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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

You have purchased a gas cylinder that is supposed to contain 5.0 mole \(\% \mathrm{Cl}_{2}(\pm 0.1 \%)\) and \(95 \%\) air. The experiments you have been running are not giving reasonable results, and you suspect that the chlorine concentration in the gas cylinder is incorrect. To check this hypothesis, you bubble gas from the suspicious cylinder through \(2.0 \mathrm{L}\) of an aqueous NaOH solution (12.0 wt\% NaOH, SG = 1.13) for exactly one hour. The inlet gas is metered at a gauge pressure of \(510 \mathrm{mm} \mathrm{H}_{2} \mathrm{O}\) and a temperature of \(23^{\circ} \mathrm{C}\). Prior to entering the vessel, the gas passes through a flowmeter that indicates a flow rate of \(2.00 \mathrm{L} / \mathrm{min}\). At the conclusion of the experiment, a sample of the residual \(\mathrm{NaOH}\) solution is analyzed and the results show that the \(\mathrm{NaOH}\) content has been reduced by \(23 \% .\) What is the concentration of \(\mathrm{Cl}_{2}\) in the cylinder gas? (Assume the \(\mathrm{Cl}_{2}\) is completely consumed in the reaction \(\mathrm{Cl}_{2}+2 \mathrm{NaOH} \rightarrow \mathrm{NaCl}+\mathrm{NaOCl}+\mathrm{H}_{2} \mathrm{O}\)

A tank in a room at \(19^{\circ} \mathrm{C}\) is initially open to the atmosphere on a day when the barometric pressure is 102 kPa. A block of dry ice (solid \(\mathrm{CO}_{2}\) ) with a mass of \(15.7 \mathrm{kg}\) is dropped into the tank, which is then sealed. The reading on the tank pressure gauge initially rises very quickly, then much more slowly, eventually reaching a value of 3.27 MPa. Assume \(T_{\text {final }}=19^{\circ} \mathrm{C}\) (a) How many moles of air were in the tank initially? Neglect the volume occupied by \(\mathrm{CO}_{2}\) in the solid state, and assume that a negligible amount of \(\mathrm{CO}_{2}\) escapes prior to the sealing of the tank. (b) Estimate the percentage error made by neglecting the volume of the block of dry ice placed in the tank. (The specific gravity of solid carbon dioxide is approximately 1.56 .) (c) What is the final density (g/L) of the gas in the tank? (d) Explain the observed variation of pressure with time. More specifically, what is happening in the tank during the initial rapid pressure increase and during the later slow pressure increase?

A distillation column is being used to separate methanol and water at atmospheric pressure. The column temperature varies from approximately \(65^{\circ} \mathrm{C}\) at the top to \(100^{\circ} \mathrm{C}\) at the bottom. Liquid enters the top of the column and flows down to the bottom; vapor is generated in a reboiler at the bottom of the column, flows upward, and leaves at the top. The molar flow rate of vapor up the column may be assumed to be constant from top to bottom. The vapor velocity is kept below \(5.0 \mathrm{ft} / \mathrm{s}\) to keep the vapor from entraining liquid (suspending and carrying away liquid droplets). (a) Where in the column is the greatest risk of liquid entrainment? Explain your answer. (b) Assuming that the liquid flowing down the column and the column internals (equipment inside the column) occupy a negligible fraction of the column cross-sectional area, estimate the minimum column diameter if the vapor flow rate is 25.0 lb-mole/min. (c) Suppose the column is constructed with a diameter \(10 \%\) greater than that determined in Part (b). What are the vapor velocities at the top and bottom of the column if the vapor molar flow rate in both locations is 25.0 ib-mole/min? How much can the vapor molar flow rate be increased without causing liquid entrainment? (d) There is a need to increase process throughput, which would require the vapor molar flow rate to be doubled. It has been suggested that increasing the pressure in the column would allow that to be done without risking excessive liquid entrainment. Again applying a vapor velocity limit of \(5 \mathrm{ft} / \mathrm{s}\) what would the new pressure be?

A gas consists of 20.0 mole \(\% \mathrm{CH}_{4}, 30.0 \% \mathrm{C}_{2} \mathrm{H}_{6},\) and \(50.0 \% \mathrm{C}_{2} \mathrm{H}_{4} .\) Ten kilograms of this gas is to be compressed to a pressure of 200 bar at \(90^{\circ} \mathrm{C}\). Using Kay's rule, estimate the final volume of the gas.

A natural gas contains 95 wt\% \(\mathrm{CH}_{4}\) and the balance \(\mathrm{C}_{2} \mathrm{H}_{6}\). Five hundred cubic meters per hour of this gas at \(40^{\circ} \mathrm{C}\) and 1.1 bar is to be burned with \(25 \%\) excess air. The air flowmeter is calibrated to read the volumetric flow rate at standard temperature and pressure. What should the meter read (in SCMH) when the flow rate is set to the desired value?
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