Question

For an adiabatic expansion of a perfect gas, the value of $\\{(\Delta \mathrm{P}) / \mathrm{P}\\}$ is equal to (A) \(-\sqrt{\gamma}\\{(\Delta \mathrm{r}) / \mathrm{v}\\}\) (B) \(-\\{(\Delta \mathrm{v}) / \mathrm{v}\\}\) (C) \(-\gamma^{2}\\{(\Delta \mathrm{v}) / \mathrm{v}\\}\) (D) \(-\gamma\\{(\Delta \mathrm{v}) / \mathrm{v}\\}\)

Short answer

The short answer is: \(\frac{\Delta P}{P} = -\gamma\frac{\Delta V}{V}\). The correct option is (D).

Step-by-step solution

Determine the Adiabatic Equation for Perfect Gases

The adiabatic equation for a perfect gas can be written in the following form: \[PV^{\gamma} = \text{constant},\] where \(P\) is the pressure, \(V\) is the volume of the gas, and \(\gamma\) is the adiabatic index. Since we want the fractional change in pressure, let's consider an infinitesimal change in the volume of the gas.

Examine the Change in Pressure and Volume

When the volume of the gas changes from \(V\) to \(V+\Delta V\), the new pressure \(P + \Delta P\) can be related to the old pressure \(P\) as: \[(P + \Delta P)(V + \Delta V)^{\gamma} = PV^{\gamma}. \]

Expanding and Rearranging the Equation

Let's expand and simplify the above equation: \[P(V + \Delta V)^{\gamma} + (\Delta P)(V + \Delta V)^{\gamma} = PV^{\gamma}. \] Divide both sides by \(PV^{\gamma}\): \[1 + \frac{\Delta P}{P} = \left(\frac{V + \Delta V}{V}\right)^{\gamma}.\] Now rewrite the left side with \(\frac{\Delta P}{P}\) as the subject: \[\frac{\Delta P}{P} = \left(\frac{V + \Delta V}{V}\right)^{\gamma} - 1.\]

Taylor Expansion

Now perform a first-order Taylor expansion of the right side of the equation assuming \(\Delta V\) is small. It gives: \[\frac{\Delta P}{P} \approx \gamma \left(\frac{\Delta V}{V}\right). \] This is the equation for the fractional change in pressure during adiabatic expansion.

Compare with Options

Now compare the above equation with the options and select the correct one: \( -\gamma\frac{\Delta V}{V}\) matches the given option (D): \[\frac{\Delta P}{P} = -\gamma\frac{\Delta V}{V}. \] Hence, the correct answer is (D).