Question

If \(42.0 \mathrm{~kJ}\) of heat is added to a 32.0 -g sample of liquid methane under 1 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 is gaseous, and the final temperature is \(688.05 \mathrm{K}\).

Step-by-step solution

Convert the given quantities into suitable units

We are given 42.0 kJ of heat, which we convert to J by multiplying by 1000: \(42.0 \mathrm{~kJ} \times 1000 = 42000 \mathrm{~J}\) We also convert the temperature from Celsius to Kelvin by adding 273.15: \(-170^{\circ} \mathrm{C} + 273.15 = 103.15 \mathrm{K}\) The normal boiling point of methane is given in Celsius, so we convert it to Kelvin as well: \(-161.5^{\circ} \mathrm{C} + 273.15 = 111.65 \mathrm{K}\)

Calculate the energy required to raise the temperature to the boiling point

To raise the liquid methane to the boiling point, we use the specific heat of liquid methane (3.48 J/g-K) and the mass of the sample (32.0 g) to find the required energy with the formula: \(Q = mc \Delta T\) Where: Q = heat energy (J) m = mass (g) c = specific heat capacity (J/g-K) ΔT = change in temperature (K) In this case, the change in temperature is: \(\Delta T = 111.65 \mathrm{K} - 103.15 \mathrm{K} = 8.5 \mathrm{K}\) Now we calculate the energy required: \(Q = (32.0 \mathrm{~g})(3.48 \mathrm{~J/g-K})(8.5 \mathrm{~K}) = 944.16 \mathrm{~J}\)

Compare the given heat with the energy required to evaporate the liquid

The heat energy given in the problem is 42000 J. We have calculated that 944.16 J is required to raise the temperature of the liquid methane to its boiling point. So now we can determine the remaining energy available for phase transition: \(42000 \mathrm{~J} - 944.16 \mathrm{~J} = 41055.84 \mathrm{~J}\) Now we check the energy required to vaporize the entire sample. We use the formula that relates the heat of vaporization and mass for a phase change: \(Q = mL\) Where: Q = heat energy (J) m = mass (g) L = latent heat of vaporization (J/g) For methane, the latent heat of vaporization is 8.17 J/g. Now we calculate the energy required for phase transition: \(Q = (32.0 \mathrm{~g})(8.17 \mathrm{~J/g}) = 261.44 \mathrm{~J}\)

Determine the final state and temperature of the methane

The remaining energy available for phase transition (41055.84 J) is significantly more than the energy required to vaporize the entire sample (261.44 J). Thus, the entire sample will become gaseous. Now we have to find the final temperature of the gaseous methane. We know that there will be \(41055.84 \mathrm{~J} - 261.44 \mathrm{~J} = 40794.4 \mathrm{~J}\) of energy left to increase the temperature of the gas. We use the specific heat capacity of gaseous methane (2.22 J/g-K) to find the final temperature: \(Q = mc \Delta T\) \(\Delta T = \frac{Q}{mc}\) \(\Delta T = \frac{40794.4 \mathrm{~J}}{(32.0 \mathrm{~g})(2.22 \mathrm{~J/g-K})} = 576.4 \mathrm{~K}\) Taking into account the initial temperature at boiling point: \(T_{final} = 111.65 \mathrm{K} + 576.4 \mathrm{K} = 688.05 \mathrm{K}\) Therefore, the final state of the methane is gaseous, and the final temperature is \(688.05 \mathrm{K}\).

Key concepts

Heat Energy and Temperature

Understanding the relationship between heat energy and temperature is essential when studying phase changes in chemistry. Heat energy is the total kinetic energy of the particles within a substance, and it is measured in joules (J). As this energy increases, so does the movement of the particles, which is perceived as an increase in temperature.However, temperature alone does not tell us how much heat energy is present. It simply measures the average kinetic energy of the particles. For instance, heating a substance might not always lead to a rise in temperature, especially during phase changes. When a substance reaches its melting or boiling point, additional heat energy is used to break intermolecular bonds rather than increasing temperature. This absorption of heat without a change in temperature is critical to understanding why materials change phase at specific temperatures.In our example of heating liquid methane, the added heat increases the temperature up to the boiling point. Once it reaches this point, any additional heat goes into converting the liquid into its gaseous state, which is why there's no immediate increase in temperature despite the addition of heat energy.

Specific Heat Capacity

The specific heat capacity is another pivotal concept in understanding how substances absorb and release heat. It indicates how much heat energy is required to raise the temperature of one gram of a substance by one degree Celsius (or Kelvin). Its units are joules per gram per degree Celsius \( J/g-\degree C \).Take for instance the specific heat capacities of liquid and gaseous methane from our exercise, 3.48 and 2.22 \( J/g-K \), respectively. These values mean that on a per gram basis, the liquid form requires more energy to increase its temperature compared to its gaseous form. We applied this information to calculate the amount of energy needed to heat the liquid methane to its boiling point, and similarly, we would use the specific heat capacity of gaseous methane to determine how much more energy would raise its temperature once it has vaporized.

Why Does Specific Heat Capacity Matter?

In everyday life, materials with high specific heat capacities are adept at storing heat, which makes them suitable for applications like storing solar energy or in heat packs. The different capacities for liquid and gaseous methane also indicate the energy efficiency aspects in cryogenics and fuel technologies.

Latent Heat of Vaporization

The latent heat of vaporization is the heat absorbed during the transformation of a liquid into a gas at constant temperature and pressure. When a substance changes from one phase to another, latent heat is involved without any change in temperature. For a specific material, the latent heat is expressed in joules per gram \( J/g \).The example problem provided demonstrates how, after heating liquid methane to its boiling point, additional heat is used to vaporize it. This process is quantified using the latent heat of vaporization for methane, which is 8.17 \( J/g \). Once the liquid methane has absorbed sufficient energy to surpass this threshold, it turns into gas – this is the 'latent' period where temperature doesn't increase, despite the addition of heat energy.

Importance of Latent Heat

Understanding latent heat is crucial for practical applications such as refrigeration, where the transfer of heat is fundamental to the cooling process, or meteorology, to comprehend the energy involved in the formation of clouds and precipitation. The concept also has implications in designing energy systems, where the energy inputs and outputs must be accurately measured during the phase change processes.