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Chapter 7: Problem 931

The bulk modulus of an ideal gas at constant temperature..... (A) is equal to its volume \(\mathrm{V}\) (B) is equal to \(\mathrm{P} / 2\) (C) is equal to its pressure \(\mathrm{P}\) (D) cannot be determined

Short Answer

Expert verified
The bulk modulus of an ideal gas at constant temperature is equal to its pressure \(\mathrm{P}\) (option C). This conclusion was reached by differentiating the ideal gas law and substituting the resulting expression for ΔP into the bulk modulus equation, which yielded \(K = P\).

Step by step solution

01

Recall the definition of bulk modulus

The bulk modulus (K) of a substance measures its resistance to compression under pressure. For an ideal gas, the bulk modulus is defined as the ratio of applied pressure (ΔP) to the fractional change in volume (ΔV/V): \[K = -\dfrac{ΔP}{ΔV/V}\]
02

Write the ideal gas law

For an ideal gas, the relationship between pressure, volume, and temperature is given by the ideal gas law: \[PV = nRT\] where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.
03

Differentiate the ideal gas law

Since bulk modulus is defined for a constant temperature, we can treat T as constant and differentiate the ideal gas law with respect to volume. Using the chain rule, we get: \[\dfrac{dP}{dV}V + P = 0\]
04

Rearrange to find ΔP

Solving for the change in pressure (ΔP), we can write: \[ΔP = - \dfrac{P}{V} ΔV\]
05

Substitute ΔP back into the bulk modulus equation

Now, we will substitute the value of ΔP we found into the bulk modulus equation: \[K = -\dfrac{-\displaystyle\dfrac{P}{V} ΔV}{ΔV/V}\]
06

Simplify the expression

We can cancel out the ΔV terms in the numerator and denominator and simplify our expression for bulk modulus: \[K = P\] This means the bulk modulus of an ideal gas at constant temperature is equal to its pressure. So, the correct answer is: (C) is equal to its pressure \(\mathrm{P}\)

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