Specific Heat — Revision Notes

⚡ 30-Second Revision

Specific Heat ($c$) — Heat to change $1\,\text{kg}$ by $1\,\text{K}$. Unit: J kg$^{-1}$ K$^{-1}$.
Formula — $Q = mcDelta T$
Heat Capacity ($C$) — Heat to change object by $1\,\text{K}$. Unit: J K$^{-1}$. $C = mc$.
Molar Specific Heat ($C_m$) — Heat to change $1\,\text{mol}$ by $1\,\text{K}$. Unit: J mol$^{-1}$ K$^{-1}$. $C_m = Mc$.
Gases — $C_v$ (constant volume), $C_p$ (constant pressure).
Mayer's Relation — $C_p - C_v = R$ (for ideal gas).
Ratio of Specific Heats — $\gamma = C_p / C_v$.

- Monatomic ($f=3$): $C_v = \frac{3}{2}R$, $C_p = \frac{5}{2}R$, $\gamma = \frac{5}{3}$. - Diatomic ($f=5$ at moderate T): $C_v = \frac{5}{2}R$, $C_p = \frac{7}{2}R$, $\gamma = \frac{7}{5}$.

Internal Energy Change — $\Delta U = nC_vDelta T$.
Calorimetry — Heat Lost = Heat Gained ($m_1c_1\Delta T_1 = m_2c_2\Delta T_2$).

2-Minute Revision

Specific heat capacity ($c$) measures a substance's thermal inertia, quantifying the heat needed to raise the temperature of a unit mass by one degree. Its formula is $Q = mcDelta T$, with SI units of J kg$^{-1}$ K$^{-1}$. This differs from heat capacity ($C=mc$), which is for a specific object. Water has a notably high specific heat due to hydrogen bonding, making it an excellent thermal regulator.

For ideal gases, specific heat depends on the process: $C_v$ (constant volume) and $C_p$ (constant pressure). Mayer's relation, $C_p - C_v = R$, is crucial, showing $C_p$ is greater because of work done during expansion at constant pressure.

The ratio $\gamma = C_p/C_v$ varies with the gas's atomicity (degrees of freedom, $f$). For monatomic gases, $\gamma = 5/3$; for diatomic gases at moderate temperatures, $\gamma = 7/5$. Internal energy change for an ideal gas is $\Delta U = nC_vDelta T$.

Calorimetry problems apply the principle of heat lost equals heat gained in an isolated system, using $Q = mcDelta T$ for each component.

5-Minute Revision

Specific heat ($c$) is a material property defining how much heat energy ($Q$) is required to change the temperature ($\Delta T$) of a unit mass ($m$) by one degree. The core equation is $Q = mcDelta T$. Remember that $\Delta T$ can be in Celsius or Kelvin, as the change is the same. Heat capacity ($C = mc$) refers to a specific object's ability to store heat, while molar specific heat ($C_m = Mc$) is per mole, useful for gases.

For ideal gases, we distinguish between molar specific heat at constant volume ($C_v$) and constant pressure ($C_p$). At constant volume, all heat goes to increasing internal energy ($\Delta U = nC_vDelta T$).

At constant pressure, some heat also performs work ($W = PDelta V = nRDelta T$), so $C_p$ is always greater than $C_v$. Mayer's relation elegantly connects them: $C_p - C_v = R$. The ratio $\gamma = C_p/C_v$ is vital, determined by the gas's degrees of freedom ($f$).

For monatomic gases ($f=3$), $C_v = \frac{3}{2}R$, $C_p = \frac{5}{2}R$, $\gamma = \frac{5}{3}$. For diatomic gases at moderate temperatures ($f=5$), $C_v = \frac{5}{2}R$, $C_p = \frac{7}{2}R$, $\gamma = \frac{7}{5}$.

Example: Calculate the heat required to raise the temperature of $0.5\,\text{kg}$ of water from $20^\circ\text{C}$ to $70^\circ\text{C}$. (Specific heat of water $c_w = 4186\,\text{J kg}^{-1}\text{K}^{-1}$). $m = 0.5\,\text{kg}$, $c = 4186\,\text{J kg}^{-1}\text{K}^{-1}$, $\Delta T = 70 - 20 = 50^\circ\text{C} = 50\,\text{K}$. $Q = mcDelta T = 0.5 \times 4186 \times 50 = 2093 \times 50 = 104650\,\text{J} = 104.65\,\text{kJ}$.

Calorimetry problems involve heat exchange: $Q_{lost} = Q_{gained}$. If $100\,\text{g}$ of iron ($c_{Fe} = 450\,\text{J kg}^{-1}\text{K}^{-1}$) at $100^\circ\text{C}$ is dropped into $200\,\text{g}$ of water ($c_w = 4186\,\text{J kg}^{-1}\text{K}^{-1}$) at $20^\circ\text{C}$, find the final temperature $T_f$.

$0.1 \times 450 \times (100 - T_f) = 0.2 \times 4186 \times (T_f - 20)$ $45 (100 - T_f) = 837.2 (T_f - 20)$ $4500 - 45T_f = 837.2T_f - 16744$ $21244 = 882.2T_f \implies T_f \approx 24.08^\circ\text{C}$.

Remember to differentiate specific heat from latent heat (phase change without temperature change) and be mindful of units and conversions.

Prelims Revision Notes

1
Definition — Specific heat capacity ($c$) is the heat energy required to raise the temperature of a unit mass of a substance by $1\,\text{K}$ or $1^\circ\text{C}$.
2
Formula — $Q = mcDelta T$, where $Q$ is heat, $m$ is mass, $c$ is specific heat, $\Delta T$ is temperature change.
3
Units — SI unit is J kg$^{-1}$ K$^{-1}$. Other common unit: cal g$^{-1}$ °C$^{-1}$. Conversion: $1\,\text{cal} = 4.186\,\text{J}$.
4
Heat Capacity — $C = mc$. Total heat for a given object to change temperature by $1\,\text{K}$. Unit: J K$^{-1}$.
5
Molar Specific Heat ($C_m$) — Heat required to raise $1\,\text{mol}$ of substance by $1\,\text{K}$. Unit: J mol$^{-1}$ K$^{-1}$. $C_m = Mc$, where $M$ is molar mass.
6
Specific Heat of Water — Exceptionally high, approx. $4186\,\text{J kg}^{-1}\text{K}^{-1}$ or $1\,\text{cal g}^{-1}\text{°C}^{-1}$.
7
Specific Heats of Gases — Not unique, depends on process.

* **Constant Volume ($C_v$)**: All heat increases internal energy. $\Delta U = nC_vDelta T$. * **Constant Pressure ($C_p$)**: Heat increases internal energy and does work. $Q_p = \Delta U + W$.

1
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 implies $C_p > C_v$.
2
Ratio of Specific Heats ($\gamma$) — $\gamma = C_p / C_v = 1 + \frac{2}{f}$, where $f$ is degrees of freedom.

* Monatomic Gas (e.g., He, Ne): $f=3$. $C_v = \frac{3}{2}R$, $C_p = \frac{5}{2}R$, $\gamma = \frac{5}{3} \approx 1.67$. * **Diatomic Gas (e.g., O$_2$, N$_2$)**: $f=5$ (3 translational + 2 rotational at moderate T). $C_v = \frac{5}{2}R$, $C_p = \frac{7}{2}R$, $\gamma = \frac{7}{5} = 1.4$. * **Polyatomic Gas (e.g., CO$_2$, NH$_3$)**: $f=6$ (non-linear) or $f=5$ (linear) at moderate T. $\gamma$ values are lower, e.g., $\gamma \approx 1.33$ for non-linear.

1
Calorimetry Principle — In an isolated system, heat lost by hot bodies = heat gained by cold bodies. $\sum (mcDelta T)_{lost} = \sum (mcDelta T)_{gained}$.
2
Dulong-Petit Law (for solids) — For many solids at high temperatures, molar specific heat $C_v \approx 3R \approx 24.9\,\text{J mol}^{-1}\text{K}^{-1}$.
3
Distinction from Latent Heat — Specific heat involves temperature change without phase change. Latent heat involves phase change without temperature change.

Vyyuha Quick Recall

For Specific Heat, remember 'Q = MCAT' (pronounced 'Q equals M-Cat').

Q — Heat energy
M — Mass
C — Specific Heat Capacity
$\Delta$T — Change in Temperature

This helps recall the primary formula. For gases, remember 'Cp is Greater than Cv by R' (Mayer's relation: $C_p - C_v = R$) because at constant pressure, the gas does 'R' amount of work per mole per Kelvin.