Question

A sample of hydrogen chloride gas occupies \(0.750 \mathrm{~L}\) at STP. What is the pressure in atm if the volume is \(0.100 \mathrm{~L}\) at \(25^{\circ} \mathrm{C} ?\)

Short answer

The pressure is approximately 8.19 atm.

Step-by-step solution

Identify the Given Data (Initial Condition)

We have a sample of hydrogen chloride gas at standard temperature and pressure (STP). At STP, the initial volume is given: \( V_1 = 0.750 \, \text{L} \), the pressure \( P_1 \) is \( 1 \, \text{atm} \), and the temperature \( T_1 \) is \( 273.15 \, \text{K} \).

Identify the Given Data (Final Condition)

For the final condition, the final volume is \( V_2 = 0.100 \, \text{L} \) and the temperature \( T_2 \) is \( 25^{\circ} \text{C} \). You'll need to convert \( T_2 \) to Kelvin for calculations: \( T_2 = 25 + 273.15 = 298.15 \, \text{K} \). The pressure \( P_2 \) is what we are looking to find.

Apply the Combined Gas Law

The combined gas law relates pressure, volume, and temperature: \( \frac{P_1 \cdot V_1}{T_1} = \frac{P_2 \cdot V_2}{T_2} \). We will rearrange this equation to solve for \( P_2 \): \( P_2 = \frac{P_1 \cdot V_1 \cdot T_2}{V_2 \cdot T_1} \).

Perform the Calculation

Substitute the known values into the rearranged equation: \( P_2 = \frac{1 \, \text{atm} \cdot 0.750 \, \text{L} \cdot 298.15 \, \text{K}}{0.100 \, \text{L} \cdot 273.15 \, \text{K}} \). This simplifies to \( P_2 \approx 8.19 \, \text{atm} \).

Interpret the Result

The final pressure \( P_2 \) is approximately \( 8.19 \, \text{atm} \) when the volume of the gas is reduced to \( 0.100 \, \text{L} \) at a temperature of \( 25^{\circ} \text{C} \).

Key concepts

Gas Laws

Gas laws are essential in understanding the behavior of gases under different conditions of temperature, pressure, and volume. These laws help predict how a gas will act and are very useful in fields such as chemistry and physics.
The most commonly known gas laws are the Boyle's Law, Charles's Law, and Gay-Lussac's Law. Each of these deals with the relationship between two of the gas properties, keeping the other one constant:

• Boyle's Law: This law states that the pressure of a gas is inversely proportional to its volume, provided the temperature is kept constant. This means if you increase the pressure, the volume decreases.
• Charles's Law: According to this, the volume of a gas is directly proportional to its absolute temperature when pressure remains unchanged.
• Gay-Lussac's Law: This law tells us that the pressure of a gas is directly proportional to its absolute temperature if the volume is kept constant.

The Combined Gas Law brings these three individual laws together and offers a more comprehensive view of how gases behave when both temperature and pressure change. It is invaluable in situations where a gas undergoes changes in multiple parameters at the same time.

Pressure and Volume Relationship

The relationship between pressure and volume in gases is best captured by Boyle's Law. This law states that, for a given mass of gas at constant temperature, the product of the pressure and volume is a constant. Mathematically, this is presented as:\[ P_1 \times V_1 = P_2 \times V_2 \]Where:

• \( P_1 \) and \( P_2 \) are the initial and final pressures.
• \( V_1 \) and \( V_2 \) are the initial and final volumes.

This relationship means that as one increases, the other must decrease if the temperature remains constant. When you squeeze a gas (reduce its volume), its pressure goes up, provided the temperature doesn't change. This is a direct application of the gas behavior in real-world scenarios like air pumps or even our lungs.

Temperature Conversion in Gas Calculations

Temperature conversion is a crucial step in gas calculations, given that temperature profoundly affects gas behavior. In gas law equations, temperature must always be in Kelvin. The Kelvin scale starts at absolute zero, which is the point where gas molecules have minimal kinetic energy, making it appropriate for scientific calculations.
To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature:\[ T(K) = T(^\circ C) + 273.15 \]This conversion is necessary because gas laws require temperature to be on an absolute scale where direct comparisons between temperature and volume or pressure can be accurately made. Using Celsius in calculations would lead to incorrect results as it does not start from an absolute zero point. Therefore, always remember to convert temperatures to Kelvin when dealing with gas laws to ensure precise outcomes in your calculations.