Ch. 17 - Temperature, Thermal Expansion, and the Ideal Gas Law

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 17 - Temperature, Thermal Expansion, and the Ideal Gas Law

Problem 45

Chapter 17, Problem 45

If 59.2 L of oxygen at 18.0°C and an absolute pressure of 2.45 atm are compressed to 38.8 L and at the same time the temperature is raised to 56.0°C, what will the new pressure be?

Verified step by step guidance

Start by identifying the given values: initial volume \( V_1 = 59.2 \; \text{L} \), initial temperature \( T_1 = 18.0^\circ \text{C} = 291.15 \; \text{K} \) (convert to Kelvin by adding 273.15), initial pressure \( P_1 = 2.45 \; \text{atm} \), final volume \( V_2 = 38.8 \; \text{L} \), and final temperature \( T_2 = 56.0^\circ \text{C} = 329.15 \; \text{K} \). The goal is to find the final pressure \( P_2 \).

Use the combined gas law, which relates pressure, volume, and temperature: \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \). Rearrange this equation to solve for \( P_2 \): \( P_2 = P_1 \cdot \frac{V_1}{V_2} \cdot \frac{T_2}{T_1} \).

Substitute the known values into the equation: \( P_2 = 2.45 \; \text{atm} \cdot \frac{59.2 \; \text{L}}{38.8 \; \text{L}} \cdot \frac{329.15 \; \text{K}}{291.15 \; \text{K}} \).

Simplify the fractions for volume and temperature: \( \frac{59.2}{38.8} \) and \( \frac{329.15}{291.15} \). Then multiply these ratios with the initial pressure \( P_1 \).

The result of the calculation will give you the final pressure \( P_2 \) in atm. Ensure that the units are consistent throughout the calculation to maintain accuracy.

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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 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 Kelvin. This law allows us to predict how a gas will behave under changing conditions.

Charles's Law

Charles's Law states that the volume of a gas is directly proportional to its absolute temperature when pressure is held constant. Mathematically, it can be expressed as V1/T1 = V2/T2. This principle is crucial for understanding how temperature changes affect the volume of a gas, which is relevant in the given problem.

Boyle's Law

Boyle's Law states that the pressure of a gas is inversely proportional to its volume when temperature is held constant. This relationship can be expressed as P1V1 = P2V2. In the context of the problem, this law helps to understand how the pressure of the gas will change as its volume is altered, especially when combined with temperature changes.

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Gauss' Law

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