Draw a heating curve (such as the one in Figure 11.36) for 1 mole of methanol beginning at 170 K and ending at 350 K. Assume that the values given here are constant over the relevant temperature ranges. Melting point: 176 K, Boiling point: 338 K, ΔH_fus: 2.2 kJ/mol, ΔH_vap: 35.2 kJ/mol, C_s,solid: 105 J/mol·K, C_s,liquid: 81.3 J/mol·K, C_s,gas: 48 J/mol·K.

Verified step by step guidance

1. Identify the different phases and phase transitions of methanol between 170 K and 350 K: solid, liquid, and gas phases, with melting at 176 K and boiling at 338 K.
2. Calculate the energy required to heat the solid methanol from 170 K to its melting point at 176 K using the formula: \( q = m \cdot C_s,solid \cdot \Delta T \), where \( m \) is the number of moles, \( C_s,solid \) is the specific heat capacity of solid methanol, and \( \Delta T \) is the temperature change.
3. Determine the energy needed for the phase transition from solid to liquid at the melting point using the formula: \( q = m \cdot \Delta H_{fus} \), where \( \Delta H_{fus} \) is the enthalpy of fusion.
4. Calculate the energy required to heat the liquid methanol from 176 K to its boiling point at 338 K using the formula: \( q = m \cdot C_s,liquid \cdot \Delta T \), where \( C_s,liquid \) is the specific heat capacity of liquid methanol.
5. Determine the energy needed for the phase transition from liquid to gas at the boiling point using the formula: \( q = m \cdot \Delta H_{vap} \), where \( \Delta H_{vap} \) is the enthalpy of vaporization.

Key Concepts

Heating Curve

A heating curve is a graphical representation that shows the temperature change of a substance as it is heated over time. It illustrates phase changes (solid, liquid, gas) and the corresponding temperature plateaus where the substance undergoes melting or boiling. The curve typically consists of sloped lines representing temperature increases in each phase and flat lines indicating phase transitions where temperature remains constant.

Phase Changes

Phase changes refer to the transitions between solid, liquid, and gas states of matter. In the context of the heating curve, these changes occur at specific temperatures: melting occurs at the melting point, and boiling occurs at the boiling point. During these transitions, energy is absorbed or released without a change in temperature, which is quantified by the enthalpy of fusion (ΔH_fus) and enthalpy of vaporization (ΔH_vap).

Specific Heat Capacity

Specific heat capacity is the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius (or Kelvin). Different phases of a substance have different specific heat capacities, which affect how much energy is needed to change the temperature of the substance in each phase. For methanol, the specific heat capacities for solid, liquid, and gas phases are given, and these values are crucial for calculating the energy changes during heating.