Question

You have several identical balloons. You experimentally determine that a balloon will break if its volume exceeds 0.900 L. The pressure of the gas inside the balloon equals air pressure (1.00 atm). (a) If the air inside the balloon is at a constant 22.0\(^\circ\)C and behaves as an ideal gas, what mass of air can you blow into one of the balloons before it bursts? (b) Repeat part (a) if the gas is helium rather than air.

Short answer

(a) 1.072 g of air, (b) 0.148 g of helium.

Step-by-step solution

Understand the Ideal Gas Law

The Ideal Gas Law states that \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant (0.0821 L·atm/mol·K), and \( T \) is the temperature in Kelvin. We will use this equation to find the number of moles of gas in the balloon.

Convert Temperature to Kelvin

First, convert the temperature from Celsius to Kelvin with the formula \( T(K) = T(^\circ C) + 273.15 \). The given temperature is 22.0\(^\circ\)C. Thus, \( T(K) = 22.0 + 273.15 = 295.15 \) K.

Calculate Moles of Air Using Ideal Gas Law

Use the Ideal Gas Law to find the number of moles of air that can be blown into the balloon. Given \( P = 1.00 \) atm, \( V = 0.900 \) L, and \( T = 295.15 \) K, solve for \( n \) in \( PV = nRT \): \( n = \frac{PV}{RT} = \frac{(1.00 \text{ atm}) (0.900 \text{ L})}{(0.0821 \text{ L·atm/mol·K})(295.15 \text{ K})} \approx 0.037 \) moles.

Convert Moles of Air to Mass

Find the molar mass of air, which is approximately 28.97 g/mol. Compute the mass using the moles from Step 3: \( ext{mass} = n \times ext{molar mass} = 0.037 \text{ mol} \times 28.97 \text{ g/mol} \approx 1.072 \text{ g} \).

Calculate Moles and Mass of Helium

For helium, use the same number of moles calculated in Step 3. The molar mass of helium is 4.00 g/mol. Compute the mass: \( ext{mass} = 0.037 \text{ mol} \times 4.00 \text{ g/mol} \approx 0.148 \text{ g} \).

Key concepts

Mole Calculation

When working with gases and the Ideal Gas Law, calculating moles is a central step. Moles are a unit used to express the amount of a substance. The Ideal Gas Law equation, \( PV = nRT \), comes in handy to find the number of moles, \( n \). Here, \( P \) stands for pressure, \( V \) is volume, \( R \) is the ideal gas constant (0.0821 L·atm/mol·K), and \( T \) is temperature in Kelvin. To use this equation:

• Ensure all the parameters are in the correct units: Pressure in atmospheres, Volume in liters, Temperature in Kelvin.
• Rearrange the formula to solve for the number of moles: \( n = \frac{PV}{RT} \).

For example, given the values in the exercise:

• Pressure (\( P \)) = 1.00 atm
• Volume (\( V \)) = 0.900 L
• Temperature (\( T \)) = 295.15 K

You can compute the moles of gas in the balloon: \( n = \frac{(1.00 \text{ atm}) \times (0.900 \text{ L})}{(0.0821 \text{ L·atm/mol·K}) \times (295.15 \text{ K})} \approx 0.037 \text{ moles} \). This tells us how much gas (in moles) the balloon can hold before it pops.

Temperature Conversion

In gas calculations, it is crucial to work with the correct temperature units. The Ideal Gas Law demands temperature in Kelvin rather than Celsius. Kelvin is the absolute temperature scale, which doesn't have negative values and is necessary for accurate gas calculations. To convert Celsius to Kelvin:

• Use the conversion formula: \( T(K) = T(^\circ C) + 273.15 \).

For instance, in the exercise, the given temperature is 22.0°C. Converting this to Kelvin:

• \( T(K) = 22.0 + 273.15 = 295.15 \text{ K} \).

Simple calculations like this ensure precision in your gas laws applications. Always remember, working in Kelvin is a must for using the Ideal Gas Law, as it ensures proportional and non-negative temperature readings.

Molar Mass

Understanding molar mass is key when converting moles into grams. Molar mass is the mass of one mole of a substance, expressed in grams per mole (g/mol). This concept helps us to convert between the mass of a substance and the number of moles, making it practically applicable in calculations.For air, the molar mass is approximately 28.97 g/mol. We use this value to find how much mass the computed moles translate into. You can find the mass from moles using the formula:

• \( \text{mass} = n \times \text{molar mass} \).

Similarly, for helium:

• The molar mass is 4.00 g/mol.

So, if you have 0.037 moles of a gas like air:

• \( \text{mass} = 0.037 \text{ mol} \times 28.97 \text{ g/mol} \approx 1.072 \text{ g} \).

And for helium:

• \( \text{mass} = 0.037 \text{ mol} \times 4.00 \text{ g/mol} \approx 0.148 \text{ g} \).

This approach converts theoretical gas volume into tangible, real-world mass metrics, providing a deeper understanding of chemical quantities.