Boring
\[ T_F = \frac{9}{5}T_C +32 \]
Thermal expansion
\[ \Delta L = L \alpha \Delta T \]
\[ \Delta V = V \beta \Delta T \]
First law of thermodynamics
Work done by a gas
\[ W = \int \dd W = \int_{v_i}^{v_f} p \dd{V} \]
First law of thermodynamics
\[ dE_{int} = dQ-dW \]
where W is the work done by the system
Special cases
Adiabatic process
\[ \Delta E_{int} = -W \]
when Q=0 (occurs rapidly or is very well insulated)
Constant-volume processes
\[ \Delta E_{int} = Q \]
Cyclical process
\[ Q=W \]
No change in internal energy
Free expansions
\[ \Delta E_{int} = 0 \]
Q=W=0
Adiabatic processes
Heat transfer
Conduction
\[ P_{cond} = \frac{Q}{t} = kA\frac{T_H-T_C}{L} \]
where P is the conduction rate, k is the thermal conductivity, and T is temperature of the hot and cold reservoirs. A and L is the area surface area and length of the conducting slab
R-value (Thermal Resistance) is defined as \(\frac{L}{k}\) for a material of specified thickness
For a conduction slab made up of multiple slabs:
\[ P_{cond} = \frac{A(T_H-T_C)}{\sum (L/k)} \]
assuming thermal conduction is in the steady state (all energy absorbed is conducted away)
Greater the thermal conductivity, the smaller the temperature difference between reservoirs
Convection
Trivial
Radiation
\[ P_{rad} = \sigma \epsilon A T^4 \]
where \(\sigma = 5.6704 \times 10^{-8} W/m^2 \cdot K^4\) (Stefan-Boltzmann constant), \(\epsilon\) represents the emissivity of the object’s surface, which is a value from 0-1.
A surface with a maximum emissivity of 1.0 is a blackbody radiator.
\[ P_{abs} = \sigma \epsilon A T_{env}^4 \]
A blackbody would absorb all the radiated energy it intercepts
Thus:
\[ P_{net} = P_{abs} - P_{rad} \]