Question

A 25.0-mL sample of neon gas at \(455 \mathrm{~mm}\) Hg is cooled from \(100{ }^{\circ} \mathrm{C}\) to \(10{ }^{\circ} \mathrm{C}\). If the volume remains constant, what is the final pressure?

Short answer

The final pressure of the gas is approximately 345.7 mm Hg.

Step-by-step solution

Understand the Problem

We are given the initial pressure, initial temperature, and final temperature of a gas and need to find its final pressure using the fact that the volume remains constant.

Identify the Gas Law

We will use the ideal gas law for constant volume, which relates pressure and temperature: \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \), where \( P_1 \) and \( P_2 \) are the initial and final pressures, and \( T_1 \) and \( T_2 \) are the initial and final temperatures in Kelvin.

Convert Temperatures to Kelvin

Convert the temperatures from Celsius to Kelvin using the formula \( T[K] = T[^{\circ}C] + 273.15 \). The initial temperature \( T_1 = 100 + 273.15 = 373.15 \) K, and the final temperature \( T_2 = 10 + 273.15 = 283.15 \) K.

Apply the Formula to Find Final Pressure

Substitute the known values into the formula: \( \frac{455 \text{ mm Hg}}{373.15 \text{ K}} = \frac{P_2}{283.15 \text{ K}} \). Solve for \( P_2 \): \( P_2 = \frac{455 \text{ mm Hg} \times 283.15 \text{ K}}{373.15 \text{ K}} \approx 345.7 \text{ mm Hg} \).

Key concepts

Gas Pressure Calculations

Gas pressure calculations are essential when studying how gases behave under different conditions. When dealing with problems where gas volume is held constant, the relationship between pressure and temperature comes into focus. According to the ideal gas law for constant volume, pressure changes inversely with temperature. This principle is illustrated in the formula: \[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \] where:

• \(P_1\) is the initial pressure.
• \(T_1\) is the initial temperature in Kelvin.
• \(P_2\) is the final pressure.
• \(T_2\) is the final temperature in Kelvin.

Ultimately, this formula allows us to determine how the pressure will vary when a gas is cooled or heated, while its volume remains unchanged. Always ensure temperatures are in Kelvin to correctly apply this formula.

Temperature Conversion to Kelvin

In gas computations, it's crucial to use Kelvin rather than Celsius. Kelvin is the absolute temperature scale, removing the possibility of negative values and making the math of gas laws work seamlessly.To convert Celsius to Kelvin, you use the formula:\[ T[K] = T[^{\circ}C] + 273.15 \]For example:

• The conversion of an initial temperature of \(100^{\circ}C\) is \(100 + 273.15 = 373.15\) K.
• For a final temperature of \(10^{\circ}C\), you’d compute \(10 + 273.15 = 283.15\) K.

This straightforward step is vital before further calculations, ensuring accuracy and aligning with the gas laws context.

Constant Volume Process

When a gas is cooled or heated without a change in its container's size or shape, it undergoes a constant volume process. This condition simplifies many aspects of understanding gas behavior, as the only variables affecting the gas are pressure and temperature.The ideal gas law adapts to a constant volume situation, focusing on the pressure-temperature relationship. Noteworthy here is that as temperature decreases or increases, pressure similarly drops or rises because the gas particles move slower or faster.This means that in the problem at hand, we effectively computed how a decrease in temperature from \(373.15\) K to \(283.15\) K leads to a standard pressure adjustment from \(455\) mm Hg to \(345.7\) mm Hg. This clear relationship underscores the physical intuition behind gas laws—how energy in the form of heat translates directly to pressure changes at unchanging volumes. Remember, understanding these fundamentals not only helps with calculations but allows for a deeper appreciation of how gases naturally behave.