Ch 18: A Macroscopic Description of Matter

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 18: A Macroscopic Description of Matter

Problem 21b

Chapter 18, Problem 21b

A gas at 100°C fills volume V₀. If the pressure is held constant, what is the volume if the Kelvin temperature is doubled?

Verified step by step guidance

Step 1: Understand the relationship between temperature and volume for a gas at constant pressure. This is described by Charles's Law, which states that the volume of a gas is directly proportional to its absolute temperature (Kelvin) when pressure is constant. Mathematically, Charles's Law can be expressed as:

\frac{V}{T} = \frac{V₀}{T₀}

, where V is the final volume, T is the final temperature, V₀ is the initial volume, and T₀ is the initial temperature.

Step 2: Convert the given temperature from Celsius to Kelvin. The initial temperature is 100°C, and the Kelvin temperature is calculated using the formula:

T₀ = 100 + 273.15

. This gives the initial temperature in Kelvin.

Step 3: Double the Kelvin temperature. If the initial Kelvin temperature is T₀, the final Kelvin temperature T is given by:

T = 2T₀

. This step ensures the temperature is doubled as per the problem statement.

Step 4: Use Charles's Law to find the final volume. Rearrange the formula to solve for V:

V = V₀ × \frac{T}{T₀}

. Substitute the values of T and T₀ into the equation.

Step 5: Simplify the expression for V. Since T = 2T₀, the equation becomes:

V = V₀ × 2

. This shows that the final volume is twice the initial volume.

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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 relates the pressure, volume, temperature, and number of moles of an ideal gas through the equation PV = nRT. This law is fundamental in understanding how gases behave under varying conditions of temperature and pressure, allowing us to predict changes in volume when temperature changes.

Temperature Scales

Temperature can be measured in different scales, with Celsius and Kelvin being the most common in physics. The Kelvin scale is an absolute temperature scale where 0 K represents absolute zero, the point at which molecular motion ceases. To convert Celsius to Kelvin, one adds 273.15, making it crucial to use Kelvin when applying gas laws.

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. This means that if the temperature of a gas increases, its volume will also increase, provided the pressure does not change. This principle is essential for solving problems involving changes in gas volume with temperature variations.

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Related Practice

Textbook Question

A cylinder contains nitrogen gas. A piston compresses the gas to half its initial volume. Afterward, has the mass density of the gas changed? If so, by what factor? If not, why not?

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Textbook Question

A gas at 100°C fills volume V₀. If the pressure is held constant, what is the volume if the Celsius temperature is doubled?

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Textbook Question

The total lung capacity of a typical adult is 5.0 L. Approximately 20% of the air is oxygen. At sea level and at a body temperature of 37°C, how many oxygen molecules do the lungs contain at the end of a strong inhalation?

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Textbook Question

A 20-cm-diameter cylinder that is 40 cm long contains 50 g of oxygen gas at 20°C. What is the number density of the oxygen?

Common outdoor thermometers are filled with red-colored ethyl alcohol. One thermometer has a 0.40-mm-diameter capillary tube attached to a 9.0-mm-diameter spherical bulb. On a 0°C morning, the column of alcohol stands 30 mm above the bulb. What is the temperature in °C when the column of alcohol stands 130 mm above the bulb? The expansion of the glass is much less than that of the alcohol and can be ignored.

The solar corona is a very hot atmosphere surrounding the visible surface of the sun. X-ray emissions from the corona show that its temperature is about 2×10⁶ K. The gas pressure in the corona is about 0.03 Pa. Estimate the number density of particles in the solar corona.

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