Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 19: Problem 53

Suppose \(15.0 \mathrm{~L}\) of an ideal monatomic gas at a pressure of \(1.50 \cdot 10^{5} \mathrm{kPa}\) is expanded adiabatically (no heat transfer) until the volume is doubled. a) What is the pressure of the gas at the new volume? b) If the initial temperature of the gas was \(300 . \mathrm{K},\) what is its final temperature after the expansion?

Short Answer

Expert verified
Based on the given initial conditions and the fact that the ideal monatomic gas undergoes an adiabatic expansion, calculate the final pressure and temperature of the gas. Initial volume (\(V_1\)): 15.0 L Initial pressure (\(P_1\)): 1.50 x 10^5 kPa Initial temperature (\(T_1\)): 300 K Final volume (\(V_2\)): 30.0 L Answer: Final pressure (\(P_2\)): 3.35 x 10^4 kPa Final temperature (\(T_2\)): 134 K

Step by step solution

01

Write down the initial conditions and the adiabatic equation for ideal gases

Given information: Initial volume, \(V_1 = 15.0 L\) Initial pressure, \(P_1 = 1.50 \times 10^5 kPa = 1.50 \times 10^8 Pa\) (1 kPa = 1000 Pa) Initial temperature, \(T_1 = 300 K\) The final volume is twice the initial volume, \(V_2 = 2V_1 = 30.0 L\) To solve the problem, we will use the adiabatic equation for ideal gases: \(P_1 V_1^{\gamma} = P_2 V_2^{\gamma} \Rightarrow P_2=\frac{P_1 V_1^{\gamma}}{V_2^{\gamma}}\) Where \(P_1\) and \(P_2\) are the initial and final pressures, \(V_1\) and \(V_2\) are the initial and final volumes, and \(\gamma\) is the adiabatic index, which is equal to \(\frac{5}{3}\) for monatomic ideal gases.
02

Calculate the final pressure using the adiabatic equation

Substituting the given values and \(\gamma=\frac{5}{3}\) into the adiabatic equation: \(P_2 =\frac{P_1 V_1^{\frac{5}{3}}}{V_2^{\frac{5}{3}}}=\frac{(1.50 \times 10^8 Pa)(15.0 L)^{\frac{5}{3}}}{(30.0 L)^{\frac{5}{3}}}\) Solve for \(P_2\): \(P_2 \approx 3.35 \times 10^7 Pa\) or \(3.35 \times 10^4 kPa\) The final pressure of the gas at the new volume is approximately \(3.35 \times 10^4 kPa\).
03

Use the ideal gas law to find the final temperature

The ideal gas law is given by \(PV=nRT \Rightarrow \frac{P_1 V_1}{T_1}=\frac{P_2 V_2}{T_2}\) We need to find \(T_2\), the final temperature. First, we can solve for \(\frac{P_1 V_1}{T_1}\): \(\frac{P_1 V_1}{T_1}=\frac{(1.50 \times 10^8 Pa)(15.0 L)}{300 K} \approx 7.50 \times 10^6 \frac{Pa L}{K}\) Now, we'll use the final pressure and volume to find \(T_2\): \(T_2=\frac{P_2 V_2}{\frac{P_1 V_1}{T_1}} =\frac{(3.35 \times 10^7 Pa)(30.0 L)}{7.50 \times 10^6 \frac{Pa L}{K}}\) Solve for \(T_2\): \(T_2 \approx 134 K\) The final temperature of the gas after the expansion is approximately \(134 K\).

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) What is the root-mean-square speed for a collection of helium- 4 atoms at \(300 . \mathrm{K} ?\) b) What is the root-mean-square speed for a collection of helium- 3 atoms at 300 . K?

Air at 1.00 atm is inside a cylinder \(20.0 \mathrm{~cm}\) in radius and \(20.0 \mathrm{~cm}\) in length that sits on a table. The top of the cylinder is sealed with a movable piston. A \(20.0-\mathrm{kg}\) block is dropped onto the piston. From what height above the piston must the block be dropped to compress the piston by \(1.00 \mathrm{~mm} ? 2.00 \mathrm{~mm} ? 1.00 \mathrm{~cm} ?\)

As noted in the text, the speed distribution of molecules in the Earth's atmosphere has a significant impact on its composition. a) What is the average speed of a nitrogen molecule in the atmosphere, at a temperature of \(18.0^{\circ} \mathrm{C}\) and a (partial) pressure of \(78.8 \mathrm{kPa} ?\) b) What is the average speed of a hydrogen molecule at the same temperature and pressure?

Molar specific heat at constant pressure, \(C_{p}\), is larger than molar specific heat at constant volume, \(C_{V}\), for a) a monoatomic ideal gas. b) a diatomic atomic gas. c) all of the above. d) none of the above.

At Party City, you purchase a helium-filled balloon with a diameter of \(40.0 \mathrm{~cm}\) at \(20.0^{\circ} \mathrm{C}\) and at \(1.00 \mathrm{~atm} .\) a) How many helium atoms are inside the balloon? b) What is the average kinetic energy of the atoms? c) What is the root-mean-square speed of the atoms?
See all solutions

Recommended explanations on Physics Textbooks

Electricity

Read Explanation

Scientific Method Physics

Read Explanation

Circular Motion and Gravitation

Read Explanation

Rotational Dynamics

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Electric Charge Field and Potential

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