Ch 18: A Macroscopic Description of Matter

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 18: A Macroscopic Description of Matter

Problema 21b

Capitolo 18, Problema 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?

Guida verificata passo dopo passo

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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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.

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Ideal Gases and the Ideal Gas Law

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.

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Introduction To Temperature Scales

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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Gauss' Law

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