what is the residual entropy at T=0 k for 1 mol of C6H5Br Bromobenzene?
The residual entropy at T=0 K for 1 mol of C6H5Br Bromobenzene is zero. This is because all molecular motion stops at absolute zero, and therefore there is no entropy.
To calculate the residual entropy at T=0 K for 1 mol of C6H5Br (Bromobenzene), we first need to find the number of microstates corresponding to a given macrostate.
The macrostate we are interested in is a 1 mol sample of C6H5Br. At T=0 K, all the molecules will be in their lowest energy state (ground state). Since the ground state is unique, there is only one microstate associated with this macrostate.
Therefore, the residual entropy (S_res) at T=0 K for 1 mol of C6H5Br is equal to zero.
To calculate the residual entropy at T=0 K for 1 mole of Bromobenzene (C6H5Br), we need to consider the vibrational and rotational degrees of freedom of the molecule.
The residual entropy at absolute zero temperature (T=0 K) is defined as the entropy that remains due to the degeneracy of the molecule's quantum states. At this temperature, the molecule is at its ground state, and only its zero-point energy remains.
To calculate the residual entropy at T=0 K, follow these steps:
1. Count the total number of atoms in the molecule. Bromobenzene (C6H5Br) contains 7 atoms: 6 carbon (C) atoms, 5 hydrogen (H) atoms, and 1 bromine (Br) atom.
2. Calculate the number of rotational degrees of freedom. For a nonlinear molecule like Bromobenzene, the number of rotational degrees of freedom is given by:
F_rot = 3N - 6
Where N is the total number of atoms in the molecule. In this case, F_rot = 3(7) - 6 = 15.
3. Calculate the number of vibrational degrees of freedom. A molecule with N atoms has 3N - 6 vibrational degrees of freedom for a linear molecule, and 3N - 5 for a nonlinear molecule. Bromobenzene is a nonlinear molecule, so it has 3N - 5 = 3(7) - 5 = 16 vibrational degrees of freedom.
4. Calculate the total residual entropy. The residual entropy (S_res) is given by the equation:
S_res = R * ln(g)
Where R is the gas constant (8.314 J/(mol K)), and g is the degeneracy of the molecule's quantum states, which can be calculated using:
g = 2^(F_rot + F_vib)
Where F_rot is the number of rotational degrees of freedom, and F_vib is the number of vibrational degrees of freedom.
Plugging in the values:
- R = 8.314 J/(mol K)
- F_rot = 15
- F_vib = 16
Calculate g = 2^(15 + 16) = 2^31 = 2,147,483,648
Finally, calculate the residual entropy:
S_res = 8.314 J/(mol K) * ln(2,147,483,648)
The answer will be the calculated result of S_res in J/(mol K).
NOTE: The residual entropy is typically very small or even zero for simple molecules like Bromobenzene at T=0 K as most molecular vibrations and rotations are frozen out at low temperatures.