Heat Capacity and Enthalpy of Heating Solids & Liquids

Welcome back! In our last session, we formulated the general steady-state energy balance, $q=\Delta H$, and saw that the core of the problem lies in determining the total enthalpy ($H$) of each substance entering and leaving the reactor. We left the molar enthalpy, $H(T)$, as a value you could simply look up in a database.

Today, we will open that black box. This lesson is dedicated to calculating the enthalpy of a substance at the high temperatures typical of metallurgical processes. This is the essential skill needed to turn our abstract energy balance equation into a concrete quantitative model for predicting energy costs or process temperatures.

Specifically, this lesson covers the learning outcome: Calculate heat capacities and enthalpy changes for heating solids and liquids to high temperatures. We will break this down into two key components:

• Sensible Heat: The energy absorbed by a substance as its temperature increases.
• Latent Heat: The energy absorbed during a phase change (like melting) at a constant temperature.

By the end of this lesson, you'll be able to calculate the total energy required to take a substance from room temperature, melt it, and bring it to a final high temperature, which is a fundamental calculation in pyrometallurgy.

1. Defining Enthalpy at High Temperatures

The absolute enthalpy of a substance is not a practical quantity. Instead, we always work with enthalpy changes relative to a universally agreed-upon reference state. In thermodynamics, this standard state is defined as the most stable form of the pure elements at a pressure of 1 bar and a temperature of 298.15 K (25 °C).

The total molar enthalpy of a compound at a given temperature $T$, denoted $H_T$, can be expressed as the sum of two components:

1. Standard Enthalpy of Formation ($\Delta H_{f,298}^{\circ }$): The enthalpy change when one mole of the compound is formed from its constituent elements in their standard states. This is the chemical energy inherent in the substance.
2. Enthalpy Increment ($H_T-H_{298}$): The change in enthalpy from heating the substance from the reference temperature (298 K) to the target temperature $T$. This is the thermal energy you've added.

So, the full expression is:
$H_T=\Delta H_{f,298}^{\circ }+(H_T-H_{298})$

We will cover the Standard Enthalpy of Formation ($\Delta H_{f,298}^{\circ }$) in a future lesson when we discuss the thermodynamics of chemical reactions. For today, we will focus entirely on calculating the enthalpy increment, $(H_T-H_{298})$. This term consists of sensible and latent heat.

2. Sensible Heat and Heat Capacity

When you heat a substance without changing its phase (e.g., heating solid iron from 25°C to 1000°C), it absorbs "sensible heat," and its temperature rises. The property that governs this relationship is the heat capacity.

The molar heat capacity at constant pressure ($C_p$) is the amount of heat energy required to raise the temperature of one mole of a substance by one degree (e.g., in Joules per mole-Kelvin, J/mol·K).

The following video provides a great introduction to the basic concept, which you may remember from your earlier science education.

As the video explains, for a small temperature change or when heat capacity can be assumed constant, the heat absorbed $q$ is:
$q=n{C}_p\Delta T$
where $n$ is the number of moles and $\Delta T$ is the change in temperature. Since we are dealing with constant pressure processes, this heat absorbed is equal to the change in enthalpy:
$\Delta H=n{C}_p\Delta T$

Temperature-Dependent Heat Capacity

For the wide temperature ranges in pyrometallurgy, assuming $C_p$ is constant is often not accurate enough. The heat capacity of a substance changes with temperature. This relationship is typically captured by an empirical polynomial equation. A common form, known as the Kelley equation, is:

$C_p(T)=a+bT+c{T}^{-2}$

Here, $a$, $b$, and $c$ are experimentally determined constants specific to each substance, and $T$ is the absolute temperature in Kelvin.

To find the enthalpy change when heating a substance from $T_1$ to $T_2$, we must integrate the heat capacity function over that temperature range. Your calculus background will make this straightforward.

$\Delta H=H_{T_2}-H_{T_1}={\int }_{T_1}^{T_2}{C}_p(T)dT={\int }_{T_1}^{T_2}(a+bT+c{T}^{-2})dT$

Solving the integral gives:
$H_{T_2}-H_{T_1}={\left[aT+\frac{b}{2}{T}^2-\frac{c}{T}\right]}_{T_1}^{T_2}$
$=a(T_2-T_1)+\frac{b}{2}(T_2^2-T_1^2)-c\left(\frac{1}{T_2}-\frac{1}{T_1}\right)$

This equation is fundamental for accurate high-temperature energy calculations. The following resource shows this exact formulation in the context of metallurgy.

3. Latent Heat: The Energy of Phase Changes

What happens when a substance reaches its melting or boiling point? As you continue to add heat, the temperature stops rising until the entire substance has changed phase (e.g., from solid to liquid). The energy absorbed during this process is called latent heat.

• Latent Heat of Fusion ($\Delta H_m$): Energy required to melt one mole of a solid at its melting point.
• Latent Heat of Vaporization ($\Delta H_v$): Energy required to vaporize one mole of a liquid at its boiling point.

This energy is used to overcome the intermolecular forces holding the substance in a solid or liquid state, increasing the potential energy of the molecules rather than their kinetic energy (temperature).

The "heating curve" is the classic way to visualize this.

4. Putting It All Together: The Full Calculation

To calculate the total enthalpy change to heat a substance across a phase transition, we simply sum the sensible and latent heat terms.

For example, to calculate the enthalpy required to heat one mole of a substance from a solid at $T_1$ (below the melting point, $T_m$) to a liquid at $T_2$ (above the melting point), you follow these three steps:

1. Heat the solid: Calculate the sensible heat to bring the solid from $T_1$ to the melting point $T_m$.
   $\Delta H_1={\int }_{T_1}^{T_m}{C}_{p,solid}(T)dT$
2. Melt the solid: Add the latent heat of fusion at $T_m$.
   $\Delta H_2=\Delta H_m$
3. Heat the liquid: Calculate the sensible heat to bring the liquid from $T_m$ to the final temperature $T_2$. Note that the heat capacity of the liquid, $C_{p,liquid}$, is different from that of the solid.
   $\Delta H_3={\int }_{T_m}^{T_2}{C}_{p,liquid}(T)dT$

The total enthalpy increment is the sum of these parts:
$(H_{T_2}-H_{T_1})=\Delta H_1+\Delta H_2+\Delta H_3$

The "Sensible Heat and Enthalpy Calculations" slides provide a perfect worked example for zinc.

Conclusion

In this lesson, we have demystified the calculation of enthalpy at high temperatures, a critical input for the energy balance models we discussed previously. You now have a complete method for quantifying the thermal energy of materials in a process.

• Key Takeaways:
  • The change in enthalpy from heating, $(H_T-H_{298})$, is the sum of sensible heat (from temperature changes) and latent heat (from phase changes).
  • Sensible heat is calculated by integrating the molar heat capacity, $C_p(T)$, over the temperature range: $\Delta H=\int C_p(T)dT$. For high-temperature processes, $C_p$ is itself a function of temperature.
  • Latent heat ($\Delta H_m$, $\Delta H_v$) is the energy absorbed at a constant temperature during a phase transition.
  • The total enthalpy change is found by breaking the process into steps and summing the $\Delta H$ for each step.

You can now calculate one of the two key components of a substance's total enthalpy. We now have the tools to calculate the $H(T)$ for all the non-reacting components in our mass and energy balance.

The next piece of the puzzle is to account for the energy released or consumed by chemical reactions themselves. In the next lesson, "Apply Gibbs free energy to determine the thermodynamic feasibility of metal reduction reactions," we will introduce the concepts of standard enthalpy of reaction and entropy. Combining these will allow us to calculate the Gibbs free energy change, $\Delta G$, which is the ultimate arbiter of whether a reaction is thermodynamically possible.