Question

If a sample of hydrogen gas occupies \(2.00 \mathrm{~L}\) at \(-50{ }^{\circ} \mathrm{C}\) and \(155 \mathrm{~mm} \mathrm{Hg}\), what is the volume at \(75^{\circ} \mathrm{C}\) and \(365 \mathrm{~mm} \mathrm{Hg} ?\)

Short answer

The volume of the hydrogen gas is approximately 1.32 L at the new conditions.

Step-by-step solution

Convert Temperature to Kelvin

The temperature must be in Kelvin for gas law calculations. Convert both temperatures using the formula: \[ T(K) = T(°C) + 273.15 \]For \(-50^{ ext{°C}}\): \[ T_1 = -50 + 273.15 = 223.15 ext{ K} \] For \(75^{ ext{°C}}\):\[ T_2 = 75 + 273.15 = 348.15 ext{ K} \]

Use Combined Gas Law

The combined gas law relates pressure, volume, and temperature between two states. It is given by: \[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]We need to find the final volume \(V_2\).

Substitute Known Values

Substitute the given values into the combined gas law equation:\[ \frac{155 \, \text{mmHg} \times 2.00 \, \text{L}}{223.15 \, \text{K}} = \frac{365 \, \text{mmHg} \times V_2}{348.15 \, \text{K}} \]

Solve for Final Volume \(V_2\)

Rearrange the equation to solve for \(V_2\):\[ V_2 = \frac{365 \, \text{mmHg} \times V_2 \times 223.15 \, \text{K}}{155 \, \text{mmHg} \times 348.15 \, \text{K}} \]Calculate:\[ V_2 = \frac{155 \, \text{mmHg} \times 2.00 \, \text{L} \times 348.15 \, \text{K}}{365 \, \text{mmHg} \times 223.15 \, \text{K}} \]\[ V_2 = \frac{107660.5}{81442.075} \]\[ V_2 \approx 1.32 \, \text{L} \]

Conclusion

The final volume of the hydrogen gas at \(75^{\circ} \text{C}\) and \(365 \, \text{mmHg}\) is approximately \(1.32 \, \text{L}\).

Key concepts

Gas Behavior

Understanding gas behavior is crucial for making sense of how gases like hydrogen change under different conditions. Gases are made up of tiny particles that are in constant motion. This movement is random, and the particles collide with each other and their container's walls. These collisions cause pressure, which we measure to understand a gas's behavior.

When studying gases, three primary factors are often considered: pressure, volume, and temperature. These factors are closely linked and are described by gas laws.

• Pressure (P): This is the force that a gas exerts on the walls of its container due to the collisions of its particles.
• Volume (V): This refers to the amount of space the gas occupies.
• Temperature (T): This affects the speed of the gas particles—higher temperatures mean faster particles and often greater pressure if the volume is constant.

By understanding these properties and how they interact, you can predict how a gas will behave when conditions change. This knowledge is essential in many industries, like chemical production and environmental science.

Temperature Conversion to Kelvin

Before using any gas law equations, it's necessary to convert temperatures to Kelvin. Kelvin is the standard unit of temperature for scientific calculations involving gases because it is an absolute scale. The Kelvin scale starts from absolute zero, the point at which particles theoretically stop moving.

To convert Celsius to Kelvin, you simply add 273.15 to the Celsius temperature. This conversion is straightforward and critical for correct calculations. For instance, the conversion formulas used here are:

• Converting \( -50^{\circ}C \) to Kelvin: \( T_1 = -50 + 273.15 = 223.15 ext{ K} \)
• Converting \( 75^{\circ}C \) to Kelvin: \( T_2 = 75 + 273.15 = 348.15 ext{ K} \)

This ensures consistency in units when applying the gas laws, allowing you to successfully calculate other variables like pressure and volume changes.

Pressure and Volume Relationship

The combined gas law provides a fundamental relationship between pressure, volume, and temperature. It is useful in calculating how a gas reacts to changes in these conditions. The formula is expressed as \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \), where subscripts 1 and 2 refer to initial and final states respectively.

In the exercise described, we use this formula to find the new volume of the hydrogen gas when its pressure and temperature change. The steps are as follows:

• Identify all known values and convert temperatures to Kelvin.
• Substitute these values into the equation to find the variable you are solving for, in this case, the final volume \( V_2 \).
• Solve the equation by rearranging for \( V_2 \), rendering: \[ V_2 = \frac{155 \, \text{mmHg} \times 2.00 \, \text{L} \times 348.15 \, \text{K}}{365 \, \text{mmHg} \times 223.15 \, \text{K}} \]

These steps help to find how changes in temperature and pressure affect the volume of a gas, illustrating the basic principles of gas behavior.