Log out Go to App
Go to App

Learning Materials

Features

Discover

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) \(\mathrm{P}\)

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

If du represents the increase in internal energy of a thermodynamic system and dw the work done by the system, which of the following statement is true? (A) \(\mathrm{du}=\mathrm{dw}\) in isothermal process (C) \(\mathrm{du}=-\mathrm{dw}\) in an aidabatic process (B) \(\mathrm{du}=\mathrm{dw}\) in aidabatic process (D) \(\mathrm{du}=-\mathrm{dw}\) in an isothermal process

A monoatomic ideal gas, intially at temperature \(1_{1}\) is enclosed in a cylinders fitted with a frictionless piston. The gas is allowed to expand adiabatically to a temperature \(\mathrm{T}_{2}\) by releasing the piston suddenly If \(\mathrm{L}_{1}\) and \(\mathrm{L}_{2}\) the lengths of the gas column before and after expansion respectively, then then \(\left(\mathrm{T}_{1} / \mathrm{T}_{2}\right)\) is given by (A) \(\left\\{\mathrm{L}_{1} / \mathrm{L}_{2}\right\\}^{(2 / 3)}\) (B) \(\left\\{\mathrm{L}_{2} / \mathrm{L}_{1}\right\\}^{(2 / 3)}\) (C) \(\left\\{\mathrm{L}_{1} / \mathrm{L}_{2}\right\\}\) (D) \(\left\\{\mathrm{L}_{2} / \mathrm{L}_{1}\right\\}\)

For an adiabatic process involving an ideal gas (A) \(\mathrm{P}^{\gamma-1}=\mathrm{T}^{\gamma-1}=\) constant (B) \(\mathrm{P}^{1-\gamma}=\mathrm{T}^{\gamma}=\) constant (C) \(\mathrm{PT}^{\gamma-1}=\) constant (D) \(\mathrm{P}^{\gamma-1}=\mathrm{T}^{\gamma}=\) constant

One \(\mathrm{kg}\) of adiatomic gas is at a pressure of $5 \times 10^{5}\left(\mathrm{~N} / \mathrm{m}^{2}\right)$ The density of the gas is \(\left\\{(5 \mathrm{~kg}) / \mathrm{m}^{3}\right\\}\) what is the energy of the gas due to its thermal motion ? (A) \(2.5 \times 10^{5} \mathrm{~J}\) (B) \(3.5 \times 10^{5} \mathrm{~J}\) (C) \(4.5 \times 10^{5} \mathrm{~J}\) (D) \(1.5 \times 10^{5} \mathrm{~J}\)

A gas expands from 1 liter to 3 liter at atmospheric pressure. The work done by the gas is about (A) \(200 \mathrm{~J}\) (B) \(2 \mathrm{~J}\) (C) \(300 \mathrm{~J}\) (D) \(2 \times 10^{5} \mathrm{~J}\)
See all solutions

Recommended explanations on English Textbooks

Sociolinguistics

Read Explanation

Language Acquisition

Read Explanation

English Language Study

Read Explanation

Phonology

Read Explanation

Global English

Read Explanation

Language Analysis

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