Specific Heat and Molar Specific Heat of a Gas: MDCAT Physics notes
Specific Heat and Molar Specific Heat of a Gas for MDCAT: heat capacity, specific heat, molar heat capacity, Q = mcΔT, Q = nCΔT, units and special cases.
Heat capacity
The heat capacity of a body is the heat required to raise its temperature by one degree (1 K or 1 °C):
$$C = \frac{\Delta Q}{\Delta T}$$
Its SI unit is J K$^{-1}$. It depends on both the material and the amount of material.
Specific heat capacity
The specific heat $c$ is the heat required to raise the temperature of 1 kg of a substance by 1 K:
$$\Delta Q = mc\Delta T \qquad c = \frac{\Delta Q}{m\Delta T}$$
Unit: J kg$^{-1}$ K$^{-1}$. It is a property of the material alone. Water has an unusually high specific heat, about 4200 J kg$^{-1}$ K$^{-1}$, much larger than metals such as copper (about 390) and iron (about 450) or glass. This is why water is used as a coolant and why coastal climates are mild.
Rate of heat transfer
If the temperature changes at a rate $R = \Delta T/\Delta t$, then the rate of heat flow is $\Delta Q/\Delta t = mcR$.
Comparing heat needed
For the same material only the product $m\Delta T$ matters. For example, 2 kg heated by 6 °C, 4 kg heated by 3 °C and 1 kg heated by 12 °C all have $m\Delta T = 12$, so they need the same heat.
Molar specific heat
For gases it is more useful to work per mole. The molar specific heat is the heat needed to raise the temperature of 1 mole by 1 K:
$$\Delta Q = nC\Delta T$$
Unit: J mol$^{-1}$ K$^{-1}$. In differential form $C = \frac{1}{n}\frac{dQ}{dT}$.
A gas has two principal molar heat capacities because the heat needed depends on whether the gas is allowed to expand.
| Quantity | Condition | Meaning |
|---|---|---|
| $C_v$ | Constant volume | No work done, so $Q = \Delta U$ and $C_v = \frac{1}{n}\frac{dU}{dT}$ |
| $C_p$ | Constant pressure | Heat raises $U$ and also does work $P\Delta V$, so $C_p = \frac{1}{n}\frac{dQ}{dT}$ and $C_p > C_v$ |
Example: 3 mol of a gas with $C_v = 10$ J mol$^{-1}$ K$^{-1}$ heated at constant volume through 20 K needs $Q = nC_v\Delta T = 3\times10\times20 = 600$ J.
Specific heat in special processes
Since $C = \Delta Q/\Delta T$:
- Adiabatic: $\Delta Q = 0$ while $T$ changes, so the specific heat is zero.
- Isothermal: heat flows but $\Delta T = 0$, so the specific heat is infinite.
Key formulas
- $C = \Delta Q/\Delta T$ (J K$^{-1}$)
- $\Delta Q = mc\Delta T$ (J kg$^{-1}$ K$^{-1}$)
- $\Delta Q = nC_v\Delta T$ at constant volume, $\Delta Q = nC_p\Delta T$ at constant pressure
- $\Delta U = nC_v\Delta T$ for an ideal gas in any process
Common MDCAT traps
- Heat capacity (J K$^{-1}$) vs specific heat (J kg$^{-1}$ K$^{-1}$) vs molar heat capacity (J mol$^{-1}$ K$^{-1}$): check the unit asked.
- Molar heat capacity is defined for 1 mole, not 1 kg or 2 moles.
- Adiabatic gives zero specific heat; isothermal gives infinite. Students often swap them.
- $C_v$ corresponds to $dU/dT$; $C_p$ to $dQ/dT$ at constant pressure.
- When only $m\Delta T$ differs, calculate the product before choosing; equal products mean equal heat.
Quick revision
- Water has the highest specific heat among common substances (about 4200 J kg$^{-1}$ K$^{-1}$).
- $C_p > C_v$ for a gas because at constant pressure some heat does work.
- Heat capacity depends on mass; specific heat does not.
- Specific heat is zero in an adiabatic change and infinite in an isothermal one.
- Rate of heat transfer $= mc \times$ rate of temperature change.