Main variables:

Main Equations:

To convert from Fahrenheit (\(T_F\)) to Celsius (\(T_C\)):

\begin{equation*}
T_C = \frac{5}{9} ( T_F – 32 ).
\end{equation*}

To convert Celsius (\(T_C\)) to Kelvin (\(T_K\)):

\begin{equation*}
T_K = T_C + 273.15.
\end{equation*}

Linear thermal expansion:

\begin{equation*}
\Delta \ell = \alpha \ell_0 \Delta T.
\end{equation*}

Volume thermal expansion:

\begin{equation*}
\Delta V = \beta V_0 \Delta T,
\end{equation*}

where \(\beta \approx 3 \alpha \).

Heat by a temperature difference:

\begin{equation*}
Q = mc \Delta T.
\end{equation*}

In a closed system, heat is conserved:

\begin{equation*}
\sum Q_i = 0.
\end{equation*}

The latent heat of a system changes its state (i.e. from liquid to gas):

\begin{equation*}
Q = \pm mL.
\end{equation*}

The ideal gas equation is:

\begin{equation*}
P V = nRT.
\end{equation*}

The molar heat capacity at constant volume for an ideal monoatomic gas is:

\begin{equation*}
C_V = \frac{3}{2} R.
\end{equation*}

The molar heat capacity at constant volume for an ideal diatomic gas is:

\begin{equation*}
C_V = \frac{5}{2} R.
\end{equation*}

The molar heat capacity at constant pressure can be related to \(C_V\) as follows:

\begin{equation*}
C_P = C_V + R.
\end{equation*}

First law of thermodynamics:

\begin{equation*}
\Delta U = Q – W,
\end{equation*}

where \(W\) is the work done by the system.

The internal change of energy for any process is:

\begin{equation*}
\Delta U = n C_V \Delta T.
\end{equation*}

The work done by the system, in differential form, is:

\begin{equation*}
W = \int P dV.
\end{equation*}

For an isobaric process, where the pressure is constant, the work is:

\begin{equation*}
W = P \Delta V.
\end{equation*}

Heat for an isobaric process:

\begin{equation*}
Q = n C_P \Delta T.
\end{equation*}

For an isochoric process, where the volume is constant, the work is:

\begin{equation*}
W = 0.
\end{equation*}

Heat for an isochoric process:

\begin{equation*}
Q = n C_V \Delta T.
\end{equation*}

For an isothermic process, where the temperature is constant, the work is:

\begin{equation*}
W = n R T \ln \left( \frac{V_f}{V_i} \right).
\end{equation*}

Heat for an isothermic process:

\begin{equation*}
Q = n R T \ln \left( \frac{V_f}{V_i} \right).
\end{equation*}

An adiabatic process is an isolated process where the characteristic equation is:

\begin{equation*}
P V ^\gamma = \text{constant}.
\end{equation*}

The adiabatic constant is:

\begin{equation*}
\gamma = \frac{C_P}{C_V}.
\end{equation*}

For an adiabatic process, the work is:

\begin{equation*}
W = \frac{1}{\gamma – 1} (P_i V_i – P_f V_f).
\end{equation*}

Heat for an adiabatic process:

\begin{equation*}
Q = 0.
\end{equation*}

The efficiency of a heat engine that takes heat \(Q_H\) from a source an transforms some of it into work \(W\) is:

\begin{equation*}
e = \frac{W}{Q_H}.
\end{equation*}

The efficiency can be written as:

\begin{equation*}
e = 1 – \frac{|Q_C|}{Q_H}.
\end{equation*}

The work can be related to the absorbed heat \(Q_H\) and the discarded heat \(Q_C\) as:

\begin{equation*}
W = Q_H – |Q_C|.
\end{equation*}

The entropy of a system can be expressed as:

\begin{equation*}
\Delta S = \int \frac{dQ}{T}.
\end{equation*}

Entropy for an adiabatic process:

\begin{equation*}
\Delta S = 0.
\end{equation*}

Entropy for an isotherm process:

\begin{equation*}
\Delta S = \frac{ \Delta Q} {T}.
\end{equation*}

Entropy for an isobaric process:

\begin{equation*}
\Delta S = n C_P \ln \left( \frac{T_f}{T_i} \right).
\end{equation*}

Entropy for an isochoric process:

\begin{equation*}
\Delta S = n C_V \ln \left( \frac{T_f}{T_i} \right).
\end{equation*}