Equation of Boyle’s law:A) dP/P = - dV/V B) dP/P = + dV/V C) {d^2}P/P = - dV/V D) {d^2}P/P = + dV/V

To understand Boyle's law and the options presented, we first need to recall what Boyle's law states. Boyle's law describes the relationship between the pressure (P) and volume (V) of a gas at constant temperature. It states that the pressure of a gas is inversely proportional to its volume when the temperature is held constant. Mathematically, this can be expressed as:

\[
PV = \text{constant}
\]

or

\[
P \propto \frac{1}{V}
\]

Now, let's analyze the differential form of Boyle's law:

1. Starting from \( PV = k \) (a constant), we can differentiate both sides with respect to volume \( V \):

\[
P \cdot dV + V \cdot dP = 0
\]

Rearranging gives us:

\[
dP = -\frac{P}{V} \cdot dV
\]

2. Dividing both sides by \( P \) and multiplying by \( V \):

\[
\frac{dP}{P} = -\frac{dV}{V}
\]

This equation indicates that an increase in volume (dV > 0) leads to a decrease in pressure (dP < 0), which is consistent with Boyle's law.

### Interpretation of the Options

Now let's analyze the given options in light of our derivation:

A) \(\frac{dP}{P} = -\frac{dV}{V}\)
**This is correct**. It directly represents the relationship between changes in pressure and volume as per Boyle's law.

B) \(\frac{dP}{P} = +\frac{dV}{V}\)
**This is incorrect**. It suggests that an increase in volume leads to an increase in pressure, which contradicts Boyle's law.

C) \(\frac{d^2P}{P} = -\frac{dV}{V}\)
**This is incorrect**. The second derivative does not apply here and doesn't represent the relationship.

D) \(\frac{d^2P}{P} = +\frac{dV}{V}\)
**This is also incorrect** for the same reasons as option C.

### Conclusion

The correct answer is **A) \(\frac{dP}{P} = -\frac{dV}{V}\)**. This equation captures the essence of Boyle’s law, showing that pressure decreases as volume increases, and vice versa, when the temperature is constant.