/*! 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 36 A neon sign is made of glass tub... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 36

A neon sign is made of glass tubing whose inside diameter is \(2.5 \mathrm{~cm}\) and whose length is \(5.5 \mathrm{~m}\). If the sign contains neon at a pressure of \(1.78\) torr at \(35^{\circ} \mathrm{C}\), how many grams of neon are in the sign? (The volume of a cylinder is \(\pi r^{2} h .\)

Short Answer

Expert verified
There are approximately \(6.3\times 10^{-3}\mathrm{~g}\) of neon gas in the sign.

Step by step solution

01

Calculate the volume of the cylinder

Given: Inside diameter = \(2.5\mathrm{~cm}\) Length = \(5.5\mathrm{~m}\) First, we need to find the radius of the tube. The radius will be half of the diameter. Then, we will use the formula for the volume of a cylinder \(V = \pi r^{2}h\). Radius = Inside diameter/2 = \(2.5\mathrm{~cm}/2\) = \(1.25\mathrm{~cm}\) Length = \(5.5\mathrm{~m}\) (Convert this to centimeters, as radius is in centimeters) = \(550\mathrm{~cm}\) Now, we can find the volume: Volume (V) = \(\pi (1.25\mathrm{~cm})^{2} \times 550\mathrm{~cm}\) = \(2692.81\mathrm{~cm^3}\) or \(2.6928\mathrm{~L}\) (Convert cubic centimeters to Liters; \(1\mathrm{~L} = 1000\mathrm{~cm^3}\))
02

Convert pressure and temperature to appropriate units

Given: Pressure (P) = \(1.78\) torr Temperature (T) = \(35^{\circ} \mathrm{C}\) We need to convert the pressure from torr to atm and temperature from Celsius to Kelvin. Pressure in atm = Pressure in torr / 760 = \(1.78\mathrm{~torr}/760\) = \(0.002342\mathrm{~atm}\) Temperature in Kelvin = Temperature in Celsius + 273.15 = \(35^{\circ} \mathrm{C} + 273.15\) = \(308.15\mathrm{~K}\)
03

Use the Ideal Gas Law to find the number of moles of neon

Using the Ideal Gas Law (PV = nRT), we can solve for the number of moles (n). Here, R is the Ideal Gas Constant and it is equal to \(0.0821 \frac{\mathrm{L\cdot atm}}{\mathrm{mol \cdot K}}\). Rearrange the equation to solve for n: n = \(\frac{PV}{RT}\) Now substitute the values we found in Step 2: n = \(\frac{(0.002342\mathrm{~atm}) \times (2.6928\mathrm{~L})}{(0.0821 \frac{\mathrm{L\cdot atm}}{\mathrm{mol \cdot K}})(308.15\mathrm{~K})}\) n = 0.0003125 mol
04

Convert the number of moles to grams

Given: Molar mass of neon (Ne) = \(20.18\mathrm{~g/mol}\) Now, we can calculate the mass of neon gas in the tube by using the molar mass and the number of moles (n) calculated in Step 3: Mass of neon = Number of moles (n) 脳 Molar mass of neon = \(0.0003125\mathrm{~mol} \times 20.18\mathrm{~g/mol}\) = \(6.30625 \times 10^{-3} \mathrm{~g}\) Therefore, there are approximately \(6.3\times 10^{-3}\mathrm{~g}\) of neon gas in the sign.

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Key Concepts

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

Molar Mass of Neon
Understanding the molar mass of neon is crucial for solving various chemistry problems, particularly those involving gas calculations. The molar mass is the mass of one mole of a substance, measured in grams per mole (g/mol). For neon, a noble gas with the atomic number 10, the molar mass is approximately 20.18 g/mol. This value is essential when converting between grams and moles of neon, as required in gas law problems.

For instance, if you have a certain number of moles of neon gas and want to find out the mass in grams, you would multiply the number of moles by the molar mass of neon. Conversely, to convert mass to moles, you would divide the mass by the molar mass. Molar mass also plays a key role in stoichiometry calculations in chemical reactions where neon may be a reactant or product.
Gas Law Calculations
Gas law calculations involve equations that relate the pressure, volume, temperature, and number of moles of a gas. The Ideal Gas Law, represented by the equation PV = nRT, is one of the most vital equations in understanding gas behavior under various conditions. In this equation, P stands for pressure, V for volume, n for the number of moles, R for the gas constant (0.0821 L路atm/mol路K), and T for temperature in Kelvin (K).

When applying the Ideal Gas Law, it's important to consistently use the right units. Pressure should be in atmospheres (atm), volume in liters (L), temperature in Kelvin (K), and the amount of gas in moles (mol). By manipulating the Ideal Gas Law, one can solve for an unknown variable if the other three are known. This versatility makes the equation a powerful tool for chemists and physicists alike.
Conversion of Units
Correctly converting units is a foundational skill in both chemistry and physics, essential for solving problems related to the Ideal Gas Law and other scientific calculations. To avoid errors and ensure accurate results, all units must be consistent with the equation you're using.

For example, when working with pressure, you may need to convert torr to atmospheres since the Ideal Gas Law requires pressure to be in atm. This is done by dividing the pressure in torr by 760. Similarly, with temperature, Celsius must be converted to Kelvin by adding 273.15. Volume units may also require conversion; for instance, the volume in cubic centimeters (cm鲁) can be converted to liters (L) by dividing by 1,000, as there are 1,000 cm鲁 in a liter. Mastering unit conversions is not only important for answering textbook questions but is also crucial for lab work and real-world applications.

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Most popular questions from this chapter

A sample of \(1.42 \mathrm{~g}\) of helium and an unweighed quantity of \(\mathrm{O}_{2}\) are mixed in a flask at room temperature. The partial pressure of helium in the flask is \(42.5\) torr, and the partial pressure of oxygen is 158 torr. What is the mass of the oxygen in the container?

Which of the following statements best explains why nitrogen gas at STP is less dense than Xe gas at STP? (a) Because Xe is a noble gas, there is less tendency for the Xe atoms to repel one another, so they pack more densely in the gas state. (b) Xe atoms have a higher mass than \(\mathrm{N}_{2}\) molecules. Because both gases at STP have the same number of molecules per unit volume, the Xe gas must be denser. (c) The Xe atoms are larger than \(\mathrm{N}_{2}\) molecules and thus take up a larger fraction of the space occupied by the gas. (d) Because the Xe atoms are much more massive than the \(\mathrm{N}_{2}\) molecules, they move more slowly and thus exert less upward force on the gas container and make the gas appear denser.

The density of a gas of unknown molar mass was measured as a function of pressure at \(0{ }^{\circ} \mathrm{C}\), as in the table below. (a) Determine a precise molar mass for the gas. Hint: Graph \(d / P\) versus \(P\). (b) Why is \(d / P\) not a constant as a function of pressure? $$ \begin{array}{llllll} \hline \text { Pressure } & & & & & \\ \begin{array}{l} \text { (atm) } \end{array} & 1.00 & 0.666 & 0.500 & 0.333 & 0.250 \\ \text { Density } & & & & & \\ \begin{array}{l} \text { (g/L) } \end{array} & 2.3074 & 1.5263 & 1.1401 & 0.7571 & 0.5660 \\ \hline \end{array} $$

What property or properties of gases can you point to that support the assumption that most of the volume in a gas is empty space?

Carbon dioxide, which is recognized as the major contributor to global warming as a "greenhouse gas," is formed when fossil fuels are combusted, as in electrical power plants fueled by coal, oil, or natural gas. One potential way to reduce the amount of \(\mathrm{CO}_{2}\) added to the atmosphere is to store it as a compressed gas in underground formations. Consider a 1000-megawatt coalfired power plant that produces about \(6 \times 10^{6}\) tons of \(\mathrm{CO}_{2}\) per year. (a) Assuming ideal gas behavior, \(1.00 \mathrm{~atm}\), and \(27^{\circ} \mathrm{C}\), calculate the volume of \(\mathrm{CO}_{2}\) produced by this power plant. (b) If the \(\mathrm{CO}_{2}\) is stored underground as a liquid at \(10^{\circ} \mathrm{C}\) and \(120 \mathrm{~atm}\) and a density of \(1.2 \mathrm{~g} / \mathrm{cm}^{3}\), what volume does it possess? (c) If it is stored underground as a gas at \(36{ }^{\circ} \mathrm{C}\) and \(90 \mathrm{~atm}\), what volume does it occupy?
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