Write a mathematical statement for the change in entropy, ds, for the case where no work
is done irrespective of the path.
The change in entropy, denoted as dS, for a process where no work is done irrespective of the path can be represented by the equation:
dS = δQ / T
where δQ is the heat transfer during the process, and T is the temperature at which the process occurs.
To understand how this equation is derived, consider the second law of thermodynamics, which states that the change in entropy of a closed system is determined by the heat transfer and the temperature. Mathematically, it can be expressed as:
dS = δQ / T + δS_surr
where δS_surr is the change in entropy of the surroundings.
However, when no work is done irrespective of the path, the process is considered to be internally reversible. In this case, the entropy change of the system and surroundings cancel each other out, resulting in:
δS_surr = -δS_sys
Therefore, the equation simplifies to:
dS = δQ / T
This equation represents the change in entropy for a process where no work is done, regardless of the path taken.