Problem 9 - Chemistry Matter and Change

Question

Helium gas in a 2.00-L cylinder is under 1.12 atm pressure. At 36.5°C, that same gas sample has a pressure of 2.56 atm. What was the initial temperature of the gas in the cylinder?

Step-by-Step Solution

Answer

The initial temperature of the helium gas in the cylinder was 135.47 K.

Step 1: Write down the given information

We are given that: - Initial volume: \(V_1 = 2.00 L\) - Initial pressure: \(P_1 = 1.12 atm\) - Final temperature: \(T_2 = 36.5 °C\) - Final pressure: \(P_2 = 2.56 atm\) We need to find the initial temperature, \(T_1\).

Step 2: Convert temperatures to Kelvins

We need to convert the given temperature, which is in Celsius, to Kelvin: \(T_{2(K)} = T_{2(°C)} + 273.15\) \(T_{2(K)} = 36.5 + 273.15\) \(T_{2(K)} = 309.65 K\)

Step 3: Express the combined gas law and rearrange to find T_1

We are given that the volume of the gas remains constant, so \(V_1 = V_2\). Thus, the combined gas law can be simplified as: \(\frac{P_1}{T_1} = \frac{P_2}{T_2}\) We need to find \(T_1\), so we can rearrange the formula as: \(T_1 = \frac{P_1T_2}{P_2}\)

Step 4: Substitute the values and solve for initial temperature

Now, we can substitute the given values and solve for the initial temperature: \(T_1 = \frac{1.12 \times 309.65}{2.56}\) \(T_1 = \frac{346.808}{2.56}\) \(T_1 = 135.47 K\)

Step 5: Write the answer

The initial temperature of the helium gas in the cylinder was 135.47 K.

Key Concepts

Helium Gas

Helium is an inert, lightweight gas commonly used in various scientific and industrial applications. It is colorless, odorless, and tasteless, making it ideal for applications where reactive gases would pose a problem. Helium is found in the atmosphere, but its concentration is very low. Hence, it is usually extracted from natural gas fields. Helium is used to fill balloons, in cryogenics, and as a protective gas in welding.
Understanding the behavior of helium gas, especially under varying pressure and temperature conditions, is essential for many applications. The behavior of helium, like other gases, can be described by the Combined Gas Law, which portrays the relationship between pressure, volume, and temperature.

Temperature Conversion

In gas law calculations, temperatures must always be in Kelvin. Celsius temperatures cannot be used directly because they do not begin at absolute zero, making Kelvin the standard unit of thermodynamic temperature.
The conversion is simple: add 273.15 to the Celsius temperature. For the exercise, the conversion of 36.5°C to Kelvin is done by:

• Taking 36.5°C and adding 273.15.
• This results in 309.65 K.

This conversion ensures accuracy and consistency across scientific calculations, as Kelvin provides a direct measure of thermal energy.

Pressure Calculations

Pressure in gases is the force applied by gas molecules colliding with the walls of its container. It is often measured in atmospheres (atm), but can also be denoted in pascals (Pa) or bars. In this exercise, pressure calculations are crucial to determine the initial temperature of helium.
The Combined Gas Law is used, which, at constant volume, simplifies to \( \frac{P_1}{T_1} = \frac{P_2}{T_2} \). Here, the variables stand for:

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

To find the initial temperature \( T_1 \), rearrange the formula: \( T_1 = \frac{P_1T_2}{P_2} \).
Using the given information and solving, you get: \( T_1 = \frac{1.12 \times 309.65}{2.56} \), yielding an initial temperature of 135.47 K. This calculation highlights how pressure change affects the temperature of a gas when the volume remains constant.