Heat Capacities — Revision Notes

⚡ 30-Second Revision

Heat Capacity (C) — $C = \frac{Q}{\Delta T}$ (J/K). Extensive property.
Specific Heat Capacity (c) — $c = \frac{Q}{m\Delta T}$ (J/(kg·K)). Intensive property.
Molar Heat Capacity ($C_m$) — $C_m = \frac{Q}{n\Delta T}$ (J/(mol·K)). Intensive property.
Mayer's Relation (Ideal Gas) — $C_P - C_V = R$.
Degrees of Freedom (f)

* Monoatomic: $f=3$ (translational) * Diatomic: $f=5$ (3 translational + 2 rotational, at moderate T) * Polyatomic (non-linear): $f=6$ (3 translational + 3 rotational)

Molar Heat Capacities from f (Ideal Gas)

* $C_V = \frac{f}{2}R$ * $C_P = (\frac{f}{2}+1)R$

Ratio of Specific Heats (Ideal Gas) — $\gamma = \frac{C_P}{C_V} = 1 + \frac{2}{f}$.
Heat Transfer — $Q_V = nC_V\Delta T$ (constant volume), $Q_P = nC_P\Delta T$ (constant pressure).
Internal Energy Change — $\Delta U = nC_V\Delta T$ (for ideal gas, any process).

2-Minute Revision

Heat capacity is a measure of how much heat energy a substance absorbs for a given temperature change. It comes in three main forms: general heat capacity (C, for a specific object), specific heat capacity (c, per unit mass), and molar heat capacity ($C_m$, per unit mole).

For gases, the molar heat capacity depends on the process. At constant volume, it's $C_V$, where all heat goes to internal energy. At constant pressure, it's $C_P$, where some heat also does work, making $C_P > C_V$.

Mayer's relation, $C_P - C_V = R$, quantifies this difference for ideal gases. The values of $C_V$ and $C_P$ for ideal gases are determined by their degrees of freedom ($f$) through the equipartition theorem: $C_V = \frac{f}{2}R$ and $C_P = (\frac{f}{2}+1)R$.

Monoatomic gases have $f=3$, diatomic $f=5$, and non-linear polyatomic $f=6$. The ratio $\gamma = C_P/C_V = 1 + \frac{2}{f}$ is crucial for adiabatic processes. Remember that internal energy change for an ideal gas is always $\Delta U = nC_V\Delta T$, regardless of the process.

5-Minute Revision

A thorough understanding of heat capacities is vital for NEET. Start by solidifying the definitions: Heat capacity (C) is for a body, specific heat capacity (c) is per unit mass, and **molar heat capacity ($C_m$)** is per unit mole. Their units are J/K, J/(kg·K), and J/(mol·K) respectively. For solids and liquids, $Q = mc\Delta T$ is the primary formula for heat transfer.

For ideal gases, the concept becomes more nuanced due to the possibility of work done. We distinguish between **molar heat capacity at constant volume ($C_V$) and at constant pressure ($C_P$)**.

When volume is constant, no P-V work is done, so all heat supplied ($Q_V$) increases internal energy ($\Delta U$). Thus, $Q_V = nC_V\Delta T = \Delta U$. When pressure is constant, the gas expands and does work ($W = P\Delta V$).

So, $Q_P = nC_P\Delta T = \Delta U + W$. This means $C_P$ is always greater than $C_V$, and for ideal gases, Mayer's relation states $C_P - C_V = R$, where $R$ is the universal gas constant ($8.314 \, \text{J/(mol·K)}$).

Crucially, the values of $C_V$ and $C_P$ for ideal gases depend on the degrees of freedom (f) of the gas molecules, which are the independent ways a molecule can store energy (translational, rotational, vibrational). According to the equipartition theorem:

Monoatomic gas — (e.g., He, Ne): $f=3$ (3 translational). So, $C_V = \frac{3}{2}R$, $C_P = \frac{5}{2}R$, and $\gamma = \frac{5}{3} \approx 1.67$.
Diatomic gas — (e.g., O$_2$, N$_2$): $f=5$ (3 translational + 2 rotational, at moderate temperatures). So, $C_V = \frac{5}{2}R$, $C_P = \frac{7}{2}R$, and $\gamma = \frac{7}{5} = 1.4$.
Polyatomic gas — (non-linear, e.g., NH$_3$): $f=6$ (3 translational + 3 rotational). So, $C_V = \frac{6}{2}R = 3R$, $C_P = 4R$, and $\gamma = \frac{4}{3} \approx 1.33$.

Remember that the change in internal energy for an ideal gas depends *only* on temperature and is always given by $\Delta U = nC_V\Delta T$, irrespective of the process. This is a common point of confusion. Practice problems involving calculating heat, work, and internal energy for different processes, and be adept at using Mayer's relation and the degrees of freedom concept.

Prelims Revision Notes

1
Definitions — Heat capacity (C) is $dQ/dT$. Specific heat capacity (c) is $C/m$. Molar heat capacity ($C_m$) is $C/n$. Units: J/K, J/(kg·K), J/(mol·K) respectively.
2
Ideal Gases - Molar Heat Capacities — For ideal gases, heat capacity depends on the process.

* **Constant Volume ($C_V$)**: All heat goes to internal energy. $Q_V = nC_V\Delta T$. $\Delta U = nC_V\Delta T$. * **Constant Pressure ($C_P$)**: Heat goes to internal energy and work done. $Q_P = nC_P\Delta T$. $W = P\Delta V$.

1
Mayer's Relation — For an ideal gas, $C_P - C_V = R$, where $R = 8.314 \, \text{J/(mol·K)}$ is the universal gas constant. This implies $C_P > C_V$.
2
Degrees of Freedom (f) — The number of independent ways a molecule can store energy.

* Monoatomic (He, Ne, Ar): $f=3$ (translational). * Diatomic (O$_2$, N$_2$, H$_2$): $f=5$ (3 translational + 2 rotational, at moderate T). At high T, $f=7$ (vibrational modes activate). * Polyatomic (non-linear, e.g., NH$_3$): $f=6$ (3 translational + 3 rotational).

1
Equipartition Theorem — Each degree of freedom contributes $\frac{1}{2}RT$ to the molar internal energy.

* Molar Internal Energy $U = \frac{f}{2}RT$. * $C_V = \frac{dU}{dT} = \frac{f}{2}R$. * $C_P = C_V + R = (\frac{f}{2}+1)R$.

1
Ratio of Specific Heats ($\gamma$) — $\gamma = \frac{C_P}{C_V} = 1 + \frac{2}{f}$.

* Monoatomic: $\gamma = \frac{5}{3} \approx 1.67$. * Diatomic: $\gamma = \frac{7}{5} = 1.4$. * Polyatomic (non-linear): $\gamma = \frac{4}{3} \approx 1.33$.

1
Internal Energy Change (General for Ideal Gas) — $\Delta U = nC_V\Delta T$ is always true for an ideal gas, regardless of the process (isochoric, isobaric, isothermal, adiabatic). This is a critical point for problem-solving.
2
First Law of Thermodynamics — $\Delta U = Q - W$. Combine this with heat capacity formulas to solve problems involving heat, work, and internal energy changes.

Vyyuha Quick Recall

For ideal gases, remember 'My Dear Parents, Can Volume Really Profit?'. This helps recall: Monoatomic ($f=3$, $\gamma=5/3$), Diatomic ($f=5$, $\gamma=7/5$), Polyatomic ($f=6$, $\gamma=4/3$). And CP - CV = R.