Heat Capacity — Revision Notes

⚡ 30-Second Revision

Heat Capacity ($C$) — $C = \frac{dq}{dT}$ (extensive property)
Specific Heat Capacity ($c$) — $c = \frac{C}{m}$ (intensive, $J,g^{-1},K^{-1}$)
Molar Heat Capacity ($C_m$) — $C_m = \frac{C}{n}$ (intensive, $J,mol^{-1},K^{-1}$)
Heat Transfer — $q = mc\Delta T$ or $q = nC_m\Delta T$
Constant Volume ($C_V$) — $C_V = (\frac{\partial U}{\partial T})_V$. For ideal gas, $\Delta U = nC_V\Delta T$.
Constant Pressure ($C_P$) — $C_P = (\frac{\partial H}{\partial T})_P$. For ideal gas, $\Delta H = nC_P\Delta T$.
Mayer's Formula (Ideal Gas) — $C_P - C_V = nR$ (for $n$ moles) or $C_{P,m} - C_{V,m} = R$ (for 1 mole).
Ratio of Heat Capacities ($\gamma$) — $\gamma = \frac{C_P}{C_V}$

- Monatomic: $\gamma = 5/3 \approx 1.67$ - Diatomic: $\gamma = 7/5 = 1.40$ - Polyatomic (non-linear): $\gamma = 4/3 \approx 1.33$

Ideal Gas Constant — $R = 8.314,J,mol^{-1},K^{-1}$

2-Minute Revision

Heat capacity is a measure of how much heat energy a substance can absorb for a given temperature rise. It comes in three main forms: total heat capacity ($C$, extensive), specific heat capacity ($c$, per unit mass, intensive), and molar heat capacity ($C_m$, per unit mole, intensive). The fundamental equation for heat transfer is $q = mcDelta T$ or $q = nC_mDelta T$.

Crucially, heat capacity depends on the conditions. At constant volume, it's $C_V = (\frac{partial U}{partial T})_V$, relating to internal energy change. At constant pressure, it's $C_P = (\frac{partial H}{partial T})_P$, relating to enthalpy change.

For ideal gases, $C_P$ is always greater than $C_V$ because at constant pressure, some heat is used for expansion work. This difference is quantified by Mayer's formula: $C_{P,m} - C_{V,m} = R$. The ratio $gamma = C_P/C_V$ is also important, with characteristic values for monatomic ($approx 1.

67$), diatomic ($approx 1.40$), and polyatomic ($approx 1.33$) gases, reflecting their molecular degrees of freedom. Remember to use consistent units and pay attention to whether the problem specifies constant volume or constant pressure.

5-Minute Revision

Heat capacity is a critical thermodynamic property indicating a substance's ability to store thermal energy. It's defined as $C = dq/dT$. We distinguish between total heat capacity ($C$), which is extensive, and specific heat capacity ($c = C/m$) or molar heat capacity ($C_m = C/n$), which are intensive properties.

The heat absorbed or released ($q$) for a temperature change ($Delta T$) is calculated as $q = mcDelta T$ or $q = nC_mDelta T$. Always ensure units are consistent (e.g., $J$, $g$, $K$ or $J$, $mol$, $K$).

For gases, the conditions under which heat is added are vital. **Heat capacity at constant volume ($C_V$)** is related to the change in internal energy: $Delta U = nC_VDelta T$. Here, no P-V work is done.

**Heat capacity at constant pressure ($C_P$)** is related to the change in enthalpy: $Delta H = nC_PDelta T$. At constant pressure, the system can expand, doing work against the surroundings. This additional work means that $C_P$ is always greater than $C_V$ for gases.

The relationship is given by Mayer's formula: $C_{P,m} - C_{V,m} = R$ for one mole of an ideal gas, where $R = 8.314,J,mol^{-1},K^{-1}$.

The heat capacities are also influenced by the molecule's degrees of freedom (translational, rotational, vibrational). This leads to characteristic values for the **ratio of heat capacities, $gamma = C_P/C_V$**: $\approx 1.67$ for monatomic gases (e.g., He), $1.40$ for diatomic gases (e.g., O$_2$), and $\approx 1.33$ for non-linear polyatomic gases (e.g., H$_2$O) at room temperature. These values are important for understanding adiabatic processes and molecular structure.

Worked Example: $2.5,\text{mol}$ of an ideal diatomic gas is heated from $300,\text{K}$ to $400,\text{K}$ at constant pressure. Calculate the heat absorbed. (Given: $R = 8.314,J,mol^{-1},K^{-1}$)

1
For a diatomic gas, $C_{V,m} = \frac{5}{2}R$.
2
Using Mayer's formula, $C_{P,m} = C_{V,m} + R = \frac{5}{2}R + R = \frac{7}{2}R$.
3
Substitute $R$: $C_{P,m} = \frac{7}{2} \times 8.314,J,mol^{-1},K^{-1} = 29.1,J,mol^{-1},K^{-1}$.
4
Calculate $Delta T = 400,\text{K} - 300,\text{K} = 100,\text{K}$.
5
Heat absorbed $q_P = nC_{P,m}Delta T = 2.5,\text{mol} \times 29.1,J,mol^{-1},K^{-1} \times 100,\text{K} = 7275,J = 7.275,kJ$.

Prelims Revision Notes

Heat capacity ($C$) is an extensive property, defined as $C = dq/dT$. For NEET, focus on intensive forms: specific heat capacity ($c$, per gram) and molar heat capacity ($C_m$, per mole). The key formula for heat transfer is $q = mcDelta T$ (for mass $m$) or $q = nC_mDelta T$ (for moles $n$). Ensure $Delta T$ is in Kelvin or Celsius (as $Delta T$ values are identical). Units are crucial: $J,g^{-1},K^{-1}$ for $c$ and $J,mol^{-1},K^{-1}$ for $C_m$.

Distinguish between $C_V$ (constant volume) and $C_P$ (constant pressure). $C_V = (\frac{partial U}{partial T})_V$ means heat goes solely into internal energy. $C_P = (\frac{partial H}{partial T})_P$ means heat goes into internal energy AND P-V work. For ideal gases, this leads to Mayer's formula: $C_{P,m} - C_{V,m} = R$, where $R = 8.314,J,mol^{-1},K^{-1}$. Thus, $C_P > C_V$ for gases. For solids and liquids, $C_P \approx C_V$ as volume changes are negligible.

The ratio $gamma = C_P/C_V$ is important for ideal gases based on molecular structure and degrees of freedom:

Monatomic (e.g., He, Ne) — 3 translational degrees of freedom. $C_{V,m} = \frac{3}{2}R$, $C_{P,m} = \frac{5}{2}R$, $\gamma = \frac{5}{3} \approx 1.67$.
Diatomic (e.g., O$_2$, N$_2$) — 3 translational + 2 rotational degrees of freedom (at room temp). $C_{V,m} = \frac{5}{2}R$, $C_{P,m} = \frac{7}{2}R$, $\gamma = \frac{7}{5} = 1.40$.
Polyatomic (non-linear, e.g., H$_2$O, CH$_4$) — 3 translational + 3 rotational degrees of freedom (at room temp). $C_{V,m} = 3R$, $C_{P,m} = 4R$, $\gamma = \frac{4}{3} \approx 1.33$.

Remember that vibrational degrees of freedom become active at higher temperatures, increasing heat capacities. Practice numerical problems involving these formulas and conceptual questions differentiating the various forms and their implications.

Vyyuha Quick Recall

To remember the order of $\gamma$ values for gases: Many Donkeys Play. Monatomic (highest $\gamma$), Diatomic (middle $\gamma$), Polyatomic (lowest $\gamma$).

Monatomic: $\gamma \approx 1.67$ Diatomic: $\gamma = 1.40$ Polyatomic: $\gamma \approx 1.33$