Question

If \(42.0 \mathrm{~kJ}\) of heat is added to a \(32.0-\mathrm{g}\) sample of liquid methane under \(101.3 \mathrm{kPa}\) 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 methane is gaseous at 199.76°C after equilibrium.

Step-by-step solution

Calculate the Heat Required for Phase Change

First, determine the amount of heat required to convert the liquid methane to gas at its boiling point. The heat required can be calculated using the formula for heat of vaporization.To find the heat of vaporization, refer to methane's properties, where the enthalpy change is given as 510 J/g. Calculate the total heat required for vaporization: \[ q = ext{mass} \times ext{heat of vaporization} = 32.0 ext{ g} \times 510 \frac{J}{g} = 16,320 \text{ J} (or 16.32 kJ) \]

Check Available Heat Post Vaporization

From the total energy of 42.0 kJ provided, subtract the heat used for vaporizing the methane to see the heat remaining: \[ ext{Remaining Heat} = 42.0 ext{ kJ} - 16.32 ext{ kJ} = 25.68 ext{ kJ} \]

Calculate Temperature Change for Gaseous Methane

Using the remaining energy, calculate the temperature change for the now-gaseous methane. Use the specific heat capacity of gaseous methane to find the temperature increase:\[ q = m \cdot c \cdot \Delta T \quad \Rightarrow \quad \Delta T = \frac{q}{m \cdot c}\]where \(q=25.68 \text{ kJ} = 25680 \text{ J} \), \(m = 32.0 \text{ g}\), and \(c = 2.22 \text{ J/g-K}\):\[ \Delta T = \frac{25680}{32.0 \times 2.22} = 361.26 \text{ K}\]

Determine Final Temperature

Add the temperature increase to the boiling point to find the final temperature:\[ T_{final} = -161.5^{\circ}C + 361.26 \text{ K} = 199.76^{\circ}C \]Thus, the final temperature of the methane when equilibrium is reached is 199.76°C, with methane in a gaseous state.

Key concepts

Understanding Phase Change

When a substance transitions from one state of matter to another, it undergoes a phase change. In thermochemistry, the most common phase changes include melting, freezing, vaporization, condensation, sublimation, and deposition. Each phase change involves an energy exchange that does not change the temperature of the substance until the entire sample has transitioned into the new phase.
For example, vaporization is the phase change from liquid to gas, which occurs without any temperature change until all the liquid is converted. This requires adding energy, often in the form of heat, to break intermolecular forces. In the exercise, methane moves from liquid to gas at its boiling point, requiring energy input to achieve this transition.

Heat of Vaporization

The heat of vaporization is the amount of energy required to convert a liquid into a gas at its boiling point without increasing its temperature. This is a crucial property in phase change calculations as it indicates the energy needed to overcome intermolecular attractions in a liquid.
In the context of the exercise, methane's heat of vaporization is given as 510 J/g. This tells us that every gram of liquid methane requires 510 Joules of heat to become gas at the boiling point. Understanding this helps calculate the energy needed for the phase transition, ensuring accurate thermochemical evaluations.

The Role of Specific Heat Capacity

Specific heat capacity is a material's heat retention ability. It is defined as the energy required to increase the temperature of one gram of a substance by one degree Celsius (or Kelvin).
In the exercise, both liquid and gaseous methane are given specific heat values: 3.48 J/g-K for liquid and 2.22 J/g-K for gas. These values help predict how much energy is required to change the temperature of the substance once it is no longer undergoing a phase change. Knowing these values can allow us to accurately calculate temperature changes post phase transition in thermochemical assessments.

Calculating Temperature Change

Temperature change is calculated when a substance absorbs heat, aside from during a phase change. The formula used is based on the concept of specific heat capacity and is given by:

• \( q = m \cdot c \cdot \Delta T \)

where \( q \) is the amount of heat added, \( m \) is the mass, \( c \) is the specific heat capacity, and \( \Delta T \) is the temperature change.
From the problem, after the methane vaporizes, we need to know how much the temperature can increase using remaining heat energy. The specific heat capacity of gaseous methane allows us to determine the temperature rise post phase change, providing insights into the final state of the substance.

Properties of Gaseous Methane

Methane, a simple hydrocarbon, typically exists as a gas at room temperature. Gaseous methane has properties that are important to grasp when performing thermochemical calculations.

• Low Specific Heat Capacity: Gaseous methane's specific heat capacity (2.22 J/g-K) indicates that it requires relatively little energy to increase its temperature.
• Vaporization Requirement: Methane vaporizes at its boiling point of -161.5°C under standard pressure, and this transition needs significant energy input.

The exercise illustrates these aspects, showing the behavior of methane as it transitions from a liquid to a gas and subsequently checks how the gaseous form's temperature rises with additional heat.