Molar Heat Capacities — Revision Notes

⚡ 30-Second Revision

Molar Heat Capacity ($C$): — Heat for 1 mole, $1\text{ K}$ temp rise. Units: $\text{J mol}^{-1}\text{ K}^{-1}$.
Constant Volume ($C_v$): — $C_v = \frac{1}{n} (\frac{dU}{dT})_v$. All heat increases internal energy. For ideal gas: $dU = n C_v dT$.
Constant Pressure ($C_p$): — $C_p = \frac{1}{n} (\frac{dQ}{dT})_p$. Heat increases internal energy AND does work.
Mayer's Relation: — For ideal gas, $C_p - C_v = R$.
Degrees of Freedom ($f$): — Monoatomic $f=3$, Diatomic $f=5$ (at room temp), Polyatomic $f=6$ (at room temp).
Equipartition Theorem: — $U = n \frac{f}{2} RT$.
$C_v$ from $f$: — $C_v = \frac{f}{2} R$.
$C_p$ from $f$: — $C_p = (\frac{f}{2} + 1) R$.
Ratio $\gamma$: — $\gamma = \frac{C_p}{C_v} = 1 + \frac{2}{f}$.
Universal Gas Constant $R$: — $8.314 \text{ J mol}^{-1}\text{ K}^{-1}$.

2-Minute Revision

Molar heat capacity is the heat required to change the temperature of one mole of a substance by one Kelvin. For gases, we distinguish between $C_v$ (constant volume) and $C_p$ (constant pressure). At constant volume, all heat goes into increasing internal energy, so $Q = nC_v\Delta T$.

At constant pressure, the gas expands and does work, so $Q = nC_p\Delta T$, and $C_p$ is always greater than $C_v$. Mayer's relation, $C_p - C_v = R$, quantifies this difference for ideal gases, where $R$ is the universal gas constant.

The values of $C_v$ and $C_p$ depend on the number of active degrees of freedom ($f$) of the gas molecules. For monoatomic gases, $f=3$, leading to $C_v = \frac{3}{2}R$. For diatomic gases at room temperature, $f=5$, giving $C_v = \frac{5}{2}R$.

The ratio $\gamma = C_p/C_v$ is also important, and it relates to $f$ as $\gamma = 1 + \frac{2}{f}$. Remember to use the correct $f$ values and apply Mayer's relation accurately.

5-Minute Revision

Molar heat capacity ($C$) is a measure of how much heat energy is needed to raise the temperature of one mole of a substance by one Kelvin. For ideal gases, this concept is critical and leads to two distinct values: $C_v$ and $C_p$.

$C_v$ is the molar heat capacity at constant volume. When heat is added at constant volume, no work is done by the gas, so all the heat directly increases the internal energy. Thus, $dQ = dU$, and $C_v = \frac{1}{n}(\frac{dU}{dT})_v$.

$C_p$ is the molar heat capacity at constant pressure. Here, the gas expands and does work ($dW = P dV$), so the heat supplied not only increases internal energy but also provides energy for this work.

Hence, $C_p$ is always greater than $C_v$. The relationship between them for an ideal gas is given by Mayer's relation: $C_p - C_v = R$, where $R$ is the universal gas constant ($8.314 \text{ J mol}^{-1}\text{ K}^{-1}$).

The values of $C_v$ and $C_p$ are determined by the number of active degrees of freedom ($f$) of the gas molecules, according to the Law of Equipartition of Energy. Each degree of freedom contributes $\frac{1}{2}RT$ to the molar internal energy. Therefore, the total internal energy for $n$ moles is $U = n \frac{f}{2} RT$. From this, we derive $C_v = \frac{f}{2}R$ and $C_p = (\frac{f}{2} + 1)R$. The ratio $\gamma = C_p/C_v$ is also important, given by $\gamma = 1 + \frac{2}{f}$.

Example: For a monoatomic gas (like He, Ne), $f=3$ (3 translational). So, $C_v = \frac{3}{2}R$, $C_p = \frac{5}{2}R$, and $\gamma = \frac{5}{3} \approx 1.67$.

Example: For a diatomic gas (like $O_2, N_2$) at room temperature, $f=5$ (3 translational + 2 rotational). So, $C_v = \frac{5}{2}R$, $C_p = \frac{7}{2}R$, and $\gamma = \frac{7}{5} = 1.4$.

Remember that vibrational degrees of freedom are typically ignored at room temperature for NEET problems unless explicitly mentioned. Always check if the question refers to specific heat capacity (per unit mass) or molar heat capacity (per mole) to avoid errors.

Prelims Revision Notes

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Definition: — Molar heat capacity ($C$) is the heat required to raise the temperature of one mole of a substance by $1\text{ K}$. Units: $\text{J mol}^{-1}\text{ K}^{-1}$.
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**Molar Heat Capacity at Constant Volume ($C_v$):**

* Process: Volume is kept constant ($dV=0$). * Work done ($dW$): $0$. * First Law of Thermodynamics: $dQ = dU$. * Definition: $C_v = \frac{1}{n} (\frac{dU}{dT})_v$. * Internal Energy Change: For an ideal gas, $dU = n C_v dT$ (always true, even if volume changes).

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**Molar Heat Capacity at Constant Pressure ($C_p$):**

* Process: Pressure is kept constant ($dP=0$). * Work done ($dW$): $P dV$. * First Law of Thermodynamics: $dQ = dU + P dV$. * Definition: $C_p = \frac{1}{n} (\frac{dQ}{dT})_p$. * Why $C_p > C_v$: At constant pressure, extra heat is needed to do work against external pressure during expansion.

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Mayer's Relation (for Ideal Gases):

* $C_p - C_v = R$, where $R$ is the universal gas constant ($8.314 \text{ J mol}^{-1}\text{ K}^{-1}$). This is a crucial formula.

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**Degrees of Freedom ($f$):**

* Number of independent ways a molecule can store energy. * Translational: 3 (for all molecules). * Rotational: 2 (for linear molecules like diatomic), 3 (for non-linear molecules like polyatomic). * Vibrational: Generally ignored at room temperature for NEET unless specified (active at high temperatures).

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

* Total Internal Energy: $U = n \frac{f}{2} RT$.

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**Relation of $C_v$, $C_p$, $\gamma$ to $f$ (for Ideal Gases):**

* $C_v = \frac{f}{2} R$ * $C_p = (\frac{f}{2} + 1) R$ * Ratio of Heat Capacities (Adiabatic Index): $\gamma = \frac{C_p}{C_v} = 1 + \frac{2}{f}$

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Values for Common Ideal Gases (at room temperature, no vibration):

* **Monoatomic ($f=3$):** $C_v = \frac{3}{2}R$, $C_p = \frac{5}{2}R$, $\gamma = \frac{5}{3} \approx 1.67$. * **Diatomic ($f=5$):** $C_v = \frac{5}{2}R$, $C_p = \frac{7}{2}R$, $\gamma = \frac{7}{5} = 1.4$. * **Polyatomic (non-linear, $f=6$):** $C_v = 3R$, $C_p = 4R$, $\gamma = \frac{4}{3} \approx 1.33$.

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Heat Transfer:

* Isochoric process: $Q = n C_v \Delta T$. * Isobaric process: $Q = n C_p \Delta T$.

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Specific vs. Molar Heat Capacity: — $C = M_m \times c$, where $M_m$ is molar mass.

Vyyuha Quick Recall

To remember the degrees of freedom and $\gamma$ for common gases:

My Dog Plays 3 5 6 Rounds.

Monoatomic: 3 degrees of freedom ($f=3$).
Diatomic: 5 degrees of freedom ($f=5$).
Polyatomic: 6 degrees of freedom ($f=6$).
Remember $C_v = \frac{f}{2}R$ and $\gamma = 1 + \frac{2}{f}$ to quickly calculate the rest!