Ch 18: Thermal Properties of Matter

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 18: Thermal Properties of Matter

Problem 9

Chapter 18, Problem 9

A large cylindrical tank contains $0.750$ m³ of nitrogen gas at $27$ °C and $7.50 times 10 cubed$ Pa (absolute pressure). The tank has a tight-fitting piston that allows the volume to be changed. What will be the pressure if the volume is decreased to $0.410$ m³ and the temperature is increased to $157$ °C?

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Start by identifying the initial and final states of the gas. The initial state has a volume $ V_1 = 0.750 \, \text{m}^3 $, temperature $ T_1 = 27^\circ \text{C} $, and pressure $ P_1 = 7.50 \times 10^3 \, \text{Pa} $. The final state has a volume $ V_2 = 0.410 \, \text{m}^3 $ and temperature $ T_2 = 157^\circ \text{C} $.

Convert the temperatures from Celsius to Kelvin, as the ideal gas law requires temperatures in Kelvin. Use the formula $ T(K) = T(^\circ C) + 273.15 $. Thus, $ T_1 = 27 + 273.15 = 300.15 \, \text{K} $ and $ T_2 = 157 + 273.15 = 430.15 \, \text{K} $.

Apply the combined gas law, which relates the pressure, volume, and temperature of a gas: $ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} $. This equation assumes the amount of gas remains constant.

Rearrange the combined gas law to solve for the final pressure $ P_2 $: $ P_2 = P_1 \times \frac{V_1}{V_2} \times \frac{T_2}{T_1} $.

Substitute the known values into the equation: $ P_2 = 7.50 \times 10^3 \, \text{Pa} \times \frac{0.750 \, \text{m}^3}{0.410 \, \text{m}^3} \times \frac{430.15 \, \text{K}}{300.15 \, \text{K}} $. Calculate $ P_2 $ using these values to find the final pressure.

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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 physics that relates the pressure, volume, and temperature of an ideal gas. It is expressed as PV = nRT, where P is pressure, V is volume, T is temperature, n is the number of moles, and R is the ideal gas constant. This law helps predict how a gas will respond to changes in pressure, volume, and temperature.

Charles's Law

Charles's Law states that the volume of a gas is directly proportional to its temperature when pressure is held constant. This means that if the temperature of a gas increases, its volume increases, provided the pressure remains unchanged. This concept is crucial when analyzing how temperature changes affect gas volume in a closed system.

Boyle's Law

Boyle's Law describes the inverse relationship between the pressure and volume of a gas at constant temperature. According to this law, if the volume of a gas decreases, its pressure increases, assuming the temperature remains constant. This principle is essential for understanding how volume reduction impacts gas pressure in a confined space.

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