Question

A 5.00-L sample of air is collected at \(500^{\circ} \mathrm{C}\) and 5.00 atm. What is the volume of air at STP?

Short answer

The volume of air at STP is approximately 8.83 L.

Step-by-step solution

Understanding Standard Temperature and Pressure (STP)

STP refers to conditions where the temperature is 0°C (273.15 K) and the pressure is 1 atm. We will convert the given conditions to STP conditions to find the new volume of air.

Convert Celsius to Kelvin

The temperature must be in Kelvin to use the ideal gas law. Convert the given temperature from Celsius to Kelvin: \[ T(K) = T(^{\circ}C) + 273.15 = 500 + 273.15 = 773.15 \text{ K} \]

Apply the Combined Gas Law

The combined gas law is used to find the new volume when the initial and final conditions of a gas are given: \[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]Here, \( P_1 = 5.00 \text{ atm} \), \( V_1 = 5.00 \text{ L} \), \( T_1 = 773.15 \text{ K} \), \( P_2 = 1.00 \text{ atm} \), \( T_2 = 273.15 \text{ K} \). We need to find \( V_2 \).

Solve for V2

Rearrange the combined gas law to solve for \( V_2 \):\[ V_2 = \frac{P_1 V_1 T_2}{T_1 P_2} = \frac{5.00 \times 5.00 \times 273.15}{773.15 \times 1.00} \approx 8.83 \text{ L} \]

Verify the Approach and Units

Double-check the calculation and ensure that all units were properly converted and used. We maintained consistency with Kelvin for temperature and atm for pressure throughout the calculation.

Key concepts

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure, or STP, is often used as a reference point in chemistry for gas calculations. It allows scientists to compare different gas behaviors under a common condition. At STP, the temperature is set to 0°C, which is equivalent to 273.15 Kelvin, and the pressure is standardized to 1 atmosphere (atm). This standardization simplifies calculations, making it easier to predict how a gas will behave when conditions change.
Using STP is particularly useful when working with the Ideal Gas Law, as it provides a point of reference. Moreover, it aids in the conversion between different gas conditions by allowing accurate predictions about volume changes when converting to or from STP. Thus, knowing how to adjust from any given condition to STP can be critical in understanding gas behavior effectively.

Combined Gas Law

The Combined Gas Law is a powerful tool for understanding how gases behave when subjected to multiple changing conditions—specifically, changes in pressure, volume, and temperature. This law amalgamates three individual gas laws: Boyle's Law, Charles's Law, and Gay-Lussac's Law.
The formula for the Combined Gas Law is:

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

where:

• \(P_1\) and \(P_2\) are the initial and final pressures of the gas.
• \(V_1\) and \(V_2\) are the initial and final volumes of the gas.
• \(T_1\) and \(T_2\) are the initial and final temperatures, in Kelvin.

Using this formula, you can solve for any one of these variables if the others are known. It's crucial to keep all temperatures in Kelvin during calculations to maintain accuracy. The Combined Gas Law allows you to handle real-world problems in which gas conditions are altered, making it an indispensable tool in chemistry.

Temperature Conversion

Temperature conversion, particularly between Celsius and Kelvin, is essential in scientific calculations involving gases. The Celsius scale is commonly used in everyday life, but the Kelvin scale is preferred in scientific calculations because it starts at absolute zero, the point at which all thermal motion ceases.
To convert from Celsius to Kelvin, add 273.15 to the Celsius temperature. For instance, a temperature of 500°C becomes:

• \(T(K) = 500 + 273.15 = 773.15 \text{ K}\)

This conversion is necessary because gas laws require temperatures to be in Kelvin, as it ensures that the calculations acknowledge absolute temperature changes rather than relative ones. Failing to convert temperatures can lead to incorrect results, making this a crucial step in working with gas laws.