Degrees of Freedom

In the discussion of translational kinetic energy above, the molecule was implicitly free to move anywhere in three-dimensional space, needing three coordinates ($x$, $y$, $z$) and three velocity components ($v_x$, $v_y$, $v_z$) to fully describe its motion.

More generally, the degrees of freedom of a system are defined as the total number of independent coordinates (or independent quantities) required to completely describe its position and configuration.

• A molecule restricted to move on a flat, two-dimensional surface needs only two coordinates ($x$, $y$) and two velocity components ($v_x$, $v_y$) -- it has 2 translational degrees of freedom.
• A molecule restricted to move along a single straight line needs only one coordinate ($x$) and one velocity component ($v_x$) -- it has 1 translational degree of freedom.
• A molecule completely free to move in three-dimensional space (the general case) has 3 translational degrees of freedom. …