Log out Go to App
Go to App

Learning Materials

Features

Discover

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 \(\mathrm{P}\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A body floats in water with one-third of its volume above the surface of water. It is placed in oil it floats with half of: Its volume above the surface of the oil. The specific gravity of the oil is. (A) \((5 / 3)\) (B) \((4 / 3)\) (C) \((3 / 2)\) (D) 1

What is the possible value of posson's ratio? (A) 1 (B) \(0.9\) (C) \(0.8\) (D) \(0.4\)

Read the assertion and reason carefully and mark the correct option given below. (a) If both assertion and reason are true and the reason is the correct explanation of the assertion. (b) If both assertion and reason are true but reason is not the correct explanation of the assertion. (c) If assertion is true but reason is false. (d) If the assertion and reason both are false. Assertion: When height of a tube is less then liquid rise in the capillary tube the liquid does not overflow. Reason: Product of radius of meniscus and height of liquid incapilling tube always remains constant. (A) a (B) \(b\) (C) c (D) d

Read the assertion and reason carefully to mark the correct option out of the option given below. (a) If both assertion and reason are true and the reason is the correct explanation of the reason. (b) If both assertion and reason are true but reason is not the correct explanation of the assertion. (c) If assertion is true but reason is false. (d) If the assertion and reason both are false. (e) If assertion is false but reason is true. Assertion: The molecules of \(0^{\circ} \mathrm{C}\) ice and $0^{\circ} \mathrm{C}$ water will have same potential energy. Reason: Potential energy depends only on temperature of the system. (A) a (B) \(b\) (C) \(\mathrm{c}\) (D) d (E) e

A soap bubble of radius \(r\) is blown up to form a bubble of radius $2 \mathrm{r}\( under isothermal conditions if the \)\mathrm{T}$ is the surface tension of soap solution the energy spent in the slowing is. (A) \(3 \pi \mathrm{Tr}^{2}\) (B) \(6 \pi \mathrm{Tr}^{2}\) (C) \(12 \pi \mathrm{Tr}^{2}\) (D) \(24 \pi \mathrm{Tr}^{2}\)
See all solutions

Recommended explanations on English Textbooks

Key Concepts in Language and Linguistics

Read Explanation

Creative Story

Read Explanation

Text Comparison

Read Explanation

Global English

Read Explanation

Language Analysis

Read Explanation

Graphology

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free