Step 1: Understanding the Topic
This question asks for the derivation of the isothermal compressibility ($\kappa_T$) for an ideal gas. Isothermal compressibility is a thermodynamic property that measures the fractional change in volume of a substance in response to a change in pressure, while the temperature is held constant.
Step 2: Key Formula or Approach
The derivation starts with the equation of state for an ideal gas. We will then mathematically follow the definition of $\kappa_T$ by performing the required differentiation.
1. Start with the Ideal Gas Law: $PV = nRT$.
2. Express Volume $V$ as a function of Pressure $P$.
3. Calculate the partial derivative $\left(\frac{dV}{dP}\right)_T$.
4. Substitute this derivative and the expression for $V$ back into the formula for $\kappa_T$.
Step 3: Detailed Derivation
A. Express V in terms of P:
From the ideal gas law, $PV = nRT$, we can solve for $V$:
\[
V = \frac{nRT}{P}
\]
B. Differentiate V with respect to P:
We need to find the derivative of $V$ with respect to $P$ at a constant temperature T. We can write $V = (nRT)P^{-1}$.
\[
\left(\frac{dV}{dP}\right)_T = nRT \frac{d}{dP}(P^{-1}) = nRT(-1 \cdot P^{-2}) = -\frac{nRT}{P^2}
\]
C. Substitute into the Compressibility Formula:
The definition of isothermal compressibility is:
\[
\kappa_T = -\frac{1}{V}\left(\frac{dV}{dP}\right)_T
\]
Now, we substitute the expressions for $V$ and the derivative:
\[
\kappa_T = -\frac{1}{\left(\frac{nRT}{P}\right)} \left(-\frac{nRT}{P^2}\right)
\]
The negative signs cancel out:
\[
\kappa_T = \left(\frac{P}{nRT}\right) \left(\frac{nRT}{P^2}\right)
\]
The terms $nRT$ cancel, and one factor of $P$ cancels:
\[
\kappa_T = \frac{1}{P}
\]
Step 4: Final Answer
The isothermal compressibility of an ideal gas is equal to the reciprocal of its pressure.
\[
\boxed{\kappa_T = \frac{1}{P}}
\]