The following information is for mercury: specific heat capacity (solid) s.hc (liquid) = 139.5 J/(kg·K), s.h.c (vapor) = 104.6 J/(kg·K), AHfus = 2.29 kJ/mol, AHvap = 59.2 kJ/mol, melting point: 38.9°C, boiling point: 356.6°C. 25.0 g of Hg is initially at -50.08°C. Calculate the heat when the solid is converted to liquid at 25.09°C. The liquid is warmed from 25.09°C to 100.0°C. The sample is warmed from liquid at 100.0°C to vapor at 356.6°C.
The Correct Answer and Explanation is:
To calculate the total heat required to convert 25.0 g of mercury (Hg) from solid at –50.08°C to vapor at 356.6°C, we break the process into five steps:
Step-by-step Calculations:
Given:
- Mass = 25.0 g = 0.0250 kg
- Specific heat capacity (solid) = assumed to be same as liquid = 139.5 J/(kg·K)
- Specific heat capacity (liquid) = 139.5 J/(kg·K)
- Specific heat capacity (vapor) = 104.6 J/(kg·K)
- ΔH_fus = 2.29 kJ/mol = 2290 J/mol
- ΔH_vap = 59.2 kJ/mol = 59200 J/mol
- Melting point = 38.9°C
- Boiling point = 356.6°C
- Molar mass of Hg = 200.59 g/mol
Step 1: Heating solid from –50.08°C to 38.9°C
ΔT = 38.9 – (–50.08) = 88.98°C
q₁ = m × c × ΔT
q₁ = 0.0250 kg × 139.5 J/(kg·K) × 88.98 K = 310.0 J
Step 2: Melting at 38.9°C
n = 25.0 g / 200.59 g/mol = 0.1247 mol
q₂ = n × ΔH_fus = 0.1247 mol × 2290 J/mol = 285.6 J
Step 3: Heating liquid from 38.9°C to 100.0°C
ΔT = 100.0 – 38.9 = 61.1°C
q₃ = 0.0250 kg × 139.5 J/(kg·K) × 61.1 K = 2130.6 J
Step 4: Heating liquid from 100.0°C to 356.6°C
ΔT = 356.6 – 100.0 = 256.6°C
q₄ = 0.0250 kg × 139.5 J/(kg·K) × 256.6 K = 8942.9 J
Step 5: Vaporization at 356.6°C
q₅ = n × ΔH_vap = 0.1247 mol × 59200 J/mol = 7382.2 J
Total Heat (q_total) = q₁ + q₂ + q₃ + q₄ + q₅
= 310.0 + 285.6 + 2130.6 + 8942.9 + 7382.2 = 19051.3 J
Final Answer: 19.1 kJ (rounded to 3 significant figures)
Explanation
To determine the total heat required to convert 25.0 g of mercury from a solid at –50.08°C to a vapor at 356.6°C, we analyze each phase and temperature transition separately. The process involves heating the solid, melting, heating the liquid, vaporizing, and then possibly heating the vapor. However, in this case, the final temperature is exactly the boiling point, so the final phase change directly ends with vaporization.
In the first step, we calculate the energy needed to warm the solid mercury to its melting point using the specific heat formula. While mercury’s solid-phase heat capacity is not directly provided, it is reasonably approximated by the liquid value because the difference is minimal near the melting point.
Next, we calculate the enthalpy of fusion, which is the energy required to convert solid mercury to liquid at its melting point. This depends on the number of moles and the molar enthalpy of fusion.
Then, we heat the liquid mercury from its melting point to 100.0°C, and again from 100.0°C to the boiling point at 356.6°C. These two stages use the specific heat formula again, with the liquid specific heat capacity.
Finally, we determine the energy required for vaporization at the boiling point using the enthalpy of vaporization and the number of moles.
Each step contributes to the total energy, and when added together, gives the total heat absorbed: approximately 19.1 kJ. This calculation demonstrates the importance of understanding phase changes and specific heat in thermodynamics, particularly when a substance undergoes multiple physical transformations across a wide temperature range.
