/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 If palladium metal (density \(\l... [FREE SOLUTION] | 91Ó°ÊÓ

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If palladium metal (density \(\left.12.0 \mathrm{~g} / \mathrm{cm}^{3}\right)\) dissolves 935 times its own volume of \(\mathrm{H}_{2}\) at \(\mathrm{STP}\), what is the value of \(x\) in the formula \(\mathrm{PdH}_{x}\) ? What is the density of hydrogen in \(\mathrm{PdH}_{x}\) in units of \(\mathrm{g} / \mathrm{cm}^{3}\) ? Assume that the volume of palladium is unchanged when the \(\mathrm{H}\) atoms go into the interstices.

Short Answer

Expert verified
The value of \(x\) is approximately 0.37, and the density of hydrogen in \(\mathrm{PdH}_x\) is 0.0841 g/cm³.

Step by step solution

01

Determine the Mass of Palladium

To find the mass of palladium, consider a sample with a volume of 1 cm³. The density of palladium is given as 12.0 g/cm³. Therefore, the mass of palladium is simply:\[ \text{mass of Pd} = 12.0 \text{ g/cm}^3 \times 1 \text{ cm}^3 = 12.0 \text{ g} \]
02

Calculate the Volume of Hydrogen at STP

Palladium absorbs 935 times its own volume of \( \mathrm{H}_2 \) gas. Thus, the volume of \( \mathrm{H}_2 \) absorbed by 1 cm³ of palladium is:\[ \text{Volume of } \mathrm{H}_2 = 935 \text{ cm}^3 \]
03

Calculate the Moles of Hydrogen Gas

At STP, 1 mole of a gas occupies 22.4 L, which is 22400 cm³. The volume of hydrogen absorbed is 935 cm³, which is a fraction of this molar volume:\[ \text{Moles of } \mathrm{H}_2 = \frac{935 \text{ cm}^3}{22400 \text{ cm}^3/\text{mol}} \approx 0.0417 \text{ mol} \]
04

Determine the Moles of Palladium

Since we have 1 cm³ of palladium and its molar mass is 106.42 g/mol, the number of moles of palladium is:\[ \text{Moles of Pd} = \frac{12.0 \text{ g}}{106.42 \text{ g/mol}} \approx 0.1128 \text{ mol} \]
05

Calculate the Value of x in PdH_x

Using the moled quantities of Pd and \( \mathrm{H}_2 \), compute \( x \) in \( \mathrm{PdH}_x \):\[ x = \frac{\text{moles of } \mathrm{H}_2}{\text{moles of Pd}} = \frac{0.0417}{0.1128} \approx 0.369 \]Thus, \( x \approx 0.37 \).
06

Calculate the Density of Hydrogen in PdH_x

To find the density of hydrogen present in the metallic structure, calculate its mass based on moles and divide by the volume it occupies:1. Find the mass of hydrogen: 0.0417 mol of \( \mathrm{H}_2 \) weighs \( 0.0417 \text{ mol} \times 2.016 \text{ g/mol} = 0.0841 \text{ g} \).2. Since all this hydrogen is in 1 cm³ of Pd, the density is:\[ \text{Density of } \mathrm{H}_2 = \frac{0.0841 \text{ g}}{1 \text{ cm}^3} = 0.0841 \text{ g/cm}^3 \]

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Key Concepts

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

Palladium Metal
Palladium is a lustrous, silver-white metal that belongs to the platinum group. It is highly valued for its ability to absorb hydrogen gas, a property that is vital in various applications such as hydrogen purification, chemical catalysis, and in the development of fuel cells. Palladium's unique characteristic is its ability to dissolve hydrogen at room temperature, forming a compound known as palladium hydride (PdH). This ability to absorb hydrogen is not only useful in industrial applications, but also provides a fascinating field of study in chemistry and materials science. The ability to absorb 935 times its own volume in hydrogen gas, as stated in this exercise, illustrates its significant storage capacity, highlighting its importance in energy-related applications.
Palladium's role doesn't stop at being a simple receptacle for hydrogen; it actually facilitates the process of dissociating diatomic hydrogen into atomic hydrogen, which can then diffuse into the metal lattice. This behavior underpins the potential of palladium in creating efficient energy systems.
Density Calculation
Density is a fundamental property of matter that describes the ratio of mass to volume. It’s expressed as grams per cubic centimeter (g/cm³) in the metric system. In the exercise, we are given that the density of palladium is 12.0 g/cm³. This means that one cubic centimeter of palladium has a mass of 12 grams. With this information, you can easily begin any calculations involving the mass and volume of palladium, like determining how much H₂ it can absorb.
To calculate the mass of any material when its density and volume are known, use the formula:
  • Mass = Density × Volume
For example, given that the density of palladium is 12.0 g/cm³ and considering a volume of 1 cm³, the mass is simple to determine as 12 g. Density calculations are crucial not only in determining mass, but also in various engineering and packaging applications, showing how compact or expansive a material can be.
STP Conditions
Standard Temperature and Pressure (STP) are a set of conditions for experiments to have a common reference. The standard conditions are a temperature of 0°C (273.15 K) and a pressure of 1 atmosphere (101.3 kPa). At these conditions, one mole of an ideal gas occupies 22.4 liters (22,400 cm³), which is an essential reference point for gas volume calculations.
Knowing that palladium metal can absorb hydrogen gas at STP allows scientists and engineers to predict and design systems where gas behavior follows predictable and consistent patterns. The exercise's reference to hydrogen absorption at STP helps provide a straightforward understanding of how much gas can be absorbed, with 1 mole of hydrogen occupying 22,400 cm³. By using a standardized approach like STP conditions, calculations in chemistry can be made more easily and compared across different studies and scenarios.
Gas Absorption
Gas absorption is the physical or chemical process where one substance takes in another substance's particles at its surface or volume. Palladium's capacity to absorb hydrogen is a prime example of materials science, playing a significant role in the creation of palladium hydride (PdHâ‚‚).
The exercise explains that 1 cm³ of palladium can dissolve 935 cm³ of hydrogen gas. This large absorption capacity makes palladium an essential material for hydrogen storage solutions and applications where gas must be stored under minimal volume efficiently. This property is due to palladium's open crystal structure, which allows hydrogen atoms to occupy interstitial sites within the lattice without significantly changing the metal's volume.
Understanding palladium's hydrogen absorption capability can lead to breakthroughs in clean energy and battery technology, emphasizing a key area in modern research for energy efficiency and sustainability.

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Most popular questions from this chapter

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