A cylindrical tank has a tight-fitting piston that allows the volume of the tank to be changed. The tank originally contains 0.1100.110 m3 of air at a pressure of 0.3550.355 atm. The piston is slowly pulled out until the volume of the gas is increased to 0.3900.390 m3. If the temperature remains constant, what is the final value of the pressure?

Ch 18: Thermal Properties of Matter

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 18: Thermal Properties of Matter

Problem 3

Chapter 18, Problem 3

A cylindrical tank has a tight-fitting piston that allows the volume of the tank to be changed. The tank originally contains $0.110$ m³ of air at a pressure of $0.355$ atm. The piston is slowly pulled out until the volume of the gas is increased to $0.390$ m³. If the temperature remains constant, what is the final value of the pressure?

Verified step by step guidance

Identify the initial conditions: The initial volume $ V_1 $ is 0.110 m$^3$ and the initial pressure $ P_1 $ is 0.355 atm.

Recognize that the problem involves an isothermal process, meaning the temperature remains constant. Therefore, Boyle's Law applies, which states that $ P_1 V_1 = P_2 V_2 $.

Determine the final volume $ V_2 $, which is given as 0.390 m$^3$.

Rearrange Boyle's Law to solve for the final pressure $ P_2 $: $ P_2 = \frac{P_1 V_1}{V_2} $.

Substitute the known values into the equation: $ P_2 = \frac{0.355 \text{ atm} \times 0.110 \text{ m}^3}{0.390 \text{ m}^3} $. Calculate $ P_2 $ using these values.

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Key Concepts

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

Boyle's Law

Boyle's Law states that for a given amount of gas at constant temperature, the pressure of the gas is inversely proportional to its volume. This means that if the volume of the gas increases, the pressure decreases, and vice versa, as long as the temperature remains unchanged. The relationship is expressed as P1V1 = P2V2.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in physics that relates the pressure, volume, temperature, and number of moles of a gas. It is 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. In this scenario, the temperature is constant, simplifying the relationship to Boyle's Law.

Isothermal Process

An isothermal process is a thermodynamic process in which the temperature of the system remains constant. In the context of the question, the expansion of the gas in the tank is isothermal, meaning that the temperature does not change as the volume increases. This allows the use of Boyle's Law to determine the final pressure of the gas.

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A $20.0$ -L tank contains $4.86$\times$10^{-4}$ kg of helium at $18.0$ °C. The molar mass of helium is $4.00$ g/mol. What is the pressure in the tank, in pascals and in atmospheres?

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Textbook Question

Helium gas with a volume of $3.20$ L, under a pressure of $0.180$ atm and at $41.0$ °C, is warmed until both pressure and volume are doubled. How many grams of helium are there? The molar mass of helium is $4.00$ g/mol.

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