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Chapter 8: Problem 1147

The isothermal Bulk modulus of an ideal gas at pressure \(\mathrm{P}\) is (A) \(\mathrm{vP}\) (B) \(\mathrm{P}\) (C) \((\mathrm{p} / 2)\) (D) \((\mathrm{p} / \mathrm{v})\)

Short Answer

Expert verified
The isothermal bulk modulus of an ideal gas at pressure P is given by the formula \(B = \frac{PV}{V}\), which simplifies to \(B = P\). Therefore, the correct answer is (B) \(\mathrm{P}\).

Step by step solution

01

Define the isothermal bulk modulus

The isothermal bulk modulus (B) is defined as the ratio of the infinitesimal change in pressure (dP) to the fractional change in volume (-dV/V) at constant temperature: \[B = - V\frac{dP}{dV}\Bigg|_{T=constant}\]
02

Ideal Gas Equation of State

For an ideal gas, the equation of state is given by the Ideal Gas Law: \[PV = nRT\] Where P is pressure, V is volume, n is the number of moles of gas, R is the universal gas constant, and T is temperature.
03

Taking the derivative

Differentiate both sides of the ideal gas equation with respect to volume at a constant temperature: \[\frac{dP}{dV} = -\frac{nRT}{V^2}\]
04

Calculate the bulk modulus

Plug the expression for the derivative of pressure with respect to volume into the expression for bulk modulus derived earlier: \[B = - V\frac{dP}{dV}\Bigg|_{T=constant} = - V (-\frac{nRT}{V^2}) = \frac{nRT}{V}\]
05

Express the bulk modulus in terms of pressure

Since \(PV = nRT\), we can replace the \(nRT\) term in the expression for bulk modulus with \(PV\): \[B = \frac{PV}{V} = P\] Hence, the correct answer is: (B) \(\mathrm{P}\)

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