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Chapter 11: Problem 104

An 84.0-g sample of an unknown gas occupies \(60.8 \mathrm{~L}\) at \(23{ }^{\circ} \mathrm{C}\) and \(1.2 \mathrm{~atm}\). What is the molar mass of the gas?

Short Answer

Expert verified
The molar mass of the unknown gas is approximately 27.9 g/mol, calculated by using the Ideal Gas Law formula with the given values for mass, volume, temperature, and pressure, and then converting the temperature to Kelvin and calculating the number of moles.

Step by step solution

01

Write down the Ideal Gas Law formula

The Ideal Gas Law formula is: \[PV = nRT\] where, P = Pressure in atmospheres (atm) V = Volume in Liters (L) n = Number of moles R = Gas constant (0.0821 L atm K^{-1} mol^{-1}) T = Temperature in Kelvin (K)
02

Convert the given temperature from Celsius to Kelvin

Given that the temperature is 23°C, convert it to Kelvin using the formula: \[T(K) = T(^{\circ}C) + 273.15\] \[T(K) = 23 + 273.15\] \[T(K) = 296.15\]
03

Rearrange the Ideal Gas Law formula to solve for the number of moles

We know the values of P, V, R, and T. Rearrange the formula to solve for n: \[n = \frac{PV}{RT}\]
04

Substitute the given values and calculate the number of moles

Substitute the known values into the formula and solve for n: \[n = \frac{(1.2\,\mathrm{atm})(60.8\,\mathrm{L})}{(0.0821\,\mathrm{L\,atm\,K^{-1}\,mol^{-1}})(296.15\,\mathrm{K})}\] \[n \approx 3.007\,\mathrm{moles}\]
05

Calculate the molar mass of the unknown gas

Molar mass is the mass of the sample divided by the number of moles. Since we know the sample mass (84.0 g) and calculated the number of moles, we can find the molar mass using the formula: Molar mass = \(\frac{mass}{moles}\) Molar mass = \(\frac{84.0\,\mathrm{g}}{3.007\,\mathrm{moles}}\) Molar mass ≈ 27.9 g/mol The molar mass of the unknown gas is approximately 27.9 g/mol.

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Key Concepts

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

Ideal Gas Law
The Ideal Gas Law is a fundamental equation in chemistry that describes the relationship between the pressure (P), volume (V), number of moles (n), and temperature (T) of a gas. The formula is expressed as PV = nRT, where R is the gas constant.

The law allows us to calculate one of the four variables if we know the other three. This is crucial in determining properties of gases under different conditions. It's important to remember that this law assumes ideal behavior, where the gas particles have no volume of their own and do not exert attractive or repulsive forces on each other. While real gases don't perfectly follow this law, it is a close approximation under many conditions.
Gas Constant
The gas constant (R) is a vital part of the Ideal Gas Law and is essentially the same for all gases under ideal conditions. Its value is 0.0821 L atm K-1 mol-1.

This constant provides the necessary link between the energy scale (temperature in Kelvin) and the physical scale (volume in Liters and pressure in atmospheres). It can be seen as a conversion factor that relates the various units involved in gas calculations to each other. Knowing the value of R is essential for all calculations involving the Ideal Gas Law.
Mole Concept
The mole concept is a bridge between the microscopic world of atoms and molecules and the macroscopic world we observe. One mole is 6.022 x 1023 entities (usually atoms or molecules) — this number is also known as Avogadro's number.

When we talk about the molar mass of a substance, we're referring to the mass of one mole of that substance. The number of moles (n) in a given mass of a substance is a critical component for a variety of chemical calculations, including the Ideal Gas Law.
Celsius to Kelvin Conversion
Temperature must be in Kelvin when using the Ideal Gas Law because it is an absolute temperature scale. To convert from Celsius to Kelvin, simply add 273.15 to the Celsius temperature.

For example, room temperature (approximately 23°C) in Kelvin would be 23 + 273.15 = 296.15 K. This conversion ensures accuracy in gas law calculations and reflects the fact that the Kelvin scale accounts for absolute zero, the lowest possible temperature where all molecular motion stops.
Molar Mass Formula
To calculate the molar mass of a substance, you divide the mass of the sample by the number of moles of the substance present. The formula is Molar mass = mass (in grams) / moles.

This helps identify unknown gases by comparing the calculated molar mass to known values. In our exercise, for instance, dividing the mass of the gas sample (84.0 g) by the number of moles (calculated using the Ideal Gas Law) gave us a molar mass, which we could then compare to the molar masses of known substances to identify the gas.

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Most popular questions from this chapter

Consider a container of gas with the pressure inside the container the same as the room pressure outside the container. If a tiny hole is punched in the side of the container, will the gas leak out? Explain your answer.

Suppose you have a sample of \(\mathrm{CO}_{2}\) gas and want to know its mass without bothering to use a balance. How could you do this?

An automobile tire at \(22^{\circ} \mathrm{C}\) with an internal volume of \(20.0 \mathrm{~L}\) is filled with air to a total pressure of 30 psi (pounds per square inch). \(\left[1 \mathrm{~atm}=14.696 \mathrm{lb} / \mathrm{in} .^{2}\right]\) (a) What is the amount in moles of air in the tire? (b) If the air were entirely nitrogen \(\left(\mathrm{N}_{2}\right)\), how many grams of it would be in the tire? How many pounds of it would be in the tire? \([453.6 \mathrm{~g}=1 \mathrm{lb}]\)

A gas is in a container whose volume is variable. The container is in an ice bath at \(0.00^{\circ} \mathrm{C}\), and there are \(2.0\) moles of gas in it. What must the volume in liters be if the gas has a pressure of \(2.5 \mathrm{~atm}\) ?

Consider the reaction \(2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) \rightarrow 2 \mathrm{SO}_{3}(g)\) (a) What volume in liters of oxygen gas at STP is required to produce 2 moles of sulfur trioxide gas? (b) What volume in liters of oxygen gas at \(25.0^{\circ} \mathrm{C}\) and 1 atm pressure is required to produce 2 moles of sulfur trioxide gas?
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