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Chapter 18: Problem 8

\(\mathrm{A}\) welder using a tank of volume \(0.0750 \mathrm{~m}^{3}\) fills it with oxygen (molar mass \(32.0 \mathrm{~g} / \mathrm{mol}\) ) at a gauge pressure of \(3.00 \times 10^{5} \mathrm{~Pa}\) and temperature of \(37.0^{\circ} \mathrm{C}\). The tank has a small leak, and in time some of the oxygen leaks out. On a day when the temperature is \(22.0^{\circ} \mathrm{C},\) the gauge pressure of the oxygen in the tank is \(1.80 \times 10^{5} \mathrm{~Pa}\). Find (a) the initial mass of oxygen and (b) the mass of oxygen that has leaked out.

Short Answer

Expert verified
With all these steps, one can find the initial mass of oxygen in the tank and the mass of oxygen that has leaked out.

Step by step solution

01

Convert temperatures to Kelvin

The first step requires converting the given temperatures which are in Celsius to Kelvin. This can be done as: \(T_1\) in kelvin = \(37.0 + 273.15 = 310.15 K\) and \(T_2\) in kelvin = \(22.0 + 273.15 = 295.15 K\).
02

Calculate the initial mass

To find the initial mass, first calculate the initial number of moles: \(n_1 = \frac{P_1V}{RT_1}\), using the given values \(P_1 = 3.00 \times 10^{5} Pa\), \(V = 0.0750m^3\), and \(R = 8.314 J/(mol \cdot K)\). Then, convert the number of moles to mass using the molar mass: \(Mass_1 = n_1 \times M\), where \(M = 32.0 g/mol\).
03

Calculate the final mass

To calculate the final mass of oxygen, first calculate the final number of moles: \(n_2 = \frac{P_2V}{RT_2}\), using the given values \(P_2 = 1.80 \times 10^{5} Pa\). Then, convert the number of moles to mass using the molar mass as in the previous step: \(Mass_2 = n_2 \times M\).
04

Calculate the mass of leaked oxygen

The mass of oxygen that has leaked out can be calculated by subtracting the final mass from the initial mass: \(Leaked \; Mass = Mass_1 - Mass_2\).

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Key Concepts

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

Molar Mass
Molar mass is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). It helps us convert between the mass of a substance and the amount in moles. For oxygen, the molar mass is given as 32.0 g/mol.
To calculate the mass of oxygen, we use the number of moles. This is why it's important to know the molar mass 鈥� it acts as the bridge between mass and moles.
Key points to remember:
  • Molar mass allows conversion between grams and moles.
  • It helps in determining the quantity of a substance when dealing with chemical equations.
Gauge Pressure
Gauge pressure measures pressure relative to atmospheric pressure. Unlike absolute pressure, gauge pressure doesn鈥檛 include atmospheric pressure.
In this problem, we have a gauge pressure that tells us how much more pressure is in the tank compared to the surrounding air. We use this to find the number of moles using the Ideal Gas Law.
Remember:
  • Gauge pressure = Absolute pressure - Atmospheric pressure.
  • This is particularly useful for practical applications such as in tanks and tires.
Kelvin Temperature
The Kelvin scale is the absolute temperature scale. It starts at absolute zero. This makes it ideal for calculations involving gases, as temperature must be in Kelvin when using the Ideal Gas Law.
To convert Celsius to Kelvin, add 273.15 to the Celsius temperature. In this exercise, the temperatures were converted as follows:
  • From 37.0掳C to 310.15 K.
  • From 22.0掳C to 295.15 K.
Using Kelvin ensures that the proportions in calculations stay linear and consistent.
Oxygen Leakage Calculation
To find how much oxygen leaked, we need to calculate the initial and final mass, then find the difference.
Here's how it's done:
  • Calculate initial moles using the initial pressure, volume, and temperature in Kelvin.
  • Convert these moles to mass using the molar mass.
  • Repeat for the final conditions (lower pressure due to leakage).
  • Subtract the final mass from the initial mass to find the leaked mass.
This approach helps us understand how gases behave with changes in pressure and temperature.

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Most popular questions from this chapter

Consider an ideal gas at \(27^{\circ} \mathrm{C}\) and 1.00 atm. To get some idea how close these molecules are to each other, on the average, imagine them to be uniformly spaced, with each molecule at the center of a small cube. (a) What is the length of an edge of each cube if adjacent cubes touch but do not overlap? (b) How does this distance compare with the diameter of a typical molecule? (c) How does their separation compare with the spacing of atoms in solids, which typically are about \(0.3 \mathrm{nm}\) apart?

A person at rest inhales \(0.50 \mathrm{~L}\) of air with each breath at a pressure of 1.00 atm and a temperature of \(20.0^{\circ} \mathrm{C}\). The inhaled air is \(21.0 \%\) oxygen. (a) How many oxygen molecules does this person inhale with each breath? (b) Suppose this person is now resting at an elevation of \(2000 \mathrm{~m}\) but the temperature is still \(20.0^{\circ} \mathrm{C}\). Assuming that the oxygen percentage and volume per inhalation are the same as stated above, how many oxygen molecules does this person now inhale with each breath? (c) Given that the body still requires the same number of oxygen molecules per second as at sea level to maintain its functions, explain why some people report "shortness of breath" at high elevations.

Helium gas with a volume of \(3.20 \mathrm{~L}\), under a pressure of 0.180 atm and at \(41.0^{\circ} \mathrm{C}\), is warmed until both pressure and volume are doubled. (a) What is the final temperature? (b) How many grams of helium are there? The molar mass of helium is \(4.00 \mathrm{~g} / \mathrm{mol}\).

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

Modern vacuum pumps make it easy to attain pressures of the order of \(10^{-13}\) atm in the laboratory. Consider a volume of air and treat the air as an ideal gas. (a) At a pressure of \(9.00 \times 10^{-14}\) atm and an ordinary temperature of \(300.0 \mathrm{~K}\), how many molecules are present in a volume of \(1.00 \mathrm{~cm}^{3} ?\) (b) How many molecules would be present at the same temperature but at 1.00 atm instead?
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