If 42.0 kJ of heat is added to a 32.0-g sample of liquid meth-ane under 1 atm of pressure at a temperature of -170°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 °C. The specific heats of liquid and gaseous methane are 3.48 and 2.22 J/g-K, respectively. [Section 11.4]

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Convert the heat added from kJ to J by multiplying by 1000, since 1 kJ = 1000 J.

Calculate the amount of heat required to raise the temperature of the liquid methane from -170°C to its boiling point at -161.5°C using the formula q = m * c * ΔT, where m is the mass, c is the specific heat of liquid methane, and ΔT is the change in temperature.

Determine the heat required for the phase change from liquid to gas at the boiling point using the heat of vaporization (if provided) or by considering the energy balance.

Calculate the remaining heat after the phase change and use it to raise the temperature of the gaseous methane using the specific heat of gaseous methane and the formula q = m * c * ΔT.

Determine the final temperature of the methane by adding the temperature change of the gaseous methane to the boiling point temperature.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Heat Transfer and Temperature Change

Heat transfer involves the movement of thermal energy from one object to another, which can change the temperature of a substance. The relationship between heat added (q), mass (m), specific heat capacity (c), and temperature change (ΔT) is given by the equation q = mcΔT. Understanding this concept is crucial for calculating how the temperature of the methane will change when heat is added.

Phase Changes and Boiling Point

Phase changes occur when a substance transitions between solid, liquid, and gas states, often involving significant energy changes. The boiling point is the temperature at which a liquid turns into vapor, and for methane, this occurs at -161.5 °C. Recognizing that the added heat may cause the methane to reach its boiling point and potentially vaporize is essential for determining its final state.

Specific Heat Capacities

Specific heat capacity is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius (or Kelvin). Different phases of a substance have different specific heat capacities; for methane, the specific heats are 3.48 J/g-K for liquid and 2.22 J/g-K for gas. This concept is vital for calculating the temperature change of methane in both its liquid and gaseous states after heat is added.

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