Question

A 100.0 -mL sample of air is collected at \(21{ }^{\circ} \mathrm{C}\) and \(772 \mathrm{~mm} \mathrm{Hg} .\) What is the volume of air at STP?

Short answer

The air volume at STP is approximately 94.6 mL.

Step-by-step solution

Understand the Problem

We have been given a sample of air with initial conditions of volume, temperature, and pressure. We need to find the volume of air at standard temperature and pressure (STP), which is defined as 0°C and 1 atm (or 760 mmHg).

Identify Given Values and Desired Conditions

The initial volume, \( V_1 = 100.0 \, \text{mL} \), initial temperature, \( T_1 = 21^{\circ} \text{C} = 294 \, \text{K} \) (converted from Celsius to Kelvin by adding 273), and initial pressure, \( P_1 = 772 \, \text{mmHg} \). The conditions at STP are \( P_2 = 760 \, \text{mmHg} \) and \( T_2 = 273 \, \text{K} \). The final volume \( V_2 \) is what we need to find.

Use the Combined Gas Law

The combined gas law formula is \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \). We can rearrange this formula to solve for \( V_2 \), the final volume at STP: \( V_2 = \frac{P_1 V_1 T_2}{P_2 T_1} \).

Plug in the Values and Solve

Substitute the known values into the rearranged formula: \( V_2 = \frac{772 \, \text{mmHg} \times 100.0 \, \text{mL} \times 273 \, \text{K}}{760 \, \text{mmHg} \times 294 \, \text{K}} \). Calculate: \( V_2 = \frac{77200 \, \text{mL} \cdot \text{K}}{223440} \approx 94.6 \, \text{mL} \).

Conclusion

The volume of the air sample at STP conditions is approximately 94.6 mL.

Key concepts

STP conditions

In the world of chemistry, STP stands for Standard Temperature and Pressure. It's a common reference point that allows scientists to easily compare gas behaviors under uniform conditions. Generally, STP is defined as a temperature of 0°C (or 273 K) and a pressure of 1 atm (or 760 mmHg). These conditions provide a baseline for experiments and calculations involving gases.
Using STP, chemists can calculate how much a given volume of a gas will fill at these standard conditions, which is crucial for understanding gas reactions or transports.
For example, in the problem above, determining the air volume at STP helps us know how conditions affect gas volume under real conditions. This is fundamental in fields like meteorology and environmental science, where predicting the behavior of gases is essential.

Temperature conversion

Temperature plays a critical role in gas calculations because gases expand or contract depending on their temperature. In chemistry, we work with Kelvin instead of Celsius due to its absolute scale, which avoids negative temperatures.

• To convert from Celsius to Kelvin, simply add 273 to the Celsius temperature. For example, 21°C becomes 294 K.

The consistent use of Kelvin ensures accuracy in calculations involving gases, such as when applying the Combined Gas Law.
As you learned in the problem statement, STP uses 0°C, which is equivalent to 273 K. This consistent baseline makes comparing measurements much more straightforward, especially in connection with other physical laws like those governing pressure and volume.

Gas volume calculations

Understanding gas volume calculations involves using the Combined Gas Law, which combines three individual gas laws: Boyle’s, Charles’s, and Gay-Lussac’s laws. This law helps predict how a gas's volume changes in response to changes in temperature and pressure.

• Boyle's Law: Holds volume and pressure inversely proportional when temperature is constant.
• Charles's Law: Maintains volume and temperature directly proportional when pressure is constant.
• Gay-Lussac's Law: Ensures pressure and temperature are directly proportional when volume is constant.

The Combined Gas Law equation is: \[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \]Here, changes in both pressure and temperature are considered simultaneously. In the provided problem, you calculated the air's final volume under STP conditions using this equation by rearranging it to solve for the unknown volume:\[ V_2 = \frac{P_1 V_1 T_2}{P_2 T_1} \]By plugging the known values into this equation, you could determine the changes in the gas's volume from the initial to the standard conditions. This understanding is vital for many scientific applications involving gases.