Helium gas with a volume of 3.203.20 L, under a pressure of 0.1800.180 atm and at 41.041.0°C, is warmed until both pressure and volume are doubled. What is the final temperature?

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 2a

Chapter 18, Problem 2a

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. What is the final temperature?

Verified step by step guidance

Start by converting the initial temperature from Celsius to Kelvin. Use the formula: $ T(K) = T(°C) + 273.15 $. For 41.0°C, calculate the temperature in Kelvin.

Apply the ideal gas law in the form $ PV = nRT $ to relate the initial and final states of the gas. Since the number of moles $ n $ and the gas constant $ R $ remain constant, you can use the combined gas law: $ \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} $.

Substitute the known values into the combined gas law. The initial pressure $ P_1 = 0.180 $ atm, initial volume $ V_1 = 3.20 $ L, and initial temperature $ T_1 $ in Kelvin. The final pressure $ P_2 = 2 \times 0.180 $ atm and final volume $ V_2 = 2 \times 3.20 $ L.

Rearrange the combined gas law to solve for the final temperature $ T_2 $: $ T_2 = \frac{P_2V_2T_1}{P_1V_1} $.

Substitute the values into the equation and solve for $ T_2 $ to find the final temperature in Kelvin. Remember to check your units to ensure they are consistent throughout the calculation.

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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, PV=nRT, relates the pressure (P), volume (V), and temperature (T) of a gas with its amount in moles (n) and the gas constant (R). It is crucial for understanding how changes in pressure, volume, and temperature affect a gas. In this problem, it helps determine the final temperature when both pressure and volume are doubled.

Temperature Conversion

Temperature conversion is essential when dealing with gas laws, as calculations require temperatures in Kelvin. To convert from Celsius to Kelvin, add 273.15 to the Celsius temperature. This conversion is necessary to apply the Ideal Gas Law correctly, ensuring accurate calculations of the final temperature.

Direct and Inverse Proportionality

Understanding direct and inverse proportionality helps analyze how variables in the Ideal Gas Law interact. When pressure and volume are doubled, the relationship between these variables and temperature must be considered. Direct proportionality implies that if pressure and volume increase, temperature must also increase to maintain the equation's balance.

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Textbook Question

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. How many moles of helium are in the tank?

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Textbook Question

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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Textbook Question

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?

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