Thermal Equilibrium and Heat: MDCAT Physics notes

Thermodynamics and heat

Thermodynamics is the branch of physics that deals with the relationship between heat and mechanical energy (work), and the conversion of one into the other.

Heat is energy in transit from one body to another because of a temperature difference. It flows spontaneously from a body at higher temperature to one at lower temperature, whatever their internal energies or pressures. When heat leaves a hot body, its internal energy decreases; the cold body's internal energy increases.

Thermal equilibrium

When two bodies in contact reach the same temperature, no net heat flows between them; they are in thermal equilibrium. If A and B are each in equilibrium with C, then A and B are in equilibrium with each other (the zeroth law), which is the basis of thermometers.

Internal energy

Internal energy $U$ is the sum of all forms of molecular energy (kinetic and potential) of a substance. It is a function of the state of the system.

• For an ideal gas, there are no intermolecular forces, so no molecular PE; $U$ is purely molecular KE and depends only on temperature.
• Heating an ideal gas at fixed volume raises only the kinetic energy of its molecules.

Work into internal energy

Internal energy can be increased by heating or by doing work. Rubbing hands, drilling metal, or hammering a nail: work done against friction becomes internal energy, and temperature rises.

Kinetic theory of an ideal gas

Assumptions:

• A gas consists of a very large number of molecules in random motion; they may change direction after each collision.
• Molecular size is negligible compared with the separation between them.
• No forces act between molecules except during collisions.
• Collisions with each other and with the walls are perfectly elastic; at a wall the molecule's momentum changes by $2mv$ (it is not negligible). Collision time is negligible.

Pressure: $P = \tfrac23 N_0\langle\tfrac12 mv^2\rangle$, where $N_0$ is molecules per unit volume. So pressure is directly proportional to the average translational KE of the molecules (per unit volume).

At constant temperature, lowering the pressure means the molecules are spread out, so the number of collisions with the walls per second decreases; their mean speed is unchanged.

Temperature and molecular energy

$$\langle\tfrac12 mv^2\rangle = \tfrac32 kT$$

Average translational KE is proportional to absolute temperature. From 27°C (300 K) to 327°C (600 K), it doubles. For any ideal gas at 17°C (290 K): $\tfrac32(1.38\times10^{-23})(290) \approx 6.0\times10^{-21}\ \text{J}$.

Gas laws

$PV = nRT$ is the ideal gas law, with $R = 8.314\ \text{J mol}^{-1}\text{K}^{-1}$. The Boltzmann constant is $k = R/N_A = 1.38\times10^{-23}\ \text{J K}^{-1}$.

Key formulas

• $PV = nRT = NkT$
• $P = \tfrac23 N_0\langle\tfrac12 mv^2\rangle$
• $\langle\tfrac12 mv^2\rangle = \tfrac32 kT$
• $k = R/N_A$

Common MDCAT traps

• Heat flows from high to low temperature, not high internal energy or pressure.
• Pressure is proportional to translational KE, not vibrational KE.
• Convert °C to K before using $KE \propto T$.
• Friction work raises internal energy.
• Ideal-gas internal energy is all kinetic.

Quick revision

• Same temperature means thermal equilibrium.
• Internal energy = molecular KE + PE.
• $k = 1.38\times10^{-23}\ \text{J K}^{-1}$.
• Double the kelvin temperature: double average KE.