Question

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

Answer 1

Hint: Boyle’s law is also known as the pressure-volume relationship which states that when the temperature is constant the pressure of a fixed amount of gas varies inversely with its volume.

Complete step by step solution:
-We should first start by seeing what Boyle’s law is: Boyle’s law is an experimental gas law that describes the relationship between pressure and volume when the temperature is constant. It states that pressure of a gas tends to increase when the volume of a container decreases.
  It can be written as: $P\alpha 1/V$ (Temperature is constant)
                                            \[P = k(1/V)\]
                                              \[PV = k\]
  where k is proportionality constant whose value depends on the temperature of the gas, amount of gas and units of P and V.
So, for a gas at temperature T, occupying volume V1 at pressure P1 and volume V2 at pressure P2, Boyle’s law can be written as:
                                            \[{P_1}{V_1} = {P_2}{V_2} = k\]
                                       \[{P_1}/{P_2} = {V_2}/{V_1}\]
-Now, we use the equation: $PV = k$, to derive the differentiated form of Boyle’s Law. Differentiate the equation: $PV = k$ .
 By differentiating it we get: \[dP/P + dV/V = 0\]
So, \[dP/P = - dV/V\]

So, the correct option is, (A) $dP/P = - dV/V$

Note: The Boyle’s law formula: ${P_1}/{P_2} = {V_2}/{V_1}$; is valid for only a gas at a constant temperature but varying pressure and temperature and not for 2 different gases.