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Chapter 5: Problem 16

A gas cylinder filled with nitrogen at standard temperature and pressure has a mass of \(37.289 \mathrm{g}\). The same container filled with carbon dioxide at STP has a mass of 37.440 g. When filled with an unknown gas at STP, the container mass is \(37.062 \mathrm{g}\). Calculate the molecular weight of the unknown gas, and then state its probable identity.

Short Answer

Expert verified
The molecular weight of the unknown gas is 0 g/mol, suggesting the container likely contains a vacuum.

Step by step solution

01

Calculate the mass of gases

First, determine how much gas was inserted into the cylinder each time by subtracting the mass of the empty cylinder from the total mass. For nitrogen this is \(37.289 \mathrm{g} - 37.062 \mathrm{g} = 0.227 \mathrm{g}\), for CO2 it's \(37.440 \mathrm{g} - 37.062 \mathrm{g} = 0.378 \mathrm{g}\), and for the unknown gas it's \(37.062 \mathrm{g} - 37.062 \mathrm{g} = 0 \mathrm{g}\).
02

Calculate number of moles for Nitrogen and CO2

Now calculate the number of moles for nitrogen and CO2 by dividing their masses by their molecular weights. For nitrogen this is: \(0.227 \mathrm{g} / 28.0134 \mathrm{g/mol} = 0.0081 \mathrm{mol}\) and for CO2 this is: \(0.378 \mathrm{g} / 44.0095 \mathrm{g/mol} = 0.0086 \mathrm{mol}\). The average number of moles is then \((0.0081 \mathrm{mol} + 0.0086 \mathrm{mol}) / 2 = 0.00835 \mathrm{mol}\). This is the number of moles of the unknown gas, since the volume and pressure are the same.
03

Calculate the molecular weight of the unknown gas

The molecular weight of the unknown gas can now be calculated by dividing the mass of the unknown gas by the number of moles. However, in this scenario the mass of unknown gas is 0g (since its mass equals the cylinder mass), which means the unknown gas has zero molecular weight.
04

Determine the identity of the unknown gas

Since the molecular weight is zero, the unknown gas is likely to be a vacuum or it contains a gas with negligible mass, such as hydrogen or helium. However, since hydrogen and helium would still have a measurable molecular weight, it's more likely that the cylinder contains a vacuum.

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Key Concepts

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

Gas Cylinder at STP
When dealing with a gas cylinder at standard temperature and pressure (STP), it's critical to understand what STP means. STP conditions are defined as a temperature of 273.15 K (0掳C) and a pressure of 1 atmosphere (atm). These standardized conditions allow for the comparison of gas behaviors under uniform circumstances.

The gas laws and calculations involving gases at STP are simplified because the volume of an ideal gas at STP is known鈥�22.414 liters per mole. This fixed volume provides a baseline from which various quantities, such as the number of moles and molecular weight, can be derived for gases under similar conditions.

In the context of gas-filled cylinders, the process of weighing the cylinder when filled with different gases helps to deduce the amount and type of gas present. By understanding how the weight changes with different gases and comparing it to known values like that of nitrogen or carbon dioxide, the characteristics of an unknown gas can be calculated.
Moles Calculation
Calculating the number of moles in a gas is a fundamental step in determining various properties of the gas. A mole is a unit that measures the amount of a substance, and it's directly related to the molecular weight of the substance.

The formula to find the number of moles ( ) is given by dividing the mass of the substance by its molecular weight ( = / molecular weight). For example, if we take nitrogen (N鈧�), with a molecular weight of approximately 28 g/mol, and we have 0.227 g of nitrogen in the cylinder, the number of moles is calculated as follows: 鈮� 0.0081 moles.

This calculation is crucial for comparing gases under standard conditions. It helps us figure out how much of the gas is present. Identifying the moles of the unknown gas through comparisons with known gases allows further analysis, as gases under STP have predictable behaviors. This step was key in the problem, using it to derive the molecular weight of the unknown gas.
Gas Identity Determination
Determining the identity of an unknown gas involves a combination of weighing, calculating, and comparing known data with new observations. The unknown gas's identity can be inferred by its molecular weight, which is found by dividing the mass of the gas by the number of moles.

However, in this problem, it was discovered that the container filled with the unknown gas had the same weight as the empty container, indicating a mass of zero grams for the gas. This zero mass led to an initially perplexing result when trying to calculate the molecular weight.

Since gases like hydrogen and helium have very low but not zero molecular weights, the identity of the unknown gas was determined to be more likely a vacuum or some error in measuring. It's crucial to ensure accurate measurements and calculations to avoid misinterpretations. This demonstrates how important small details are in calculations involving gases, as assumptions about the environment can lead to significant changes in the outcome.

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Most popular questions from this chapter

During your summer vacation, you plan an epic adventure trip to scale Mt. Kilimanjaro in Tanzania. Dehydration is a great danger on such a climb, and it is essential to drink enough water to make up for the amount you lose by breathing. (a) During your pre-trip physical, your physician measured the average flow rate and composition of the gas you exhaled (expired air) while performing light activity. The results were \(11.36 \mathrm{L} / \mathrm{min}\) at body temperature (37^) C) and 1 atm, 17.08 mole\% oxygen, 3.25\% carbon dioxide, 6.12 mole\% \(\mathrm{H}_{2} \mathrm{O},\) and the balance nitrogen. The ambient (inspired) air contained 1.67 mole\% water and a negligible amount of carbon dioxide. Calculate the rate of mass lost through the breathing process (kg/day) and the volume of water in liters you would have to drink per day just to replace the water lost in respiration. Consider your lungs to be a continuous steady-state system, with input streams being inspired air and water and \(\mathrm{CO}_{2}\) transferred from the blood and output streams being expired air and \(\mathrm{O}_{2}\) transferred to the blood. Assume no nitrogen is transferred to or from the blood. (b) You made the trip to Tanzania and completed the climb to Uhuru Peak, the summit of Kilimanjaro, at an altitude of 5895 meters above sea level. The ambient temperature and pressure there averaged \(-9.4^{\circ} \mathrm{C}\) and \(360 \mathrm{mm} \mathrm{Hg},\) and the air contained \(0.46 \mathrm{mole} \%\) water. The molar flow rate of your expired air was roughly the same as it had been at sea level, and the expired air contained \(14.86 \% \mathrm{O}_{2}\) \(3.80 \% \mathrm{CO}_{2},\) and \(13.20 \% \mathrm{H}_{2} \mathrm{O} .\) Calculate the rate of mass lost (g/day) through breathing and water you would have to drink (L/day) just to replace the water lost in respiration. (c) The equality of the molar flow rates of expired air at sea level and at Uhuru Peak is due to a cancellation of effects, one of which would tend to increase the rate at higher altitudes and the other to decrease it. What are those effects? (Hint: Use the ideal-gas equation of state in your solution, and think about how the oxygen concentration at a high altitude would likely affect your breathing rate.)

The current global reliance on fossil fuels for heating, transportation, and electric power generation raises concems regarding the release of \(\mathrm{CO}_{2}\) and \(\mathrm{CH}_{4},\) which are greenhouse gases thought to lead to climate change, and NO, which contributes to smog. One potential solution to these problems is to produce transportation fuels from renewable biomass. You have been asked to evaluate a proposed process for converting forest residues to alcohols that may be used as transportation fuels. In the first stage of the process, steam and dry wood from hybrid poplar trees (which grow between five and eight feet a year and can be harvested roughly every five years) are fed to a gasifier in which the biomass is converted to light gases in the following reactions: $$\begin{aligned} \mathrm{C}+\mathrm{H}_{2} \mathrm{O} & \rightarrow \mathrm{CO}+\mathrm{H}_{2} \\\ \mathrm{CO}+\mathrm{H}_{2} \mathrm{O} & \rightarrow \mathrm{CO}_{2}+\mathrm{H}_{2} \\ \mathrm{C}+\mathrm{CO}_{2} & \rightarrow 2 \mathrm{CO} \\ \mathrm{C}+2 \mathrm{H}_{2} & \rightarrow \mathrm{CH}_{4} \\ \mathrm{CH}_{4}+\mathrm{H}_{2} \mathrm{O} & \rightarrow \mathrm{CO}+3 \mathrm{H}_{2} \end{aligned}$$ The effluents from the reactor are a gas stream containing \(\mathrm{H}_{2}, \mathrm{CO}, \mathrm{CO}_{2}, \mathrm{CH}_{4},\) and \(\mathrm{H}_{2} \mathrm{O},\) and a solid char stream that contains only carbon and hydrogen. The char is discarded and the gases go through additional steps in which the hydrogen and carbon monoxide are converted to mixed alcohols. This problem only concerns the gasifier. \(\cdot\) Elemental composition of biomass: 51.9 mass \(\%\) C \(, 6.3 \%\) H, and \(41.8 \%\) O \(\cdot\) Pressure and temperature of entering steam: \(155^{\circ} \mathrm{C}, 4.4 \mathrm{atm}\) \(\cdot\) Feed ratio of steam to biomass: 1.1 kg steam/kg biomass \(\cdot\) Yield and dry-basis composition of product gas: 1.35 kg dry gas/kg biomass at \(700^{\circ} \mathrm{C}, 1.2\) atm; 50.7 mol\% \(\mathrm{H}_{2}, 23.8 \%\) CO, \(18.0 \% \mathrm{CO}_{2}, 7.5 \% \mathrm{CH}_{4}\) (a) Taking a basis of \(100 \mathrm{kg}\) of biomass fed, draw and completely label a flowchart for the gasifier incorporating the given data, labeling the volumes of the steam fed and the gases produced. Perform a degree-of-freedom analysis. (b) Calculate the mass and mass composition of the char and the volumes of the steam feed and product gas streams. (c) List advantages and possible drawbacks of using biomass rather than petroleum as a fuel source.

The ultimate analysis of a No. 4 fuel oil is 86.47 wt\% carbon, \(11.65 \%\) hydrogen, \(1.35 \%\) sulfur, and the balance noncombustible inerts. This oil is burned in a steam-generating furnace with \(15 \%\) excess air. The air is preheated to \(175^{\circ} \mathrm{C}\) and enters the furnace at a gauge pressure of \(180 \mathrm{mm}\) Hg. The sulfur and hydrogen in the fuel are completely oxidized to \(\mathrm{SO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O} ; 5 \%\) of the carbon is oxidized to \(\mathrm{CO}\), and the balance forms \(\mathrm{CO}_{2}\) (a) Calculate the feed ratio ( \(\mathrm{m}^{3}\) air) \(/(\mathrm{kg} \text { oil })\) (b) Calculate the mole fractions (dry basis) and ppm (parts per million on a wet basis, or moles contained in \(10^{6}\) moles of the wet stack gas) of the stack-gas species that might be considered environmental hazards.

Lewis \(^{12}\) describes the hazards of breathing air containing appreciable amounts of an asphyxiant (a gas that has no specific toxicity but, when inhaled, excludes oxygen from the lungs). When the mole percent of the asphyxiant in the air reaches \(50 \%,\) marked symptoms of distress appear, and at \(75 \%\) death occurs in a matter of minutes. A small storage room whose dimensions are \(2 \mathrm{m} \times 1.5 \mathrm{m} \times 3 \mathrm{m}\) contains a number of expensive and dangerous chemicals. To prevent unauthorized entry, the door to the room is always locked and can be opened with a key from either side. A cylinder of liquid carbon dioxide is stored in the room. The valve on the cylinder is faulty and some of the contents have escaped over the weekend. The room temperature is \(25^{\circ} \mathrm{C}\). (a) If the concentration of \(\mathrm{CO}_{2}\) reaches the lethal 75 mole \(\%\) level, what would be the mole percent of \(\mathrm{O}_{2} ?\) (b) How much \(\mathrm{CO}_{2}(\mathrm{kg})\) is present in the room when the lethal concentration is reached? Why would more than that amount have to escape from the cylinder for this concentration to be reached? (c) Describe a set of events that could result in a fatality in the given situation. Suggest at least two measures that would reduce the hazards associated with storage of this scemingly harmless substance.

A gas turbine power plant receives a shipment of hydrocarbon fuel whose composition is uncertain but may be represented by the expression \(\mathrm{C}_{x} \mathrm{H}_{y}\). The fuel is burned with excess air. An analysis of the product gas gives the following results on a moisture-free basis: \(10.5 \%(\mathrm{v} / \mathrm{v}) \mathrm{CO}_{2}, 5.3 \% \mathrm{O}_{2},\) and \(84.2 \% \mathrm{N}_{2}\) (a) Determine the molar ratio of hydrogen to carbon in the fuel ( \(r\) ), where \(r=y / x\), and the percentage excess air used in the combustion. (b) What is the air-to-fuel ratio ( \(m^{3}\) air/kg of fuel) if the air is fed to the power plant at \(30^{\circ} \mathrm{C}\) and \(98 \mathrm{kPa} ?\) (c) The specific gravity of the fuel (a petroleum product) is \(0.85 .\) Estimate the ratio standard cubic feet of gas fed to the turbine per barrel of fuel. (d) What are the issues associated with using oil as a fuel as opposed to natural gas? Consider two factors: (i) the complete composition of typical fuel oils and their resulting emissions, and (ii) the availability and global distribution of the two fuel sources.
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