Question

A gas at \(572 \mathrm{mmHg}\) and \(35.0^{\circ} \mathrm{C}\) occupies a volume of \(6.15 \mathrm{~L} .\) Calculate its volume at STP.

Short answer

The volume of the gas at STP is approximately 5.02 L.

Step-by-step solution

Understand STP Conditions

STP (Standard Temperature and Pressure) conditions refer to a temperature of 0°C (273.15 K) and a pressure of 1 atm (760 mmHg). It's important to note these conditions as they are required for our calculation.

Identify Initial Conditions

The initial conditions of the gas are given as 572 mmHg (pressure), 35.0°C (temperature), and 6.15 L (volume). We need to convert the temperature to Kelvin for calculations; thus, \[ T_1 = 35.0°C + 273.15 = 308.15 ext{ K} \]

Use the Combined Gas Law

To find the final volume of the gas at STP, use the combined gas law:\[ \frac{P_1 \times V_1}{T_1} = \frac{P_2 \times V_2}{T_2} \]Where:- \(P_1 = 572 \text{ mmHg}\)- \(V_1 = 6.15 \text{ L}\)- \(T_1 = 308.15 \text{ K}\)- \(P_2 = 760 \text{ mmHg}\) (pressure at STP)- \(T_2 = 273.15 \text{ K}\) (temperature at STP)We will solve for \( V_2 \).

Solve for Final Volume \( V_2 \)

Rearrange the combined gas law to solve for \( V_2 \):\[ V_2 = \frac{P_1 \times V_1 \times T_2}{T_1 \times P_2} \]Substitute the known values:\[ V_2 = \frac{572 \text{ mmHg} \times 6.15 \text{ L} \times 273.15 \text{ K}}{308.15 \text{ K} \times 760 \text{ mmHg}} \]Calculate:\[ V_2 = \frac{9609.51}{234376} \approx 5.02 \text{ L} \]

Confirm Units and Reasonability

Ensure that all the calculations involved were correct and the units were consistent. The volume conversion based on pressure and temperature change appears consistent with the physical laws. The volume decreased because the conditions moved from higher temperature and lower pressure to STP, confirming the result is reasonable.

Key concepts

Standard Temperature and Pressure (STP)

Understanding Standard Temperature and Pressure, commonly abbreviated as STP, is crucial in gas law calculations. STP is defined as a set of conditions for measuring gases that ensure consistency across different experiments. It is standardized to ensure accurate comparisons between results.
STP conditions are:

• Temperature: 0°C, which is equivalent to 273.15 Kelvin.
• Pressure: 1 atmosphere (atm), which is also expressed as 760 millimeters of mercury (mmHg) or 101.325 kilopascals (kPa).

These conditions are chosen because they are universally recognized and used by scientists globally. When gases are measured at STP, it allows for direct comparisons of gas behavior and properties without additional calculations for varying temperature and pressure conditions.
In gas law problems, converting initial conditions to STP often simplifies the solution, as it reduces the variables involved. Being familiar with conversion to STP is a foundational skill in solving gas-related problems.

Gas Laws

Gas laws describe the behavior of gases in various conditions of pressure, temperature, and volume. These laws are grounded in fundamental physical principles, and they allow us to predict how gases will behave when exposed to different environments.
The Combined Gas Law is a key equation because it combines Charles's Law, Boyle's Law, and Gay-Lussac's Law, covering the relationship between pressure, volume, and temperature simultaneously. The structure of the Combined Gas Law is:

• \(\frac{P_1 \times V_1}{T_1} = \frac{P_2 \times V_2}{T_2}\)

This relation tells us that the ratio of the product of pressure and volume to the temperature for a gas remains constant when only the state changes, not the quantity of gas. It is crucial for calculations where the state of gas changes between two sets of conditions.
Comprehending this law enables easy calculation of unknown variables when the other conditions are known. Mastery of gas laws clarifies diverse practical scenarios, like inflating a balloon or calculating the volume displaced by an underwater diver's exhaled bubbles.

Volume Calculation

Volume calculations in gas law problems are essential for understanding how gases expand or contract under varying conditions. The volume of a gas is directly linked to its state - described by pressure and temperature - along with its quantity.
To calculate the volume change when conditions such as pressure and temperature change, we can utilize the Combined Gas Law, rearranged to solve for the new volume:

• \(V_2 = \frac{P_1 \times V_1 \times T_2}{T_1 \times P_2}\)

Volume calculations express the extent of gas expansion or contraction in response to environmental changes. These calculations are vital in applications ranging from industrial gas storage to determining fuel combustion efficiency in engines.
Visualizing the impact of changing variables like pressure and temperature helps in predicting how gases will occupy space, underlines the importance of maintaining certain conditions, and highlights the precise nature of gases' behavior in response to changing external constraints.

Pressure Conversion

Pressure conversion is a key part of working with gas laws, as different units of pressure are often used interchangeably. Understanding how to convert between units is critical for accurate calculations.
The most common pressure units include:

• Atmospheres (atm)
• Millimeters of Mercury (mmHg)
• Kilopascals (kPa)

For example, 1 atm is equivalent to 760 mmHg or 101.325 kPa. Recognizing these conversions is essential, as calculations often require the same units for ease and accuracy.
To convert pressure values, use the appropriate conversion factors. For instance, converting mmHg to atm requires dividing the pressure in mmHg by 760. In contrast, converting atm to kPa involves multiplying by 101.325.
These simple conversion techniques ensure all pressures in calculations align correctly, avoiding errors that could skew results, thus maintaining the integrity of any gas law calculations.