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Chapter 6: Problem 51

What is the molar mass of a gas if a \(0.121-\mathrm{g}\) sample at 740 torr occupies a volume of \(21.0 \mathrm{~mL}\) at \(29^{\circ} \mathrm{C}\) ?

Short Answer

Expert verified
The molar mass of the gas is approximately 146.5 g/mol.

Step by step solution

01

Convert Units

First, convert the pressure from torr to atm, the volume from mL to L, and the temperature from Celsius to Kelvin. \(740\,\text{torr} = \frac{740}{760}\,\text{atm} \approx 0.974\,\text{atm}\). The volume is \(21.0\,\text{mL} = 0.0210\,\text{L}\). The temperature is \(29^{\circ}\,\text{C} = 273 + 29 = 302\,\text{K}\).
02

Use the Ideal Gas Law

Apply the Ideal Gas Law \(PV = nRT\) to find the number of moles \(n\). Rearrange the equation to solve for \(n\): \(n = \frac{PV}{RT}\). Substitute the given values: \(P = 0.974\,\text{atm}\), \(V = 0.0210\,\text{L}\), \(R = 0.0821\,\text{L atm mol}^{-1}\text{K}^{-1}\), and \(T = 302\,\text{K}\).
03

Calculate the Number of Moles

Substitute the values into the formula to find \(n\):\(n = \frac{(0.974\,\text{atm})(0.0210\,\text{L})}{(0.0821\,\text{L atm mol}^{-1}\text{K}^{-1})(302\,\text{K})} \approx 0.000826\,\text{mol}\).
04

Calculate Molar Mass

The molar mass \(M\) of the gas is calculated by dividing the mass of the sample by the number of moles: \(M = \frac{0.121\,\text{g}}{0.000826\,\text{mol}} \approx 146.5\,\text{g/mol}\).

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Key Concepts

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

Molar Mass Calculation
Molar mass is a fundamental concept in chemistry. It's essentially the mass of one mole of a substance, which helps in understanding the amount of matter in a given sample. In this exercise, we are tasked with finding the molar mass of a gas sample. To do this, the mass of the sample (in grams) is divided by the number of moles. The number of moles itself is derived using the Ideal Gas Law. The formula used to calculate the molar mass is: \[ M = \frac{\text{mass (g)}}{\text{number of moles (mol)}} \] For instance, with a sample mass of 0.121 g and finding the number of moles to be approximately 0.000826 mol, the molar mass comes out to be about 146.5 g/mol.
Units Conversion
Unit conversion is crucial for ensuring the accuracy of any calculation. In this context, we need to convert pressure, volume, and temperature into consistent units. * **Pressure**: From torr to atmosphere, using: \[ 1 \, \text{atm} = 760 \, \text{torr} \] \[ 740 \, \text{torr} = \frac{740}{760} \, \text{atm} \approx 0.974 \, \text{atm} \] * **Volume**: From milliliters to liters: \[ 1 \, \text{L} = 1000 \, \text{mL} \] \[ 21.0 \, \text{mL} = 0.021 \, \text{L} \] * **Temperature**: From degrees Celsius to Kelvin: \[ T(\text{K}) = T(^{\circ}\text{C}) + 273 \] \[ 29^{\circ}\text{C} = 302 \text{K} \]These conversions are pivotal for using the Ideal Gas Law correctly.
Gas Laws
The Ideal Gas Law is a key equation that relates the pressure, volume, temperature, and number of moles of a gas. It is a cornerstone of understanding gaseous behavior:\[ PV = nRT \]* **P** is pressure* **V** is volume* **n** is the number of moles* **R** is the ideal gas constant * **T** is temperatureIn this exercise, we rearranged the equation to solve for the number of moles:\[ n = \frac{PV}{RT} \]Plugging in the values for our gas sample results in discovering the number of moles, which then helps determine the molar mass.
Chemical Problem Solving
Chemical problem solving is an essential skill in chemistry that entails applying known laws and mathematical manipulations to find unknown values. In this particular problem, we tackle multiple steps: * **Identify Known Values**: Start by noting the given mass, pressure, volume, and temperature. * **Convert Units**: Ensure all units are appropriate for the gas law formula. * **Apply Gas Laws**: Use the Ideal Gas Law for calculations. * **Calculate Outcome**: Use the derived values to compute the molar mass. By following these systematic steps, we can confidently solve chemical problems using logic and mathematical operations.

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Most popular questions from this chapter

What is the pressure, in \(a t m,\) of \(0.322 \mathrm{~g} \mathrm{~N}_{2}\) gas in a \(300\) \(\mathrm{mL}\) container at \(24{ }^{\circ} \mathrm{C} ?\)

The gas hydrogen sulfide, \(\mathrm{H}_{2} \mathrm{~S}\), has the offensive smell associated with rotten eggs. It reacts slowly with the oxygen in the atmosphere to form sulfur dioxide and water. What volume of sulfur dioxide gas, in liters, forms at constant pressure and temperature from \(2.44 \mathrm{~L}\) hydrogen sulfide, and what volume of oxygen gas is consumed?

For the following pairs of gases at the given conditions, predict which one would more closely follow the ideal gas law. Explain your choice. (a) \(\mathrm{CO}_{2}\) gas at \(0.05 \mathrm{~atm}\) or at \(10 \mathrm{~atm}\) of pressure (b) Propane (boiling point \(=-45^{\circ} \mathrm{C}\) ) or neon (boiling point \(=-246^{\circ} \mathrm{C}\) ) gas at \(-20^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) (c) Sulfur dioxide at \(0^{\circ} \mathrm{C}\) or at \(50^{\circ} \mathrm{C}\), both measured at \(1 \mathrm{~atm}\)

A \(10.5-g\) sample of hydrogen is added to a \(30-\mathrm{L}\) container that also holds argon gas at \(1.53 \mathrm{~atm}\). The gases are at \(120^{\circ} \mathrm{C}\). What is the partial pressure of hydrogen gas in the mixture, and what is the total pressure in the container?

For each of the following pairs of gases at the given conditions, predict which one would more closely follow the ideal gas law. Explain your choice. (a) Oxygen (boiling point \(=-183{ }^{\circ} \mathrm{C}\) ) or sulfur dioxide (boiling point \(\left.=-10^{\circ} \mathrm{C}\right)\), both measured at \(25^{\circ} \mathrm{C}\) and 1 atm (b) Nitrogen (boiling point \(=-196^{\circ} \mathrm{C}\) ) at \(-150^{\circ} \mathrm{C}\) or at \(100^{\circ} \mathrm{C},\) both measured at \(1 \mathrm{~atm}\) (c) Argon gas at \(1 \mathrm{~atm}\) or at \(200 \mathrm{~atm}\), both measured at \(200^{\circ} \mathrm{C}\)
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