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Chapter 5: Problem 7

Starting with Charles's law (stated as an equation), obtain an equation for the final volume of a gas from its initial volume when the temperature is changed at constant pressure.

Short Answer

Expert verified
The equation for the final volume is \( V_2 = \frac{V_1 \cdot T_2}{T_1} \).

Step by step solution

01

Understand Charles's Law Equation

Charles's Law states that the volume (V) of a gas is directly proportional to its temperature (T) when pressure is held constant. The equation representing Charles's Law can be written as \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \), where \(V_1\) and \(T_1\) are the initial volume and temperature, and \(V_2\) and \(T_2\) are the final volume and temperature.
02

Rearrange to Solve For Final Volume

To find an equation for the final volume \(V_2\), we need to rearrange the equation \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \) to solve for \(V_2\). Multiply both sides by \(T_2\) to isolate \(V_2\) on one side of the equation: \( V_2 = \frac{V_1 \cdot T_2}{T_1} \).
03

Analyze the Rearranged Equation

The equation \( V_2 = \frac{V_1 \cdot T_2}{T_1} \) shows that the final volume \(V_2\) is dependent on the initial volume \(V_1\) and the ratio of the final temperature \(T_2\) to the initial temperature \(T_1\). This implies that an increase in temperature results in an increase in volume, consistent with Charles's Law.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gas Volume
In the context of Charles's Law, understanding the gas volume is crucial. The law tells us that gas volume changes based on temperature adjustments. Think of volume as the space a gas occupies. A larger volume means the gas particles have more space to move around. Under constant pressure, if the temperature of a gas goes up, the volume usually increases too. This is because gas particles move faster and need more space. By focusing on gas volume in Charles's Law, you can predict how the gas will behave when the temperature alters. This relationship means, if you know the initial volume and the temperature change, you can figure out the new volume using the equation for Charles's Law.
Temperature Change
When we talk about temperature change in Charles's Law, we're referring to how the temperature of a gas affects its volume. Temperature is a measure of how much energy the gas particles have. If we heat the gas, the particles get more energy, move faster, and spread out, causing the volume to increase. Conversely, cooling the gas makes particles slow down, reducing the energy and volume. When we're applying Charles's Law, you must remember to use an absolute temperature scale, like Kelvin, to make accurate calculations, since the behavior hinges on proportionality in terms of energy. This principle allows you to see how heating or cooling affects a gas volume at constant pressure.
Constant Pressure
Keeping the pressure constant is a key aspect of Charles's Law. Pressure is the force exerted by gas particles against the container walls. In many real-life situations, this pressure can remain the same, even as the gas volume and temperature change. When applying Charles's Law, if the pressure is not constant, the law's simple relationship between volume and temperature doesn't hold directly. Therefore, you need to ensure the pressure remains fixed to use the law effectively. Imagine inflating a balloon in a room; as long as the surrounding air pressure stays stable, you can use Charles's Law to relate changes in volume due to temperature changes. This constancy allows us to predict and understand gas behavior precisely when only temperature variations are in play.

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Most popular questions from this chapter

What is the ratio of rates of effusion of \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\) under the same conditions?

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