, the specific heat formula \(q = m \cdot c \cdot \Delta T\) is used, where \(q\) is the heat added, \(m\) is the mass, \(c\) is the specific heat, and \(\Delta T\) is the temperature change. Understanding this helps in predicting how substances behave with heat changes, essential for determining energy requirements in various chemical processes.

Boiling Point

The boiling point of a substance is the temperature at which it changes from liquid to gas. For methane, this is \(-161.5^{\circ} \mathrm{C}\). At this point, the pressure of the gas escaping the liquid is equal to the atmospheric pressure.

When analyzing a problem involving temperature change up to a boiling point, it’s important to calculate if enough heat is added to reach this state. If the added heat does not meet the requirement to reach the boiling point from an initial temperature, the substance remains in the liquid state. The calculation involves finding the heat needed to change the temperature to the boiling point using the specific heat capacity of the liquid form. This exercise illustrates the principle by requiring learners to see if \(42.0 \mathrm{kJ}\) is sufficient to bring methane to its boiling point from \(-170^{\circ} \mathrm{C}\).

• Key factor: Energy needed depends on mass, specific heat, and temperature change.
• Checking the sufficiency of added heat helps determine the final state.

Vaporization Heat

Vaporization heat, or enthalpy of vaporization, is the heat required to turn a liquid into a gas at its boiling point without an increase in temperature. This is an essential factor in phase transitions. For methane, the vaporization heat is given as \(8.17 \times 10^2 \ \frac{J}{g}\).

To determine if the methane fully vaporizes, calculate the total energy needed for this process. Use the mass of the substance and its vaporization heat in the formula \(q_{vaporization} = m \times L_v\). If the total added heat exceeds both the heat to reach the boiling point and the vaporization heat, then full vaporization and a gaseous final state occur. If insufficient, only a portion of the substance will vaporize, and the temperature can be calculated for any remaining liquid or gas using appropriate specific heats.

• Vaporization represents the phase change energy requirement.
• Calculate using the formula: mass times heat of vaporization.