To calculate the energy of the vibrational mode that gives rise to an absorption band at 1712 cm, we can use the formula:E = h * where E is the energy, h is the Planck's constant 6.626 x 10 Js , and is the frequency of the vibration in Hz.First, we need to convert the wavenumber 1712 cm to frequency Hz . We can do this using the speed of light c and the following formula: = c * wavenumberThe speed of light c is approximately 3.0 x 10 m/s. However, the wavenumber is given in cm, so we need to convert it to m:1712 cm * 1 m / 100 cm = 17120 mNow, we can calculate the frequency: = 3.0 x 10 m/s * 17120 m = 5.136 x 10 HzFinally, we can calculate the energy of the vibrational mode:E = 6.626 x 10 Js * 5.136 x 10 Hz = 3.40 x 10 JSo, the energy of the vibrational mode that gives rise to an absorption band at 1712 cm in the IR spectrum of a molecule of acetic anhydride is approximately 3.40 x 10 J.