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Chapter 19: Problem 66

In a hospital, pure oxygen may be delivered at 50 psi (gauge pressure) and then mixed with \(\mathrm{N}_{2} \mathrm{O}\). What volume of oxygen at \(20^{\circ} \mathrm{C}\) and 50 psi (gauge pressure) should be mixed with \(1.7 \mathrm{~kg}\) of \(\mathrm{N}_{2} \mathrm{O}\) to get a \(50 \% / 50 \%\) mixture by volume at \(20^{\circ} \mathrm{C} ?\) (a) \(0.21 \mathrm{~m}^{3} ;\) (b) \(0.27 \mathrm{~m}^{3}\); (c) \(1.9 \mathrm{~m}^{3} ;\) (d) \(100 \mathrm{~m}^{3}\).

Short Answer

Expert verified
The volume of oxygen required to create a 50%/50% mixture with 1.7 kg of \(\mathrm{N}_{2} \mathrm{O}\) is \(0.279 \, m鲁\), so the closest answer is (b) \(0.27 \, m鲁\).

Step by step solution

01

Calculate Molar Mass of N2O

In order to find the number of moles of \(\mathrm{N}_2 \mathrm{O}\), you need the molar mass. Nitrogen (N) has a molar mass of 14 g/mol and Oxygen (O) has a molar mass of 16 g/mol. So, the molar mass of \(\mathrm{N}_2 \mathrm{O}\) is \(2*14 + 16 = 44 \,g/mol\).
02

Calculate the Number of Moles

Using the molar mass of \(\mathrm{N}_2 \mathrm{O}\), the number of moles \(n\) in 1.7 kg of \(\mathrm{N}_{2} \mathrm{O}\) is calculated by dividing the total mass \(m\) of \(\mathrm{N}_{2} \mathrm{O}\) by its molar mass. That is: \(n = m / M = 1700 \,g / 44 \,g/mol = 38.64 \,mol\) where \(M\) is the molar mass.
03

volume of N2O

Now, using the Ideal Gas Law \(PV = nRT\), the volume \(V\) of \(\mathrm{N}_2 \mathrm{O}\) can be calculated. With pressure \(P\) = 50 psi = 344737.86 Pa, the gas constant \(R\) = 8.3145 m鲁.pa/K/mol and the temperature \(T\) = 20掳C = 293.15 K, we substitute the values to get: \(V = nRT/P = 38.64 \, mol * 8.3145 \, m鲁.pa/K/mol * 293.15 \, K / 344737.86 \, Pa = 0.279 m鲁\).
04

Volume of Oxygen

Given that the mixture should contain 50% \(\mathrm{N}_{2} \mathrm{O}\) and 50% Oxygen by volume, the volume of Oxygen should be equal to the volume of \(\mathrm{N}_{2} \mathrm{O}\): so \(V_{O_2} = V_{N_2O} = 0.279 \, m鲁\). This volume is the result when Oxygen is at a pressure of 50 psi and temperature of \(20^{\circ}C\).

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Key Concepts

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

Molar Mass
Understanding molar mass is a critical step when working with gases. The molar mass is the mass of one mole of a substance, usually given in grams per mole (g/mol). It aids in converting between the mass of a gas and the number of moles, a common requirement in chemistry problems involving gases.
For \(\mathrm{N}_2 \mathrm{O},\) it is comprised of two nitrogen atoms (each weighing 14 g/mol) and one oxygen atom (weighing 16 g/mol). Therefore, its molar mass is calculated as \(2 \times 14 + 16 = 44 \, g/mol.\) This calculation ensures we know how much mass each mole of \(\mathrm{N}_2 \mathrm{O}\) contains, a necessary step to eventually find out the volume of the gas under certain conditions.
Gas Mixture
A gas mixture involves combining different gases. In this context, we're combining pure oxygen and\(\mathrm{N}_2 \mathrm{O}\)as needed by hospitals. In many cases, different gases will be mixed in set proportions, such as a 50% Oxygen and 50% \(\mathrm{N}_2 \mathrm{O}\) by volume.
Volume proportions are essential in applications as they affect how gases behave when mixed. It's important to calculate the exact volume of each gas to maintain the desired mixture ratio. Here, we ensure both gases occupy the same volume under identical conditions (pressure and temperature) to achieve the required mixture proportion.
Pressure Measurement
Pressure measurement is a fundamental element in gas calculations. In our problem, the pressure is given in psi (pounds per square inch), which is commonly converted to pascals (Pa) for calculations using the Ideal Gas Law. The conversion is crucial because the Ideal Gas Law requires consistent SI units.
Pressure plays a critical role in the behavior of gases and their volume. It is important to remember that pressure can dramatically impact the volume of a gas鈥攁 principle depicted in the Boyle's Law within the broader Ideal Gas Law equation\(PV = nRT,\)where pressure, volume, and temperature are interconnected.
Volume Calculation
Volume calculation in the context of gases is primarily performed using the Ideal Gas Law. We have the equation \(PV = nRT,\) which connects pressure, volume, and temperature with moles of gas and the ideal gas constant.
  • \(P\) - Pressure
  • \(V\) - Volume
  • \(n\) - Number of moles
  • \(R\) - Ideal gas constant (8.3145 m鲁.pa/K/mol)
  • \(T\) - Temperature
For the \(\mathrm{N}_2 \mathrm{O}\), the calculated 38.64 moles under given atmospheric conditions (50 psi and 20掳C) results in a volume of 0.279 m鲁. This value also represents the volume of oxygen needed, considering our goal of a 50% mixture by volume. Understanding volume is vital in determining quantities of gases needed in real-world applications.

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Most popular questions from this chapter

\( \mathrm{A}\) gas in a cylinder is held at a constant pressure of \(1.80 \times 10^{5} \mathrm{~Pa}\) and is cooled and compressed from \(1.70 \mathrm{~m}^{3}\) to \(1.20 \mathrm{~m}^{3}\). The internal energy of the gas decreases by \(1.40 \times 10^{5} \mathrm{~J}\). (a) Find the work done by the gas. (b) Find the absolute value of the heat flow, \(|Q|\), into or out of the gas, and state the direction of the heat flow. (c) Does it matter whether the gas is ideal? Why or why not?

A quantity of air is taken from state \(a\) to state \(b\) along a path that is a straight line in the \(p V\) -diagram (Fig. \(\mathbf{P 1 9 . 3 5}\) ). (a) In this process, does the temperature of the gas increase, decrease, or stay the same? Explain. (b) If \(\quad V_{a}=0.0700 \mathrm{~m}^{3}\), \(V_{b}=0.1100 \mathrm{~m}^{3}, p_{a}=1.00 \times 10^{5} \mathrm{~Pa}\) and \(p_{b}=1.40 \times 10^{5} \mathrm{~Pa},\) what is the work \(W\) done by the gas in this process? Assume that the gas may be treated as ideal.

A steel cargo drum has a height of \(880 \mathrm{~mm}\) and a diameter of \(610 \mathrm{~mm}\). With its top removed it has a mass of \(17.3 \mathrm{~kg}\). The drum is turned upside down at the surface of the North Atlantic and is pulled downward into the ocean by a robotic submarine. On this day the surface temperature is \(23.0^{\circ} \mathrm{C}\) and the surface air pressure is \(p_{0}=101 \mathrm{kPa}\). The water temperature decreases linearly with depth to \(3.0^{\circ} \mathrm{C}\) at \(1000 \mathrm{~m}\) below the surface. As the drum moves downward in the ocean, the air inside the drum is compressed, reducing the upward buoyant force. (a) At what depth \(y_{\text {neutral }}\) is the barrel neutrally buoyant? (Hint: The pressure in the drum is equal to the sea pressure, which at depth \(y\) is \(p_{0}+\rho g y\) where \(\rho=1025 \mathrm{~kg} / \mathrm{m}^{3}\) is the density of seawater. The temperature at depth can be determined using the information above. Together with the ideal- gas law, you can derive a formula for the volume of air at depth \(y,\) and therefore a formula for the upward buoyant force as a function of depth.) (b) What is the volume of the air in the drum at depth \(y_{\text {neutral }} ?\)

Five moles of an ideal monatomic gas with an initial temperature of \(127^{\circ} \mathrm{C}\) expand and, in the process, absorb \(1500 \mathrm{~J}\) of heat and do \(2100 \mathrm{~J}\) of work. What is the final temperature of the gas?

\( \cdot\) Heat \(Q\) flows into a monatomic ideal gas, and the volume increases while the pressure is kept constant. What fraction of the heat energy is used to do the expansion work of the gas?
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