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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

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

Step by step solution

01

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.
02

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.
03

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.
04

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: \( ext{mass} = n \times ext{molar mass} = 0.037 \text{ mol} \times 28.97 \text{ g/mol} \approx 1.072 \text{ g} \).
05

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

A cylindrical tank has a tight-fitting piston that allows the volume of the tank to be changed. The tank originally contains 0.110 m\(^3\) of air at a pressure of 0.355 atm. The piston is slowly pulled out until the volume of the gas is increased to 0.390 m\(^3\). If the temperature remains constant, what is the final value of the pressure?

Estimate the number of atoms in the body of a 50-kg physics student. Note that the human body is mostly water, which has molar mass 18.0 g/mol, and that each water molecule contains three atoms.

Oxygen (O\(_2\)) has a molar mass of 32.0 g/mol. What is (a) the average translational kinetic energy of an oxygen molecule at a temperature of 300 K; (b) the average value of the square of its speed; (c) the root-mean-square speed; (d) the momentum of an oxygen molecule traveling at this speed? (e) Suppose an oxygen molecule traveling at this speed bounces back and forth between opposite sides of a cubical vessel 0.10 m on a side. What is the average force the molecule exerts on one of the walls of the container? (Assume that the molecule's velocity is perpendicular to the two sides that it strikes.) (f) What is the average force per unit area? (g) How many oxygen molecules traveling at this speed are necessary to produce an average pressure of 1 atm? (h) Compute the number of oxygen molecules that are contained in a vessel of this size at 300 K and atmospheric pressure. (i) Your answer for part (h) should be three times as large as the answer for part (g). Where does this discrepancy arise?

(a) Calculate the specific heat at constant volume of water vapor, assuming the nonlinear triatomic molecule has three translational and three rotational degrees of freedom and that vibrational motion does not contribute. The molar mass of water is 18.0 g/mol. (b) The actual specific heat of water vapor at low pressures is about 2000 J/kg \(\cdot\) K. Compare this with your calculation and comment on the actual role of vibrational motion.

A flask contains a mixture of neon (Ne), krypton (Kr), and radon (Rn) gases. Compare (a) the average kinetic energies of the three types of atoms and (b) the root-mean-square speeds. (Hint: Appendix D shows the molar mass (in g/mol) of each element under the chemical symbol for that element.)

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