Problem 39 - Chemistry for Allied Health

Question

A 2.50-L volume of hydrogen measured at \(-196^{\circ} \mathrm{C}\) is warmed to \(100{ }^{\circ} \mathrm{C}\). Calculate the volume of the gas at the higher temperature, assuming no change in pressure.

Step-by-Step Solution

Answer

The volume of the gas at \(100^{\circ}C\) is approximately \(12.09\,L\).

Step 1: Understand the Problem

We need to find the new volume of hydrogen gas when its temperature changes from \(-196^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C}\), assuming constant pressure. We'll use Charles's Law for this calculation.

Step 2: Charles's Law Introduction

Charles's Law states that \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\) when the pressure remains constant, where \(V\) represents volume and \(T\) represents temperature in Kelvin.

Step 3: Convert Temperatures to Kelvin

Convert the given Celsius temperatures to Kelvin:For the initial temperature: \(T_1 = -196^{\circ}C + 273.15 = 77.15\,K\).For the final temperature:\(T_2 = 100^{\circ}C + 273.15 = 373.15\,K\).

Step 4: Apply Charles's Law

Insert the known values into Charles's Law:\(\frac{2.50\,L}{77.15\,K} = \frac{V_2}{373.15\,K}\).

Step 5: Solve for Unknown Volume

Rearrange the equation to solve for \(V_2\):\(V_2 = \frac{2.50\,L \times 373.15\,K}{77.15\,K}\).

Step 6: Calculate Result

Perform the calculation:\(V_2 \approx \frac{2.50 \times 373.15}{77.15} \approx 12.09\,L\).

Key Concepts

Temperature Conversion to Kelvin

When dealing with gases in scientific studies, it's important to use the Kelvin scale for temperature. Kelvin is the global standard unit for temperature in the scientific community. This is because Kelvin is an absolute scale with no negative numbers and is directly related to the kinetic energy of particles.
Here's how you convert Celsius to Kelvin:

• Simply add 273.15 to the Celsius temperature to get the temperature in Kelvin.

For example, to convert -196°C to Kelvin, you do the following: \[ T_1 = -196 + 273.15 = 77.15 \text{ K} \]For the final temperature of 100°C, it converts like this: \[ T_2 = 100 + 273.15 = 373.15 \text{ K} \]
Remember, this temperature conversion is crucial for calculations involving gas laws, as gas laws like Charles's Law are based on absolute temperatures.

Volume and Temperature Relationship

Charles's Law is key to understanding the connection between the volume and temperature of a gas. This law tells us that the volume of a gas is directly proportional to its temperature, provided the pressure remains constant.
Essentially, if the temperature of a gas increases, its volume will increase too; likewise, if the temperature decreases, the volume will decrease. This can be mathematically expressed as: \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \] where:

• \( V_1 \) and \( T_1 \) are the initial volume and temperature
• \( V_2 \) and \( T_2 \) are the final volume and temperature

In our exercise, the gas starts at 2.50 L and 77.15 K, and the goal is to find the final volume V₂ at 373.15 K using this law.

Ideal Gas Law Applications

The Ideal Gas Law is a broader framework that Charles's Law is a part of, often expressed as \[ PV = nRT \] where:

• \( P \) is pressure
• \( V \) is volume
• \( n \) is the number of moles of the gas
• \( R \) is the ideal gas constant
• \( T \) is temperature in Kelvin

While Charles's Law focuses on the relationship between volume and temperature at constant pressure, the Ideal Gas Law encapsulates the behavior of gases under varying conditions of temperature, volume, and pressure.
In practical applications, knowing how to convert temperatures to Kelvin and apply specific laws like Charles's Law can help predict how a gas will behave when conditions change. This kind of understanding is foundational for fields like chemistry and physics where the manipulation of gases is often necessary.