Five moles of monatomic ideal gas have initial pressure 2.50×1032.50\$$times10^3 Pa $$and initial volume 2.102.10 m3. While undergoing an adiabatic expansion, the gas does 14801480 J of work. What is the final pressure of the gas after the expansion?

Ch 19: The First Law of Thermodynamics

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 19: The First Law of Thermodynamics

Problem 28

Chapter 19, Problem 28

Five moles of monatomic ideal gas have initial pressure $2.50 times 10 cubed$ Pa and initial volume $2.10$ m³. While undergoing an adiabatic expansion, the gas does $1480$ J of work. What is the final pressure of the gas after the expansion?

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Identify the type of process: The problem states that the gas undergoes an adiabatic expansion. In an adiabatic process, no heat is exchanged with the surroundings, so the first law of thermodynamics simplifies to $ \Delta U = -W $, where $ \Delta U $ is the change in internal energy and $ W $ is the work done by the gas.

Use the adiabatic condition for an ideal gas: For a monatomic ideal gas, the adiabatic condition is given by $ PV^\gamma = \text{constant} $, where $ \gamma = \frac{5}{3} $ for a monatomic gas. This relationship will help us find the final pressure.

Calculate the change in internal energy: For a monatomic ideal gas, the change in internal energy $ \Delta U $ is given by $ \Delta U = \frac{3}{2} nR \Delta T $. However, since $ \Delta U = -W $ and $ W = 1480 \text{ J} $, we can use this to find $ \Delta U $.

Relate initial and final states using the adiabatic condition: Since $ PV^\gamma = \text{constant} $, we can write $ P_i V_i^\gamma = P_f V_f^\gamma $. We know $ P_i = 2.50 \times 10^3 \text{ Pa} $ and $ V_i = 2.10 \text{ m}^3 $. We need to find $ V_f $ using the work done and then solve for $ P_f $.

Solve for the final pressure $ P_f $: Use the relationship $ P_f = P_i \left( \frac{V_i}{V_f} \right)^\gamma $ to find the final pressure after determining $ V_f $ from the work-energy relationship. This will give you the final pressure of the gas after the adiabatic expansion.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics, expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature. It describes the relationship between these variables for an ideal gas, allowing us to predict how a gas will behave under different conditions.

Adiabatic Process

An adiabatic process is a thermodynamic process in which no heat is exchanged with the surroundings. For an ideal gas, this means that any change in internal energy is due to work done on or by the system. In an adiabatic expansion, the gas does work on its surroundings, leading to a decrease in internal energy and temperature.

First Law of Thermodynamics

The First Law of Thermodynamics, also known as the law of energy conservation, states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. In an adiabatic process, since no heat is exchanged, the change in internal energy is equal to the negative of the work done by the gas.

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Textbook Question

A monatomic ideal gas that is initially at $1.50 \times 10^{5}$ Pa and has a volume of $0.0800$ m³ is compressed adiabatically to a volume of $0.0400$ m³. What is the final pressure?

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Textbook Question

A cylinder contains 0.100 mol of an ideal monatomic gas. Initially the gas is at $1.00 \times 10^{5}$ Pa and occupies a volume of $2.50 \times 10^{- 3}$ m³. If the gas is allowed to expand to twice the initial volume, find the final temperature (in kelvins) and pressure of the gas if the expansion is (i) isothermal; (ii) isobaric; (iii) adiabatic.

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Textbook Question

On a warm summer day, a large mass of air (atmospheric pressure $1.01 \times 10^{5}$ Pa) is heated by the ground to $26.0$ °C and then begins to rise through the cooler surrounding air. (This can be treated approximately as an adiabatic process; why?) Calculate the temperature of the air mass when it has risen to a level at which atmospheric pressure is only $0.850 \times 10^{5}$ Pa. Assume that air is an ideal gas, with $g equals 1.40$ . (This rate of cooling for dry, rising air, corresponding to roughly $1$ C° per $100$ m of altitude, is called the dry adiabatic lapse rate.)

Textbook Question

A monatomic ideal gas that is initially at $1.50$\times$10^5$ Pa and has a volume of $0.0800$ m³ is compressed adiabatically to a volume of $0.0400$ m³. What is the ratio of the final temperature of the gas to its initial temperature? Is the gas heated or cooled by this compression?

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Textbook Question

A player bounces a basketball on the floor, compressing it to $80.0 percent sign$ of its original volume. The air (assume it is essentially N₂ gas) inside the ball is originally at $20.0$ °C and $2.00$ atm. The ball's inside diameter is $23.9$ cm. What temperature does the air in the ball reach at its maximum compression? Assume the compression is adiabatic and treat the gas as ideal.

Textbook Question

The temperature of $0.150$ mol of an ideal gas is held constant at $77.0$ °C while its volume is reduced to $25.0 percent sign$ of its initial volume. The initial pressure of the gas is $1.25$ atm. Does the gas exchange heat with its surroundings? If so, how much? Does the gas absorb or liberate heat?

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Five moles of monatomic ideal gas have initial pressure 2.50×1032.50\(\times\)10^3 Pa and initial volume 2.102.10 m3. While undergoing an adiabatic expansion, the gas does 14801480 J of work. What is the final pressure of the gas after the expansion?

Verified video answer for a similar problem:

Key Concepts

Ideal Gas Law

Adiabatic Process

First Law of Thermodynamics

Five moles of monatomic ideal gas have initial pressure $2.50 times 10 cubed$ Pa and initial volume $2.10$ m³. While undergoing an adiabatic expansion, the gas does $1480$ J of work. What is the final pressure of the gas after the expansion?