# Heat Capacity and Specific Heat **Heat capacity** is a physical property of matter that quantifies the amount of heat required to change a substance's temperature by a given amount. It is a fundamental concept in thermodynamics for understanding how different materials store and transfer thermal energy. ## Heat Capacity ($C$) Heat capacity is the amount of heat energy required to raise the temperature of an entire object or substance by one degree (1°C or 1 K). - **Relationship:** The heat supplied ($\Delta Q$) is directly proportional to the change in temperature ($\Delta T$). - **Formula:** $$ C = \frac{\Delta Q}{\Delta T} $$ - **Unit:** Joules per Kelvin (J/K). - **Note:** Heat capacity is an *extensive* property, meaning it depends on the amount of the substance. ## Specific Heat Capacity ($c$) Specific heat capacity is a more precise measure that accounts for the mass of the substance. It is the amount of heat required to raise the temperature of **one unit of mass** (e.g., 1 kg) of a substance by one degree. - **Formula:** The heat supplied is proportional to both mass ($m$) and temperature change ($\Delta T$). $$ \Delta Q = c m \Delta T \quad \implies \quad c = \frac{\Delta Q}{m \Delta T} $$ - **Unit:** Joules per kilogram per Kelvin (J·kg⁻¹·K⁻¹). - **Note:** Specific heat capacity is an *intensive* property, meaning it is an intrinsic characteristic of a material, regardless of its mass. ## Molar Specific Heat Capacity ($C_m$) For substances where it is more convenient to work with moles (especially gases), we use molar specific heat capacity. It is the amount of heat required to raise the temperature of **one mole** of a substance by one degree. - **Formula:** $$ \Delta Q = C_m n \Delta T \quad \implies \quad C_m = \frac{\Delta Q}{n \Delta T} $$ - **Unit:** Joules per mole per Kelvin (J·mol⁻¹·K⁻¹). ## Molar Specific Heats of Gases ($C_p$ and $C_v$) For gases, the amount of heat required for a temperature change depends on the conditions under which the heat is added. The two most important cases are constant volume and constant pressure. ### Molar Specific Heat at Constant Volume ($C_v$) This is the heat required to raise the temperature of **one mole of a gas by 1 K** while keeping its **volume constant**. - **First Law Application:** In a constant volume process, the gas does no work ($W = P\Delta V = 0$). Therefore, all the heat added goes directly into increasing the internal energy of the gas. $$ \Delta Q_v = \Delta U \quad \implies \quad nC_v\Delta T = \Delta U $$ ### Molar Specific Heat at Constant Pressure ($C_p$) This is the heat required to raise the temperature of **one mole of a gas by 1 K** while keeping its **pressure constant**. - **First Law Application:** In a constant pressure process, the gas expands and does work on its surroundings. The heat added must therefore be used for two purposes: increasing the internal energy *and* doing work. $$ \Delta Q_p = \Delta U + W $$ ### Why is $C_p$ Always Greater than $C_v$? - At **constant volume**, all the heat supplied is used to increase the internal energy (and thus the temperature). - At **constant pressure**, the heat must not only increase the internal energy by the same amount for the same temperature rise but must *also* provide the energy for the gas to do expansion work. - Therefore, **more heat is required at constant pressure** to achieve the same temperature change. This means **$C_p > C_v$**. | Process | Heat Supplied ($\Delta Q$) | Change in Internal Energy ($\Delta U$) | Work Done ($W$) | | :--- | :--- | :--- | :--- | | **Constant Volume** | $\Delta Q_v = nC_v\Delta T$ | $\Delta U$ | $0$ | | **Constant Pressure** | $\Delta Q_p = nC_p\Delta T$ | $\Delta U$ (same as above for same $\Delta T$) | $P\Delta V$ | ## The Relation: $C_p - C_v = R$ (Mayer's Relation) For an ideal gas, the difference between the molar specific heats at constant pressure and constant volume is equal to the **universal gas constant ($R$)**. **Derivation:** 1. Start with the First Law for a constant pressure process: $\Delta Q_p = \Delta U + W$. 2. Substitute the known expressions: $\Delta Q_p = nC_p\Delta T$ and $W = P\Delta V$. 3. The change in internal energy ($\Delta U$) depends only on the change in temperature, so we can use the expression from the constant volume case: $\Delta U = nC_v\Delta T$. 4. The equation becomes: $nC_p\Delta T = nC_v\Delta T + P\Delta V$. 5. From the Ideal Gas Law ($PV = nRT$), a change in volume at constant pressure gives $P\Delta V = nR\Delta T$. 6. Substitute this in: $nC_p\Delta T = nC_v\Delta T + nR\Delta T$. 7. Divide the entire equation by $n\Delta T$: $$ C_p = C_v + R \quad \implies \quad C_p - C_v = R $$ The value of $R$ is approximately 8.314 J·mol⁻¹·K⁻¹. ## Possible Questions and Answers **Q:** What does the relationship $C_p - C_v = R$ physically mean? **A:** It means that the extra heat required to raise the temperature of one mole of an ideal gas by 1 K at constant pressure (compared to constant volume) is exactly the amount of energy needed for the gas to do expansion work. This work is equal to the universal gas constant, $R$. **Q:** Is $C_p > C_v$ true for liquids and solids? **A:** Yes, the principle holds, but the difference is much smaller. Liquids and solids expand very little when heated, so the amount of work done is almost negligible. For this reason, we usually only refer to a single specific heat capacity for solids and liquids.