Temperature and heat

Units of temperature

$$\begin{align*} T_F &= \frac{9}{5}T_C + 32 \\ T_K &= T_C + 273.15 \end{align*}$$

info

• Triple point of water: solid ice, liquid water and water vapour coexist at 273.16 K (= 0.01 °C) and a vapour pressure of 612 Pa (about 0.006 atm)
• Absolute zero: 0 K (= -273.15 °C)

Thermal expansion

Linear expansion

$$\begin{align*} \Delta L = L_0\alpha\Delta T \end{align*}$$

• $\alpha$: coefficient of linear expansion (K$^{-1}$)
• When an object under goes thermal expansion, any holes in the object expand as well.

Volume expansion

$$\begin{align*} \Delta V &= V_0\beta\Delta T \\ &= 3V_0\alpha\Delta T \end{align*}$$

• $\beta$: coefficient of volume expansion (K$^{-1}$)
  • $\beta = 3\alpha$

Heat

Heat capacity

$$\begin{align*} Q &= mc\Delta T \\ &= nC\Delta T \\ \end{align*}$$

• $c$ : specific heat capacity (J/kg $\cdot$ K)
• $C = Mc$ : molar heat capacity (J/mol $\cdot$ K)

Latent heat

$$\begin{align*} Q = mL \end{align*}$$

• $L_f$: latent heat of fusion (J/kg)
• $L_v$: latent heat of vaporization (J/kg)

Heat transfer

Conduction

Through physical contact from hot to cold

$$\begin{align*} H &= kA\frac{T_H-T_C}{L} \\ H &= -kA\frac{dT}{dx} = \frac{dQ}{dT} \end{align*}$$

• $k$: thermal conductivity (W/m $\cdot$ K)
• $A$: cross-sectional area
• $L$: length of the conductor
• Thermal resistance $R = \frac{L}{k}$ (m$^2$/W $\cdot$ K)

Example

To find the temperature of the junction between two different conductors:

$$\begin{align*} k_1A_1\frac{T_H-T}{L_1} = k_2A_2\frac{T-T_C}{L_2} \end{align*}$$

Radiation

Through electromagnetic wave which does not require a medium

Stephan-Boltzmann law:

$$\begin{align*} H_{rad} &= Ae\sigma T^4 \\ H_{net} &= Ae\sigma(T^4-T_S^4) \end{align*}$$

• sigma $\sigma = 5.67\times10^{-8}$ W/m$^2\cdot$ K$^4$ : Stephan-Boltzmann constant
• $e$: emissivity ($0 \le e \le 1$)
• $A$: total surface area
• $T_S$: temperature of surroundings

Convection

Through motion of fluid from one region to another