Heat Capacity

Heat capacity describes how much heat energy a substance must absorb or release to change its temperature by a given amount. It is central to understanding how materials store and transfer thermal energy.

Specific Heat Capacity

The heat required to change the temperature of a mass $m$ by $\Delta T$ is:

$Q = mc\Delta T$

Where:

• $Q$ is the heat energy added (J)
• $m$ is the mass (kg)
• $c$ is the specific heat capacity of the material (J/kg·K)
• $\Delta T$ is the temperature change (K or °C — they are equivalent for differences)

Common Specific Heats

Water's unusually high specific heat is why it is so effective as a coolant and why coastal climates are mild.

Molar Heat Capacity

For gases and chemical processes it is often more natural to work in moles. The molar heat capacity $C_v$ (at constant volume) gives:

$Q = nC_v\Delta T$

Where $n$ is the number of moles and $C_v$ has units of J/(mol·K).

Mayer's Relation for Ideal Gases

For an ideal gas, the molar heat capacity at constant pressure $C_p$ exceeds $C_v$ by exactly $R$, the universal gas constant:

$C_p = C_v + R$

Where $R = 8.314\,\text{J/(mol·K)}$. This extra energy accounts for the work the gas does when it expands at constant pressure.

For a monatomic ideal gas: $C_v = \frac{3}{2}R \approx 12.47\,\text{J/(mol·K)}$ and $C_p = \frac{5}{2}R \approx 20.79\,\text{J/(mol·K)}$.

Your Task

Implement the three functions below. Use $R = 8.314\,\text{J/(mol·K)}$ as a module-level constant.