Specific Heat Capacity — Revision Notes

⚡ 30-Second Revision

Specific Heat Capacity (c) — $Q = mcDelta T$, Unit: J/kg·K
Heat Capacity (C) — $C = mc$, Unit: J/K
Molar Specific Heat at Constant Volume ($C_v$) — $C_v = \frac{f}{2}R$
Molar Specific Heat at Constant Pressure ($C_p$) — $C_p = \left(\frac{f+2}{2}\right)R$
Mayer's Formula — $C_p - C_v = R$
Ratio of Specific Heats ($\gamma$) — $\gamma = \frac{C_p}{C_v} = 1 + \frac{2}{f}$
Degrees of Freedom (f)

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

Values of $\gamma$ — Monoatomic: $5/3 \approx 1.67$; Diatomic: $7/5 = 1.40$; Polyatomic: $4/3 \approx 1.33$

2-Minute Revision

Specific heat capacity ($c$) measures how much heat energy is needed to change the temperature of a unit mass of a substance by one degree. Its formula is $Q = mcDelta T$. For gases, we distinguish between molar specific heat at constant volume ($C_v$) and constant pressure ($C_p$).

$C_p$ is always greater than $C_v$ because at constant pressure, the gas expands and does work, requiring additional energy. Mayer's formula, $C_p - C_v = R$, quantifies this difference for ideal gases.

The equipartition theorem is key to determining $C_v$ and $C_p$ based on the molecule's degrees of freedom ($f$). Each degree of freedom contributes $\frac{1}{2}RT$ to the internal energy per mole. Monoatomic gases have $f=3$, diatomic $f=5$ (at moderate T), and polyatomic $f=6$ (at moderate T).

From these, $C_v = \frac{f}{2}R$, $C_p = \frac{f+2}{2}R$, and the ratio $\gamma = C_p/C_v = 1 + \frac{2}{f}$ can be calculated. These values are crucial for understanding gas behavior in various thermodynamic processes, especially adiabatic ones.

5-Minute Revision

Specific heat capacity is a critical concept in thermal physics, quantifying a material's resistance to temperature change upon heat transfer. It's defined as the heat required per unit mass per unit temperature change ($c = Q/(mDelta T)$). For an entire object, we use heat capacity ($C = mc$).

For ideal gases, the scenario is more complex due to the possibility of work done during heating. This leads to two distinct molar specific heats:

1
$C_v$ (Constant Volume) — All heat supplied goes into increasing the internal energy ($\Delta U$). $Q_v = nC_vDelta T = \Delta U$.
2
$C_p$ (Constant Pressure) — Heat supplied increases internal energy AND does work of expansion ($W = PDelta V$). $Q_p = nC_pDelta T = \Delta U + W$.

Mayer's Formula: For one mole of an ideal gas, $C_p - C_v = R$, where $R$ is the universal gas constant ($8.314\,\text{J/mol}\cdot\text{K}$). This formula highlights that $C_p$ is always greater than $C_v$.

**Equipartition Theorem and Degrees of Freedom ($f$)**: This theorem states that each degree of freedom contributes $\frac{1}{2}RT$ to the internal energy per mole. The total internal energy for $n$ moles is $U = n \frac{f}{2}RT$. Since $\Delta U = nC_vDelta T$, we get $C_v = \frac{f}{2}R$. Using Mayer's formula, $C_p = \left(\frac{f}{2}+1\right)R = \left(\frac{f+2}{2}\right)R$.

Types of Gases and their Specific Heats (at moderate temperatures):

Monoatomic (e.g., He) — $f=3$ (3 translational)

* $C_v = \frac{3}{2}R$ * $C_p = \frac{5}{2}R$ * $\gamma = \frac{C_p}{C_v} = \frac{5}{3} \approx 1.67$

Diatomic (e.g., O$_2$) — $f=5$ (3 translational + 2 rotational)

* $C_v = \frac{5}{2}R$ * $C_p = \frac{7}{2}R$ * $\gamma = \frac{C_p}{C_v} = \frac{7}{5} = 1.40$ * (At high temperatures, vibrational modes activate, $f=7$, leading to $C_v = \frac{7}{2}R$, $C_p = \frac{9}{2}R$, $\gamma = \frac{9}{7} \approx 1.29$)

Polyatomic (non-linear, e.g., H$_2$O) — $f=6$ (3 translational + 3 rotational)

* $C_v = \frac{6}{2}R = 3R$ * $C_p = \frac{8}{2}R = 4R$ * $\gamma = \frac{C_p}{C_v} = \frac{4}{3} \approx 1.33$

Key Takeaways: Remember Mayer's formula, the equipartition theorem, and the specific values of $f$, $C_v$, $C_p$, and $\gamma$ for different gas types. These are frequently tested in NEET.

Prelims Revision Notes

Specific Heat Capacity (c)

Definition — Heat required to raise temperature of unit mass by $1\,\text{K}$.
Formula — $Q = mcDelta T \implies c = Q/(mDelta T)$.
Units — J/kg·K (SI), cal/g·°C.
Property — Intensive (material-specific).

Heat Capacity (C)

Definition — Heat required to raise temperature of entire body by $1\,\text{K}$.
Formula — $C = Q/\Delta T = mc$.
Units — J/K.
Property — Extensive (depends on mass).

Molar Specific Heat Capacities for Ideal Gases

$C_v$ (Constant Volume)

* Heat supplied only increases internal energy ($\Delta U$). * $Q_v = nC_vDelta T = \Delta U$. * From equipartition theorem: $C_v = \frac{f}{2}R$.

$C_p$ (Constant Pressure)

* Heat supplied increases internal energy AND does work ($W = PDelta V$). * $Q_p = nC_pDelta T = \Delta U + W$. * From equipartition theorem: $C_p = \left(\frac{f+2}{2}\right)R$.

Mayer's Formula

Relation — $C_p - C_v = R$ (for 1 mole of ideal gas).
Significance — $C_p > C_v$ because of work done at constant pressure.

Degrees of Freedom ($f$) and Gas Types

Equipartition Theorem — Each degree of freedom contributes $\frac{1}{2}RT$ to molar internal energy.
Monoatomic Gas (e.g., He, Ar)

* $f=3$ (3 translational) * $C_v = \frac{3}{2}R$, $C_p = \frac{5}{2}R$, $\gamma = \frac{5}{3} \approx 1.67$

Diatomic Gas (e.g., O$_2$, N$_2$)

* Moderate T: $f=5$ (3 translational + 2 rotational) * $C_v = \frac{5}{2}R$, $C_p = \frac{7}{2}R$, $\gamma = \frac{7}{5} = 1.40$ * High T: $f=7$ (3 translational + 2 rotational + 2 vibrational) * $C_v = \frac{7}{2}R$, $C_p = \frac{9}{2}R$, $\gamma = \frac{9}{7} \approx 1.29$

Polyatomic Gas (non-linear, e.g., H$_2$O, CH$_4$)

* Moderate T: $f=6$ (3 translational + 3 rotational) * $C_v = 3R$, $C_p = 4R$, $\gamma = \frac{4}{3} \approx 1.33$

Ratio of Specific Heats ($\gamma$)

Formula — $\gamma = \frac{C_p}{C_v} = 1 + \frac{2}{f}$.
Importance — Used in adiabatic processes ($PV^\gamma = \text{constant}$).

Vyyuha Quick Recall

To remember degrees of freedom for common gases: My Dog Plays:

Monoatomic: 3 (just translational)
Diatomic: 5 (3 translational + 2 rotational)
Polyatomic: 6 (3 translational + 3 rotational)

And for Mayer's formula: Cool People Minus Cool Vegans Rejoice! ($C_p - C_v = R$)