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Chapter 3: Problem 2

Show that an ideal gas has the isothermal bulk modulus \(B=P\) when it obeys Boyle's law \(P v=\) const. For rapid changes in volume, the temperature changes in the gas do not have time to equalize The ideal gas then obeys the adiabatic equation \(P v \gamma=\) const, where \(\gamma\) is a constant. Show then that the adiabatic bulk modulus is \(B=\gamma P\).

Short Answer

Expert verified
Question: Show that for an ideal gas obeying Boyle's law and the adiabatic equation, the isothermal bulk modulus is equal to the pressure and the adiabatic bulk modulus is equal to γP. Answer: Using Boyle's law and the adiabatic equation, we have derived that the isothermal bulk modulus (B_iso) for an ideal gas is equal to its pressure (P), and the adiabatic bulk modulus (B_ad) is equal to γP, where γ is the adiabatic index.

Step by step solution

01

Define the bulk modulus

Bulk modulus (B) is a measure of the resistance of a fluid or gas to compression. It is defined as the pressure increase (ΔP) divided by the fractional decrease in volume (ΔV/V): \(B = -\dfrac{\Delta P}{\Delta V/V}\)
02

Apply Boyle's law to find the isothermal bulk modulus

According to Boyle's law, the product of pressure (P) and volume (V) is constant when the temperature is held constant: \(P V = \text{const}\) To find the isothermal bulk modulus, we need to find the change in pressure with respect to the change in volume, holding the temperature constant. Differentiate the above equation with respect to V: \(\dfrac{dP}{dV}V + P = 0\) Now, express the change in pressure divided by the change in volume as: \(-\dfrac{dP}{dV} = \dfrac{P}{V}\) Using the definition of isothermal bulk modulus, we have: \(B_{iso} = -\dfrac{\Delta P}{\Delta V/V} = P\) So, the isothermal bulk modulus for an ideal gas obeying Boyle's law is equal to its pressure.
03

Apply the adiabatic equation to find the adiabatic bulk modulus

For an ideal gas obeying the adiabatic equation, the product of pressure (P) and volume (V) raised to the power γ is constant: \(P V^{\gamma} = \text{const}\) To find the adiabatic bulk modulus, we need to find the change in pressure with respect to the change in volume, with no heat exchange between the gas and its surroundings. Differentiate the above equation with respect to V: \(\dfrac{dP}{dV}V^{\gamma} + \gamma P V^{\gamma - 1} = 0\) Now, express the change in pressure divided by the change in volume as: \(-\dfrac{dP}{dV} = \gamma \dfrac{P}{V}\) Using the definition of adiabatic bulk modulus, we have: \(B_{ad} = -\dfrac{\Delta P}{\Delta V/V} = \gamma P\) So, the adiabatic bulk modulus for an ideal gas obeying the adiabatic equation is equal to γP.

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